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FILTER BAN K TRANSCEIVER S FO R OFD M AN D DMT SYSTEM S Providing key background material together with advanced topics, this self-contained book i s writte n i n a n easy-to-rea d styl e an d i s idea l fo r newcomer s t o multicarrie r systems. Early chapters provide a review of basic digital communication, starting from th e equivalent discrete-time channel and including a detailed review of the MMSE receiver. Later chapters then provide extensive performance analysis of OFDM and DMT systems, with discussion s o f man y practica l issue s such a s implementation an d power spectrum considerations. Throughout, theoretical analysis is presented alongside practical design considerations, whilst the filter bank transceiver representation of OFDM and DMT systems opens up possibilities for further optimization such as minimum bit error rate, minimum transmission power, and higher spectral efficiency . With plenty of insightful real-world examples and carefully designed end-of-chapter problems, this is an ideal single-semester textbook for senior undergraduate and graduate students, as well as a self-study guide for researchers and professional engineers. YUAN-PEI LI N i s a Professor i n Electrical Engineering at the National Chiao Tung University, Hsinchu, Taiwan. She is a recipient of the Ta-You Wu Memorial Award, the Chinese Institute o f Electrica l Engineering’ s Outstandin g Yout h Electrical Enginee r Award, and of the Chinese Automatic Control Society’s Young Engineer in Automatic Control Award. SEE-MAY PHOON G i s
a Professor i n the Graduate Institute of Communication Engineering and the Department of Electrical Engineering at the National Taiwan University (NTU). He is a recipient o f th e Charles H. Wilts Prize for outstandin g independen t doctoral research in electrical engineering at the California Institute of Technology, and the Chinese Institute of Electrical Engineering’s Outstanding Youth Electrical Engineer Award. P . P . VAIDYANATHA N i
s a Professor i n Electrica l Engineerin g a t th e Californi a Institute o f Technology , where he has been a faculty membe r sinc e 1983. He is an IEEE Fellow and has authored over 400 technical papers, four books, and many invited chapters in leading journals, conferences, an d handbooks. He was a recipient o f th e Award for Excellence in Teaching at the California Institute of Technology three times, and he has received numerous other awards including the F. E. Terman Award of the American Society for Engineering Education and the Technical Achievement Award of the IEEE Signal Processing Society.
FILTER BAN K TRANSCEIVER S FO R OFDM AN D DM T SYSTEM S YUAN-PEI LI N National Chiao Tung University, Taiwan
SEE-MAY PHOON G National Taiwan University
P. P . VAIDYANATHA N California Institute of Technology
CAMBRIDGE UNIVERSIT Y PRES S
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi, Dubai, Tokyo, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information o n this title: www.cambridge.org/9781 107002739 © Cambridge University Press 2011 This publication is in copyright. Subject to statutory exceptio n and to the provisions of relevant collective licensing agreements , no reproduction of any part may take place without the written permission of Cambridge University Press. First published 201 1 Printed in the United Kingdom at the University Press, Cambridge A catalog record for this publication is available from the British Library ISBN 978-1-107-00273-9 Hardbac k
Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred t o in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
To our families Yuan-Pei Lin and See-May Phoong
To Usha, Vikram, Sagar, and my parents — P. P. Vaidyanathan
Contents Preface x
i
1 Introductio n 1 1.1 Notation s 7 2 Preliminarie s o f digita l communication s 9 2.1 Discrete-tim e channe l model s 9 2.2 Equalizatio n 1 2.3 Digita l modulatio n 1 2.3.1 Puls e amplitud e modulatio n (PAM ) 1 2.3.2 Quadratur e amplitud e modulatio n (QAM ) 2 2.4 Paralle l subchannel s 2 2.5 Furthe r readin g 3 2.6 Problem s 3
6 7 8 2 8 1 1
3 FI R equalizer s 3 3.1 Zero-forcin g equalizer s 3 3.2 Orthogonalit y principl e an d linea r estimatio n 3 3.2.1 Biase d an d unbiase d linea r estimate s 4 3.2.2 Estimatio n o f multipl e rando m variable s 4 3.3 MMS E equalizer s 4 3.3.1 FI R channel s 4 3.3.2 MIM O frequency-nonselectiv e channel s 4 3.3.3 Example s 5 3.4 Symbo l detectio n fo r MMS E receiver s 5 3.5 Channel-shortenin g equalizer s 5 3.6 Concludin g Remark s 6 3.7 Problem s 6
3 4 9 1 4 5 5 8 0 6 9 5 5
4 Fundamental s o f multirat e signa l processin g 7 4.1 Multirat e buildin g block s 7 4.1.1 Transfor m domai n formula s 7 4.1.2 Multirat e identitie s 7 4.1.3 Blockin g an d unblockin g 7 4.2 Decimatio n filter s 7 4.3 Interpolatio n filter s 8 4.3.1 Tim e domai n vie w o f interpolatio n filte r 8 4.3.2 Th e Nyquist(M ) propert y 8
1 1 3 5 6 9 0 2 2
vn
CONTENTS
Vlll
4.4 Polyphas e decompositio n 8 4.4.1 Decimatio n an d interpolatio n filter s 8 4.4.2 Synthesi s filte r bank s 8 4.4.3 Analysi s filte r bank s 9 4.5 Concludin g remark s 9 4.6 Problem s 9
4 7 9 0 1 1
Multirate formulatio n o f communicatio n system s 9 5.1 Filte r ban k transceiver s 9 5.1.1 Th e multiplexin g operatio n 9 5.1.2 Redundanc y i n filte r ban k transceiver s 9 5.1.3 Type s o f distortio n i n transceiver s 10 5.2 Analysi s o f filte r ban k transceiver s 10 5.2.1 ISI-fre e filte r ban k transceiver s 10 5.2.2 Polyphas e approac h 10 5.2.3 Channel-independen t ISI-fre e filte r ban k transceiver s 10 5.3 Pseudocirculan t an d circulan t matrice s 10 5.3.1 Pseudocirculant s an d blocke d version s o f scala r system s 10 5.3.2 Circulant s an d circula r convolution s 10 5.4 Redundanc y fo r IB I eliminatio n 11 5.4.1 Zero-padde d system s 11 5.4.2 Cyclic-prefixe d system s 11 5.4.3 Summar y an d compariso n 11 5.4.4 IBI-fre e system s wit h reduce d redundanc y 12 5.5 Fractionall y space d equalize r system s 12 5.5.1 Zero-forcin g F S E system s 12 5.5.2 Polyphas e approac h 12 5.6 Concludin g remark s 12 5.7 Problem s 12
5 5 7 7 0 1 1 3 5 6 6 8 1 2 5 9 1 2 4 5 9 9
D F T - b a s e d t r a n s c e i v e r s 13 6.1 O F D M system s 13 6.1.1 Nois e analysi s 14 6.1.2 Bi t erro r rat e 14 6.2 Zero-padde d O F D M system s 14 6.2.1 Zero-forcin g receiver s 14 6.2.2 Th e MMS E receive r 15 6.3 Single-carrie r system s wit h cycli c prefi x (SC-CP ) 15 6.3.1 Nois e analysis : zero-forcin g cas e 15 6.3.2 Th e MMS E receive r 15 6.3.3 Erro r analysis : MMS E cas e 15 6.4 Single-carrie r syste m wit h zero-paddin g (SC-ZP ) 16 6.5 Filte r ban k representatio n o f O F D M system s 16 6.5.1 Transmitte d powe r spectru m 16 6.5.2 Z P - O F D M system s 16 6.6 D M T system s 16 6.7 Channe l estimatio n an d carrie r frequencysynchronizatio n 17 6.7.1 Pilo t symbo l aide d modulatio n 17 6.7.2 Synchronizatio n o f carrie r frequenc y 17
5 6 0 2 7 7 0 2 5 6 7 0 3 6 8 8 8 8 9
CONTENTS 6.8 A historica l not e an d furthe r readin g 18 6.9 Problem s 18 7 P r e c o d e d O F D M s y s t e m s 19 7.1 Zero-forcin g precode d O F D M system s 19 7.2 Optima l precoder s fo r Q P S K modulatio n 19 7.3 Optima l precoders : othe r modulation s 20 7.4 MMS E precode d O F D M system s 20 7.4.1 MMS E receiver s 20 7.4.2 Optima l precoder s fo r Q P S K modulatio n 20 7.4.3 Othe r modulatio n scheme s 20 7.5 Simulatio n example s 21 7.6 Furthe r readin g 21 7.7 Problem s 22
IX
0 1 3 4 8 2 3 4 7 9 1 9 0
8 Transceive r desig n w i t h channe l informatio n a t th e transmitter22 3 8.1 Zero-forcin g bloc k transceiver s 22 3 8.1.1 Zero-forcin g Z P system s 22 5 8.1.2 Zero-forcin g Z J system s 22 6 8.2 Proble m formulatio n 22 8 8.3 Optima l bi t allocatio n 22 9 8.4 Optima l Z P transceiver s 24 0 8.4.1 Optima l G zp 24 0 8.4.2 Optima l A zp 24 1 8.4.3 Summar y an d discussion s 24 3 8.5 Optima l zero-jammin g (ZJ ) transceiver s 24 7 8.5.1 Optima l S^ - 24 7 8.5.2 Optima l A zj 24 9 8.5.3 Summar y an d discussion s 24 9 8.6 Furthe r readin g 25 3 8.7 Problem s 25 4 9 D M T s y s t e m s wit h improve d frequenc y characteristic s 25 9.1 Sidelobe s matter ! 26 9.2 Overal l transfe r matri x 26 9.3 Transmitter s wit h subfilter s 26 9.3.1 Choosin g th e subfilter s a s a D F T ban k 26 9.3.2 D F T ban k implementatio n 26 9.4 Desig n o f transmi t subfilter s 27 9.5 Receiver s wit h subfilter s 27 9.5.1 Choosin g subfilter s a s a D F T ban k 27 9.5.2 D F T ban k implementatio n 27 9.6 Desig n o f receive r subfilter s 28 9.7 Zero-padde d transceiver s 28 9.8 Furthe r readin g 28 9.9 Problem s 28
9 0 3 5 6 6 2 6 7 7 0 5 5 6
X
CONTENTS
10 Minimu m redundanc y FI R transceiver s 29 10.1 Polyphas e representatio n 29 10.2 Propertie s o f pseudocirculant s 29 10.2.1 Smit h for m decompositio n 29 10.2.2 DF T decompositio n 29 10.2.3 Propertie s derive d fro m th e tw o decomposition s 29 10.2.4 Congruou s zero s 29 10.3 Transceiver s wit h n o redundanc y 30 10.3.1 FI R minima l transceiver s 30 10.3.2 II R minima l transceiver s 30 10.4 Minimu m redundanc y 30 10.5 Smit h for m o f FI R pseudocirculant s 30 10.6 Proo f o f Theore m 10. 2 31 10.6.1 Identica l Smit h form s 31 10.6.2 Zero s fro m differen t Bi decoupl e 31 10.6.3 A n exampl e o f derivin g th e Smit h for m o f 5](z ) 31 10.6.4 Smit h for m o f £(z ) 31 10.7 Furthe r readin g 31 10.8 Problem s 31
1 2 3 4 5 6 7 1 1 1 3 8 1 2 3 3 6 9 9
A Mathematica l tool s 32
3
B Revie w o f rando m processe s 32 B.l Rando m variable s 32 B.2 Rando m processe s 32 B.3 Processin g o f rando m variable s an d rando m processe s 33 B.4 Continuous-tim e rando m processe s 33
7 7 9 2 6
References 34
1
Index
355
Preface Recent year s hav e see n th e grea t succes s o f O F D M (orthogona l frequenc y division multiplexing ) an d D M T (discret e multitone ) transceiver s i n man y applications. T h e O F D M syste m ha s foun d man y application s i n wireles s communications. I t ha s bee n adopte d i n I E E E 802.1 1 fo r wireles s loca l are a networks, DA B fo r digita l audi o broadcasting , an d DV B fo r digita l vide o broadcasting. T h e D M T syste m i s th e enablin g technolog y fo r high-spee d transmission ove r digita l subscribe r lines . I t i s use d i n ADS L (asymmetri c digital subscribe r lines ) an d VDS L (very-high-spee d digita l subscribe r lines) . T h e O F D M an d D M T system s ar e b o t h example s o f D F T transceiver s t h a t employ redundan t guar d interval s fo r equalization . Havin g a guar d interva l can greatl y simplif y th e tas k o f equalization a t th e receive r an d i t i s now on e o f the mos t effectiv e approache s fo r channe l equalization . I n thi s boo k w e wil l study th e O F D M an d D M T unde r th e framewor k o f filte r ban k transceivers . Under suc h a framework, ther e ar e numerou s possibl e extensions . T h e freedo m in th e filte r ban k transceiver s ca n b e exploite d t o bette r th e system s fo r variou s design criteria . Fo r example , transceiver s ca n b e optimize d fo r minimu m bi t error rate , fo r minimu m transmissio n power , o r fo r highe r spectra l efficiency . We wil l explor e al l thes e possibl e optimizatio n problem s i n thi s book . T h e firs t thre e chapter s describ e th e majo r buildin g block s relevan t fo r th e discussion o f signa l processin g fo r communicatio n an d giv e th e tool s usefu l fo r solving problem s i n thi s area . Chapter s 4- 5 introduc e th e multirat e buildin g blocks an d filte r ban k transceivers , an d th e basi c ide a o f guar d interval s fo r channel equalization . Chapte r 6 give s a detaile d discussio n o f O F D M an d D M T systems . Chapter s 7-1 0 conside r th e desig n o f filte r ban k transceiver s for differen t criteri a an d channe l environments . A detaile d outlin e i s give n a t the en d o f Chapte r 1 . Thi s boo k ha s bee n use d a s a textboo k fo r a first-year graduate cours e a t Nationa l Chia o Tun g University , Taiwan , an d a t Nationa l Taiwan University . Mos t o f th e chapter s ca n b e covere d i n 16-1 8 weeks . Homework problem s ar e give n fo r Chapter s 2-10 . It i s ou r pleasur e t o t h a n k ou r familie s fo r th e patienc e an d suppor t durin g all phase s o f thi s time-consumin g project . W e woul d lik e t o t h a n k ou r univer sities, Nationa l Chia o Tun g Universit y an d Nationa l Taiwa n University , an d the Nationa l Scienc e Counci l o f Taiwa n fo r thei r generou s suppor t durin g th e writing o f thi s book . W e woul d als o lik e t o t h a n k ou r student s Chien-Chan g Li, Chun-Li n Yang , Chen-Ch i Lo , an d Kuo-Ta i Chi u fo r generatin g som e o f the plots . P P V wishe s t o acknowledg e th e Californi a Institut e o f Technology , the Nationa l Scienc e Foundatio n (USA) , an d th e Offic e o f Nava l Researc h (USA), fo r al l th e suppor t an d encouragement .
XI
1 Introduction T h e goa l o f a communicatio n syste m i s t o transmi t informatio n efficientl y and accuratel y t o anothe r location . I n th e cas e o f digita l communications , the informatio n i s a sequenc e o f "ones " an d "zeros " calle d th e bi t stream . T h e transmitte r take s i n th e bi t strea m an d generate s a n information-bearin g continuous-time signa l x a(t), a s i n Fig . 1.1 . W h e n th e signa l propagate s through th e channel , suc h a s wirelines , atmosphere , etc. , distortio n i s in evitably introduce d int o th e transmitte d signa l x a(t). A s a result , th e receive d signal r a(t) a t th e receive r i s i n genera l differen t fro m th e transmitte d signa l xa(t). T h e tas k o f th e receive r i s t o mitigat e th e distortio n an d reproduc e a bit strea m wit h a s fe w error s a s possible . 01001100... bit stream
xa(t)
transmitter
ra(t)
channel
receiver
0110110 bit strea
Figure 1 . 1 . Digita l communicatio n system .
A digita l communicatio n syste m i n genera l consist s o f man y buildin g blocks. Figur e 1. 2 show s a bloc k diagra m consistin g o f th e majo r buildin g blocks t h a t ar e relevan t t o th e topi c o f signa l processin g fo r communications . At th e transmitter , w e hav e a sequenc e o f bit s t o b e sen t t o th e receiver . T h e bits-to-symbol mapping bloc k take s severa l bit s o f inpu t an d map s th e bits t o a rea l o r comple x modulation symbol s(n). Som e processin g ma y b e applied t o thes e symbol s an d th e discrete-tim e outpu t x(n) i s the n converte d to a continuous-tim e signa l x a(t). T h e transmitte d signa l x a(t) propagate s through th e channel . A t th e receiver , th e receive d signa l r a(t) i s converte d t o a discrete-tim e signa l r(n). Usuall y som e signa l processin g i s applie d t o r(n) before th e receive r make s a decisio n o n th e transmitte d symbol s an d obtain s s~(n) (symbol detection). T h e symbol-to-bits mapping bloc k map s th e symbol s s~(n) back t o bi t stream . T h e reade r ca n fin d relevan t backgroun d materia l i n [50, 67 , 120 , 137] . W h e n a signa l propagate s throug h th e channel , distortio n i s invariabl y 1
1. Introductio n transmitter 01001100... bit strea m
bits-tosymbol mapping
transmitter! signal processing
a?o(0
D/C
Pi(«)
channel
r(n)
»■-(*)
H Pa W
C/D
receiver signal processing receiver
s(n)
i(n)
symbol detection!
symbolto-bits mapping
bit stream
Figure 1.2 . Simpl e bloc k diagra m fo r a digita l communicatio n system .
introduced t o th e t r a n s m i t t e d signal . I n additio n t o channe l noise , ther e is als o interferenc e fro m othe r symbols . A t tim e n th e receive d signa l r(n) depends no t onl y o n s(n) , bu t als o o n pas t t r a n s m i t t e d symbol s s(n — 1), s(n — 2 ) , . . . Thi s dependenc y i s terme d inter symbol interference (ISI) . T h e processing applie d t o r(n) a t th e receive r i s carrie d ou t t o obtai n estimate s of th e t r a n s m i t t e d symbol s befor e symbo l detection . Th e proces s i s generall y known a s equalizatio n an d th e signa l processin g bloc k i s calle d a n equalizer . W h e n th e receive r ca n perfectl y regenerat e th e t r a n s m i t t e d symbol s s(n) i n the absenc e o f channe l noise , w e sa y th e equalizatio n i s zero-forcing. I n man y applications, th e transmitte r als o help s wit h equalization . I n thi s cas e som e signal processin g i s applie d t o th e symbol s s(n) , an d th e resultin g outpu t x(n) is t r a n s m i t t e d a s show n i n Fig . 1.2 . One wa y t h a t th e transmitte r ca n greatl y eas e th e tas k o f equalizatio n a t the receive r i s t o divid e th e t r a n s m i t t e d signa l int o block s an d ad d redundan t samples, als o calle d a guar d interval , t o eac h block . Figur e 1. 3 show s tw o examples o f guar d interval s calle d zero padding an d cyclic prefix. I n th e zero-padding scheme , th e guar d interval s consis t o f "zeros. " W i t h cycli c prefix , the las t fe w sample s o f eac h bloc k ar e copie d an d inserte d a t th e beginnin g o f the bloc k a s show n i n th e figure. Th e guar d interva l act s a s a buffe r betwee n consecutive blocks . I f th e guar d interva l i s sufficientl y long , th e interbloc k interference (IBI ) ca n b e avoide d o r ca n b e late r remove d a t th e receive r b y discarding th e redundan t samples . W h e n ther e i s n o IBI , interferenc e come s only fro m th e sam e block . I n thi s case , intrabloc k interferenc e ca n b e cancele d easily usin g matri x operations . T h e mos t notabl e exampl e o f system s t h a t us e cycli c prefi x a s a guar d interval i s th e D F T (Discret e Fourie r Transform)-base d transceive r show n i n
3 block #1 bloc
(a)
s(n)
•" lllllHIIIII
M
k #2
l
2M
zeros padding
(b)
x(n)
zeros padding
illlllll Ihlllll. copy cyclic prefix
(c)
x(n)
ll l copy
cyclic prefix
M
II■ I
illl.l..ll
F i g u r e 1 . 3 . T w o example s o f guar d intervals , (a ) A signa l s(n) w i t h sample s divide d i n t o blocks ; ( b ) t h e sequenc e x{n) afte r zero s ar e padde d t o eac h bloc k o f s ( n ) ; (c ) t h e sequence x(n) afte
r a cycli c prefi x i s inserte d i n eac h bloc k o f s(n).
Fig. 1.4 . Th e signa l processin g a t th e transmitte r applie s IDF T (Invers e Dis crete Fourie r Transform ) t o th e inpu t bloc k o f modulatio n symbol s an d add s a cycli c prefi x t o th e IDF T outputs . Th e receive r discard s th e prefi x an d performs a DF T o f eac h block . Du e t o th e combinatio n o f cycli c prefi x an d IDFT/DFT operations , zero-forcin g equalizatio n ca n b e achieve d b y onl y a set o f simpl e scalar s calle d frequenc y domai n equalizer s (FEQs) . Whe n IS I is canceled , th e overal l syste m fro m th e transmitte r input s t o th e receive r outputs i s equivalen t t o a se t o f paralle l subchannel s a s show n i n Fig . 1.5 . In genera l th e subchannel s hav e differen t nois e variances . I f th e informatio n of the subchanne l nois e variance s i s availabl e t o th e transmitter , th e symbol s Si(n) ca n b e furthe r designe d t o bette r th e performance . Fo r example , th e symbols o f different subchannel s ca n carr y differen t number s o f bits (bi t load ing) [23] , and th e powe r o f the symbol s ca n als o be differen t (powe r loading) . The transmitte r ca n optimiz e bi t loadin g an d powe r loadin g t o maximiz e th e transmission rat e [24] . The cyclic-prefixe d DFT-base d syste m i s widel y use d i n bot h wire d an d wireless communicatio n systems . I t i s generall y calle d a n OFD M (orthogo nal frequenc y divisio n multiplexing ) syste m [27 ] in wireles s transmissio n an d a DM T (discret e multitone ) syste m [24 ] i n wire d DS L (digita l subscribe r lines) transmission . Fo r wireles s applications , th e channe l stat e informatio n is usuall y no t availabl e t o th e transmitter . Th e transmitte r i s typicall y in -
1. Introductio
n
modulation symbols
s(n)
IDFT
•• •
•• •
parallel to serial conversion
s0(n)
->
cyclic prefix
x(n)
W") transmitter signa l processing
FEQ
r(n)
discard prefix
DFT
§1
receiver signa l processing
Figure 1.4 . DFT-base d transceive r wit h cycli c prefi x adde d a s a guard interval .
dependent o f th e channe l an d ther e i s n o bi t o r powe r loading . Havin g a channel-independent transmitte r i s als o a ver y usefu l featur e fo r broadcastin g systems. Fo r broadcas t applications , ther e i s onl y on e transmitte r an d ther e are man y receivers , eac h wit h a differen t transmissio n p a t h . I t i s impossi ble fo r th e transmitte r t o optimiz e fo r differen t channel s simultaneously . I n O F D M system s fo r wireles s applications , usuall y withou t bi t an d powe r al location, th e transmitter s hav e th e desirabl e channel-independenc e property . T h e channel-dependen t par t o f the transceive r i s the se t o f F E Q coefficient s a t the receiver . I n D M T system s fo r wire d DS L applications , signal s ar e trans mitted ove r coppe r lines . Th e channe l doe s no t var y rapidly . Thi s give s th e receiver tim e t o sen d bac k th e channe l stat e informatio n t o th e transmitter . T h e transmitte r ca n the n allocat e bit s an d powe r t o th e subchannel s t o max imize th e transmissio n rate . Mor e detail s o n DS L transmissio n ca n b e foun d in [14 , 122 , 144 , 145] . Both th e O F D M an d D M T system s hav e bee n show n t o b e ver y usefu l transmission systems . Th e D M T syste m wa s adopte d i n standard s fo r ADS L (asymmetric digita l subscribe r lines ) [7 ] an d VDS L (very-high-spee d digita l subscriber lines ) [8 ] transmission . Th e O F D M system s hav e bee n adopte d in standard s fo r digita l audi o broadcastin g [39] , digita l vide o broadcastin g
^0
S]
'■
Figure 1.5 . Equivalen t paralle l subchannels .
[40], wireles s loca l are a network s [54] , an d broadban d wireles s acces s [55] . A variation o f th e cyclic-prefixe d DFT-base d transceive r i s th e so-calle d cyclic prefixed single-carrie r (SC-CP ) syste m [129] . T h e modulatio n symbol s ar e directly sen t ou t afte r a cycli c prefi x i s added . A s i n th e O F D M system , th e redundant cycli c prefi x greatl y facilitate s equalizatio n a t th e receiver . T h e SC-CP syste m i s par t o f th e broadban d wireles s acces s s t a n d a r d [55] . transmitter receive
\{n) _^ | ftf
wZ
sfo) -► ! fN
W T?
w*)-Httf
7 f~\
Hb o W | H
wZ
r
t
•• •
7 (~\
H *M-\V-) \ transmitting filters
(~\
H M o\z) \
J
U
* jv w
f
^
1
-►| H x{z) \-+\ ±N
(~\
i W|
wU
w
k
i
1
\
•• •• ••
H^M-IOOM^ receiving filters
Figure 1.6 . Filte r ban k transceiver , i n whic h onl y th e transmitte r signa l processin g and receive r signa l processin g part s ar e shown .
T h e insertio n an d remova l o f redundan t sample s ca n b e represente d us ing multirat e buildin g blocks . (Definition s o f multirat e buildin g block s wil l be give n i n Chapte r 4. ) Base d o n th e multirat e formulatio n th e DFT-base d system ca n als o b e viewe d a s a discrete-tim e filter bank transceiver (Fig . 1.6) , or a transmultiplexer. T h e transmitte r an d receive r eac h consist s o f a ban k o f discrete-time filters. Suc h a formulatio n lend s itsel f t o th e frequenc y domai n analysis o f th e transceiver . Fo r example , fo r th e transmitte r sid e i t offer s additional insigh t o n th e effec t o f individua l transmittin g filters o n th e trans mitted spectrum . Fo r th e receive r side , w e ca n analyz e th e subchanne l nois e
6
1. Introductio n
using a frequenc y domai n approach . Thes e observation s ar e ver y usefu l fo r designing th e transceive r fo r differen t criteria . I n DS L applications , goo d fre quency separatio n amon g th e transmittin g filters i s important fo r reducin g th e so-called spectral leakage, whic h i s a n undesire d spectra l componen t outsid e the transmissio n band . W h e n th e transmittin g filters hav e highe r stopban d attenuation, th e t r a n s m i t t e d spectru m ha s a faste r spectra l rollof f an d les s spectral leakage . Fo r th e receivin g filters, frequenc y separatio n i s als o impor t a n t fo r th e suppressio n o f interferenc e fro m radi o frequenc y signal s whic h share th e sam e spectru m wit h DS L signals . T h e filter ban k framewor k i s als o usefu l fo r designin g transceiver s wit h better spectra l efficiency . I n th e DFT-base d transceiver , a lon g guar d interva l is required i f the channe l impuls e respons e i s long. Th e us e o f a lon g redundan t guard interva l decrease s th e spectra l efficiency , s o w e woul d lik e th e guar d interval t o b e a s shor t a s possible . O n th e othe r hand , i t i s desirabl e t h a t th e guard interva l b e lon g enoug h s o t h a t F I R equalizatio n i s possible . Th e filter bank transceive r ca n b e use d t o introduc e guar d interval s o f a ver y genera l form. I n mos t cases , zero-forcin g equalizatio n ca n b e achieve d usin g a guar d interval muc h shorte r t h a n wha t i s neede d i n th e DFT-base d transceiver .
Outline Chapter 2 give s a n overvie w o f a digita l communicatio n system . Fro m a continuous-time channe l impuls e respons e an d channe l noise , th e equivalen t discrete-time channe l an d channe l nois e wil l b e derived . Th e equivalen t discrete-time channe l mode l i s ver y usefu l i n th e analysi s an d desig n o f digita l communication systems . W i t h suc h a model , ther e i s n o nee d t o rever t t o the continuous-tim e channe l an d noise . Terminolog y an d fundamental s suc h as modulatio n symbols , equalization , an d transmissio n ove r paralle l channel s are als o reviewed . Chapter 3 i s a stud y o f channe l equalization . W e wil l discus s th e desig n of F I R equalization , i n whic h th e receive r contain s onl y F I R filters. A ver y powerful too l calle d th e orthogonalit y principl e wil l b e introduced . Th e prin ciple i s o f vita l importanc e i n th e desig n o f MMS E (minimu m mea n squar e error) receivers . I t ca n b e use d fo r th e equalizatio n o f scala r channel s a s wel l as paralle l channels . Chapter 4 give s th e basic s o f multirat e signa l processing . Multirat e build ing block s ar e introduced . Th e operation s o f blockin g an d unblockin g t h a t appear frequentl y i n digita l transmissio n ar e describe d usin g multirat e build ing blocks . I n addition , polyphas e decompositio n o f filters i s reviewed . Base d on th e decomposition , th e polyphas e representatio n o f filter bank s ca n b e de rived an d efficien t polyphas e implementatio n ca n b e obtained . Reader s wh o are familia r wit h multirat e system s an d filter bank s ca n ski p thi s chapter . Chapter 5 formulate s som e moder n digita l communicatio n system s usin g multirate buildin g blocks . Th e filter ban k transceive r i s introduce d an d con ditions o n th e transmitte r an d receive r fo r zer o IS I ar e derived . Usin g th e multirate formulation , redundan t sample s ca n b e inserte d i n th e t r a n s m i t t e d signal. Tw o type s o f redundan t sample s ar e discusse d i n detail : cycli c prefi x and zer o padding . Th e matri x for m representation s o f thes e system s ar e use d frequently i n th e discussion s o f application s i n late r chapters .
7
1.1. Notation s
Chapter 6 i s devote d t o th e stud y o f som e usefu l DFT-base d transceivers . T h e O F D M , D M T , an d SC-C P system s wil l b e presente d an d th e performanc e will b e analyzed . T h e correspondin g filte r ban k representatio n wil l als o b e derived. Thes e transceiver s hav e foun d man y practica l application s du e t o the fac t t h a t the y ca n b e implemente d efficientl y usin g fas t algorithms . Chapter 7 deal s wit h th e desig n o f optima l transceiver s whe n th e trans mitter doe s no t hav e th e channe l stat e information , whic h i s usuall y th e cas e for wireles s applications . A s th e transmitte r doe s no t hav e th e channe l knowl edge, ther e i s n o bit/powe r allocation . W e conside r th e desig n o f minimu m bi t error rat e (BER ) transceiver s b y addin g a unitar y precode r a t th e transmitte r and a post-code r a t th e receive r o f th e O F D M system . W e wil l se e t h a t th e derivation o f th e minimu m B E R transceive r nicel y tie s th e O F D M an d th e SC-CP system s together . Chapter 8 deal s wit h th e desig n o f optima l transceiver s whe n th e channe l state informatio n i s availabl e t o th e transmitter . I n additio n t o bi t an d powe r allocation, th e transmitte r an d receive r ca n b e jointl y optimized . Fo r a give n error rat e an d targe t transmissio n rate , th e transceive r wil l b e designe d t o minimize th e transmissio n power . Chapter 9 describe s a metho d t o improv e th e frequenc y separatio n amon g the subchannel s fo r th e DFT-base d transceivers . Som e shor t F I R filter s calle d subfilters ar e introduce d i n th e subchannel s t o enhanc e th e stopban d atten uation o f th e transmittin g an d receivin g filters . B y usin g a slightl y longe r guard interval , w e ca n includ e th e subfilter s withou t changin g th e ISI-fre e property. W h e n subfilter s ar e adde d t o th e receiver , th e transmissio n rat e can b e increase d considerabl y i n th e presenc e o f narrowban d R F I (radi o fre quency interference) . Fo r th e transmitte r side , th e subfilter s ca n improv e th e spectral rollof f o f th e transmitte d spectru m whil e havin g littl e effec t o n th e transmission rate . Chapter 1 0 i s a stud y o f minimu m redundanc y fo r F I R equalization . Fo r a given channel , w e consider th e minimu m redundanc y t h a t i s required t o ensur e the existenc e o f F I R equalizers . W e wil l se e t h a t th e answe r i s directl y tie d t o what w e cal l th e congruou s zero s o f th e channel . T h e minimu m redundanc y can b e determine d b y inspectio n onc e th e zero s o f th e channe l ar e known .
1.1 Notation
s
• Boldface d lowe r cas e letter s represen t vector s an d boldface d uppe r cas e letters ar e reserve d fo r matrices . T h e notatio n A T denote s th e transpos e of A , an d A ^ denote s th e transpose-conjugat e o f A . • T h e functio n E [y] denotes th e expecte d valu e o f a rando m variabl e y. • T h e notatio n 1M i s use d t o represen t th e M x M identit y matri x an d 0 m n denote s a n m x n matri x whos e entrie s ar e al l equa l t o zero . T h e subscript i s omitted whe n th e siz e of the matri x i s clear fro m th e context . • T h e determinan t o f a squar e matri x A i s denote d a s d e t ( A ) . T h e nota tion diag[A o A i . . . A M - I ] denote s a n M x M diagona l matri x wit h the fcth diagona l elemen t equa l t o Afe .
8
1. Introductio n
• Th e notatio n W i s use d t o represen t th e M x M unitar y DF T matrix , given b y [Wlfcn = —Le-'M* ™ for 0 VM
< k,n < M - 1 .
• Fo r a discrete-tim e sequenc e c(n) , th e z-transfor m i s denote d a s C(z) and th e Fourie r transfor m a s C(e^). Fo r a continuous-tim e functio n xa(t), th e Fourie r transfor m i s denote d a s X a(jQ,).
2 Preliminaries o f digita l communications In thi s chapter , w e shal l revie w som e introductor y material s t h a t ar e usefu l for ou r discussio n i n subsequen t chapters . Fo r convenience , w e reproduc e i n Fig. 2. 1 th e bloc k diagra m fo r digita l communicatio n system s introduce d i n Chapter 1 . transmitter 01001100.. bit strea m
bits-tosymbol mappingl
s(n)
transmitter! signal processing
x(n)
xa(t) D/C
PiW
channel
r«(t)
r
r(n) P2(t)
C/D
receiver signal processing
u(n)
s(n)
symbol detection
receiver
symbolto-bits mapping
bit stream
Figure 2 . 1 . Simpl e bloc k diagra m fo r a digita l communicatio n system .
2.1 Discrete-tim
e channe l model s
In th e stud y o f communicatio n systems , th e transmissio n channe l i s ofte n modeled a s a continuous-tim e linea r tim e invarian t (LTI ) syste m wit h impuls e 9
10
2. Preliminarie s o f digita l communication s Qa(t)
xa(t)
ra(t)
channel
xa{t) Ca(t)
Figure 2 . 2 . LT I channe l model .
qa(f) x(n)
D/C
T
xa(t)
Px(t)
j; r
Ca(t)
a(t)
J
p2(t)
^
C/D
r(»)
T
T
(a)
x(n)
> c(»
-► r(n)
)
(b) Figure 2.3 . (a ) Th e syste m fro m x(n) t o r(n). (b model.
) Equivalen t discrete-tim e channe l
response c a(t) an d additiv e nois e q a(t). Thi s LT I channe l mode l i s show n i n Fig. 2.2 . Give n th e inpu t x a(t), th e channe l produce s th e o u t p u t CO
/
-co
ca(r)xa(t -
r)dr + q
a(t).
Letting th e symbo l V denot e convolution , w e ca n writ e ra{t) =
(x a *c a)(t) +
q
a(t).
Though th e channe l i s a continuous-tim e system , i t i s ofte n mor e convenien t to wor k directl y o n a n equivalen t discrete-tim e system . I n man y aspect s o f digital communications , a discrete-tim e channe l mode l i s ofte n adequat e an d much easie r t o wor k with . I n thi s section , w e shal l sho w t h a t th e syste m fro m x(n) a t th e transmitte r t o r(n) a t th e receive r (Fig . 2.1 ) i s equivalen t t o a discrete-time LT I system . T h e syste m fro m x(n) t o r(n) i s show n separatel y i n Fig . 2.3(a) . Suppos e t h a t th e sample s x{n) ar e space d apar t b y T seconds . Th e D / C converte r take s
11
2.1. Discrete-tim e channe l model s
the discrete-tim e sequenc e x(n) an d produce s a continuous-time impuls e trai n spaced apar t b y T: ^x(n)6a(t-nT), n
where S a(t) i s th e continuous-tim e Dira c delt a function . Afte r th e impuls e train passe s throug h th e transmittin g puls e pi(t) , w e ge t a continuous-tim e signal x a(t): oo
xa{t)= J2 x{k)pi(t-kT). /c= —o o
The signa l x a(t) i s transmitte d throug h th e channel . A t th e receivin g end , the receive d signa l i s oo
ra{t) =
(x a * ca){t) + q a(t) =
Y, x(k)(pi*c
a){t-kT)
+
q a(t).
/ c = — oo
The receive d signa l r a(t) i s first passe d throug h a receiving pulse P2(t), whic h produces oo
Wa{t) = (r a*P2)(t)= Y,
x(k)(
Pi*ca*P2)(t-kT)
+
(q a*P2)(t). (2.1
)
k= — oo
Then w a(t) i s uniformly sample d ever y T second s to produce the discrete-tim e output r(n) = w a(nT). Thi s unifor m samplin g operatio n i s denote d b y th e box labele d C/D . Defin e th e effectiv e continuous-tim e channe l an d effectiv e noise, respectively , a s follows : ce(t) =
(pi * ca *p 2 )(*) an d q e(t) = (q a *P2)(*)-
Then th e receive d discrete-tim e signa l i s oo
r(n) = Y^ x(k)c
e(nT
-
kT) + q e{nT).
A;= —oo
The abov e expressio n ca n b e rewritte n a s oo
r(n) =
y , x(k)c(n — k) + q(n), k= — oo
where c(n) an d q(n) are , respectively , th e discrete-tim e equivalen t channe l and nois e give n b y c(n) = (p i * c a *p2)(t) q{n) = (q
a*P2)(t)
t=nT (2.2
)
t=nT
Thus, th e syste m show n i n Fig . 2.3(a ) ca n b e represente d a s i n Fig . 2.3(b) , which contain s onl y discrete-tim e signal s an d systems . Th e transfe r functio n of th e discrete-tim e channe l i s give n b y oo
C(z)= Y
<
n
>~n-
12
2. Preliminarie s o f digita l communication s
Observe t h a t c(n) i s th e sample d versio n o f th e cascad e o f th e transmittin g pulse pi (t) , th e channe l c a (t), an d th e receivin g puls e pi (t) . Choosin g differen t transmitting an d receivin g pulse s wil l affec t th e discrete-tim e channel . I n practice, th e channe l i s ofte n modele d a s a finit e impuls e respons e (FIR ) filter. Fro m (2.2) , w e ca n se e t h a t th e channe l lengt h i s inversel y proportiona l to th e samplin g perio d T . Reducin g T b y one-hal f wil l doubl e th e lengt h o f c(n). W h e n th e channe l C(z) ha s mor e t h a n on e nonzer o t a p , sa y c(0 ) an d c ( l ) , i t wil l introduc e interference betwee n th e receive d symbols . T o se e this , suppose t h a t ther e ar e n o "signa l processing " block s a t th e transmitte r an d receiver i n Fig . 2.1 , the n th e t r a n s m i t t e d signa l x(n) = s(n). Th e receive d signal wil l b e r(n) =
c(0)s(n) +
c(l)s(n —
1 ) + q(n);
the curren t symbo l s(n) i s contaminate d b y th e pas t symbo l s(n — 1) . Thi s phenomenon i s know n a s intersymbo l interferenc e (ISI) . Th e tas k o f symbo l recovery i s complicate d b y b o t h th e additiv e nois e q(n) an d ISI . Example 2. 1 Conside r a transmissio n syste m wit h effectiv e continuous-tim e channel c e(t) = (p i * c a *P2)(t) give n i n th e Fig . 2.4 . Vv (Pi*c ^- -
0
a*p2)(t)
^►
1
2t
Figure 2.4 . A n exampl e o f (p± * c a *P2)(t).
Suppose w e sen d on e sampl e o f x(n) pe r second , i.e . th e samplin g perio d T = 1 . The n th e discrete-tim e equivalen t channe l i s c(n) = 5{n — 1), a delay . T h e channe l doe s no t introduc e ISI . W h e n w e increas e ou r transmissio n rat e to tw o sample s pe r second , the n th e samplin g perio d become s T = 0. 5 an d the discrete-tim e equivalen t channe l i s C(z) = O.bz -1 + z~ 2 + 0.5z~ 3. T h e channel become s a three-ta p F I R channel . W e se e t h a t th e faste r w e sen d th e samples x(n), th e longe r th e F I R channe l c{n). ■ Note t h a t ther e i s n o carrie r modulatio n i n th e syste m show n i n Fig . 2.1 . This i s know n a s baseban d communication . I n wireles s communications , th e signal x a(t) i s modulate d t o a carrie r frequenc y f c fo r transmission , a s show n in Fig . 2.5 . Thi s i s know n a s passban d transmission . Fo r passban d commu nications, afte r carrie r modulatio n th e signa l t h a t i s t r a n s m i t t e d throug h th e channel c a(t) i s give n b y
va(t) =
2Re{x
a(t)e^^},
where i?e{« } denote s th e rea l part . A t th e receiver , th e receive d signa l i n thi s case become s y a(t) = (v a * c a)(t) + q a(f). Carrie r demodulatio n i s performe d to obtai n th e baseban d signal 1 ra{t) =
y
a(t)e-^^.
13
2 . 1 . Discrete-tim e channe l model s Va(t])
Xa(t)
x(n) D/C
t
-+
P,(t)
carrier mod.
ya(t)
channel
ra{t)
carrier w demod.
P2(t)
—> C/D
t
T
T
Figure 2.5 . Passban d communicatio n channel .
By followin g a simila r procedure , on e ca n sho w (Proble m 2.3 ) t h a t th e syste m sandwiched betwee n th e C / D an d D / C converter s i n a passban d communica tion syste m i s als o equivalen t t o a discrete-tim e LT I system . I n thi s case , th e equivalent discrete-tim e channe l an d nois e are , respectively , give n b y c(n)
(pi *c a *P2)(t)
q(n)
t=nT
(2.3)
where c a(t) = c a(t)e~j27r^ct an d q a(t) = q a(t)e~j27r^ct. Fro m thi s relation , on e can clearl y se e th e effec t o f carrie r modulation : wha t th e transmitte d signa l xa(t) see s i s a frequency-shifte d versio n o f th e origina l channe l c a(t) an d nois e qa(t). Not e t h a t b o t h th e channe l impuls e respons e c(n) an d th e channe l noise q(n) ar e comple x fo r passban d transmissio n du e t o th e t e r m e _ j 2 7 r ^ c t . For baseban d transmission , thes e quantitie s ar e real . Channel nois e Throughou t thi s book , w e wil l assum e t h a t th e channe l noise q(n) i s a zero-mean wide sense stationary (WSS) Gaussian rando m process. Fo r baseban d transmission , q(n) i s rea l an d th e probabilit y densit y function (pdf ) o f a zero-mea n Gaussia n nois e q(n) i s give n b y
fM
I
V/2A/" 0
(2.4)
where A/ o i s th e nois e variance . Figur e 2. 6 show s th e well-know n bell-shape d Gaussian pd f fo r differen t value s o f A/o, mor e widesprea d fo r a large r variance . For passban d transmission , th e channe l nois e q(n) i s i n genera l modele d a s a zero-mea n circularly symmetric complex Gaussian rando m variabl e whos e pdf i s give n b y
fq(q) = - U - ( « ) M / - o, (2.5 7TA/0
)
where q = qo + jqi an d A/ o i s th e varianc e o f q. T h e rea l an d imaginar y part s are b o t h zero-mea n Gaussia n an d the y hav e equa l variance : E[q^] = E[q\] = AA0/2. Thi s pd f i s show n i n Fig . 2. 7 fo r Af 0 = 1 . In th e following , w e will describ e som e commonl y use d model s o f equivalen t discrete-time channels . Thes e models , thoug h simplified , ar e usefu l fo r th e analysis o f digita l communicatio n systems . The y ar e als o frequentl y employe d in numerica l simulation s t o evaluat e th e syste m performance . 1 T h e high-frequenc y componen t centere d aroun d 2f c i s i n genera l eliminate d b y a low pass filte r i n th e proces s o f carrie r demodulation .
14
2. Preliminarie s o f digita l communication s
0.5 0.4
0.3 0.2 0.1 0 -
4
-
2
0
2
4
q Figure 2.6 . Th e pd f o f a zero-mean rea l Gaussia n rando m variable .
AGN an d AWG N channel s A channe l i s calle d a n AG N (additiv e Gaus sian noise ) channe l whe n th e channe l satisfie s th e followin g tw o properties . (1) It s channe l impuls e respons e i s
c(n) =| v J v= S(n) J ^
1
'U
=
°'
0 , otherwise
.
(2) Th e channe l nois e q(n) i s a Gaussia n rando m process . If i n additio n t o bein g a Gaussia n rando m process , q(n) i s als o white , t h a t is , E{q(n)q*(m)} = 0 wheneve r m ^ n , the n th e channe l i s a n AWG N (additiv e white Gaussia n noise ) channel . W h e n th e channe l i s a n AG N o r AWG N channel, ther e i s n o IS I an d th e transmissio n erro r come s fro m th e channe l noise only . FIR channel s I n man y applications , th e channe l no t onl y introduce s ad ditive nois e q(n), bu t als o distort s th e t r a n s m i t t e d signal . Th e channe l c(n) i s no longe r a n impulse , an d i n genera l i t ha s a causa l infinit e impuls e respons e (IIR). Fo r th e purpos e o f analysis , th e channe l i s ofte n modele d a s a finit e impulse respons e (FIR ) filter, t h a t i s L
C(z) =
J2c(n)z- n. (2.6 n=0
)
2.1. Discrete-tim e channe l model s
15
-5 -
5
Figure 2.7 . Th e pd f o f a circularl y symmetri c comple x Gaussia n rando m variable .
T h e impuls e respons e i s nonzer o onl y fo r a finit e numbe r o f coefficient s (o r taps). T h e integer s L an d L + 1 are , respectively , th e channe l orde r an d channel length . B y makin g L larg e enough , a causa l II R filte r ca n b e wel l approximated b y a n F I R filter . I n th e frequenc y domain , th e magnitud e re sponse o f th e F I R channe l | C ( e j a ; ) | i s no t fla t unles s c{n) ha s onl y on e nonzer o t a p . T h e channe l ha s differen t gains fo r differen t frequenc y components . T h u s such channel s ar e als o know n a s frequency-selectiv e channels . W h e n c{n) has onl y on e nonzer o t a p , i t i s calle d frequency-nonselective . Random channel s wit h uncorrelate d tap s I n man y situations , th e exac t channel impuls e respons e ma y no t b e available , an d onl y th e statistic s o f th e channel i s known . On e o f th e widel y adopte d channe l model s assume s t h a t the tap s ar e zero-mea n uncorrelate d rando m variable s wit h know n variances . In thi s case , c(n) satisfie s th e followin g conditions : (1) E[c(n)} = 0, (2) E[c{n)c*{n-k)] = a
2
J{k).
(2.7)
T h e se t o f quantitie s {cr^} i s calle d th e powe r dela y profile . W e sa y t h a t th e channel ha s a n exponentia l powe r dela y profil e whe n a^ decay s exponentiall y
16
2. Preliminarie s o f digita l communication s
with respec t t o n (se e Fig. 2.8) . I f th e channe l impuls e response s c(n) ar e in dependent identica l rando m variables , i t i s often calle d a n i.i.d . (independen t identically distributed ) channel . Thes e channe l model s ar e ofte n employe d i n numerical simulation s whe n w e want t o lear n th e syste m performanc e "aver aged ove r al l channels. " 1
1/2
2 n
1/4 , 1/ 8 1/ 16 || , 1/3 2 01
2
3
4
5
►
Figure 2.8 . Exponentia l powe r delay profil e with
2.2 Equalizatio
n
In wideban d communicatio n systems , th e transmissio n channe l ofte n intro duces som e degre e o f intersymbo l interferenc e (ISI ) t o th e receive d signal . Suppose a signa l x(n) i s transmitte d ove r a n FI R channe l wit h impuls e re sponse c(n) , fo r 0 < n < L. The n th e receive d signa l i s r(n) = c(0)x(n) + c(l)x(n — 1 ) + • • • + c{L)x(n — L) + q(n), where q(n) i s the channe l noise . A t th e receiver , w e would lik e to proces s r(n) so that it s outpu t x(n) i s "close " t o x(n — no), a delaye d versio n o f the trans mitted signa l x(n). Th e intege r n o is called the syste m delay . Suc h processin g is generally know n a s equalization . Ther e ar e man y equalizatio n techniques . One approac h i s t o us e a n LT I filte r a{n) a s show n i n Fig . 2.9 . Thi s i s als o known a s linea r equalizatio n an d th e filte r a{n) i s calle d a n equalizer . Be low w e loo k a t som e simpl e equalizatio n techniques . I n late r chapters , mor e advanced method s fo r channe l equalizatio n wil l b e explored . Consider th e FI R channe l C(z) = ^2 n= 0c(n)z~n. Whe n C(z) i s known , there ar e many method s t o eliminat e o r mitigate ISI . One simple way is to us e an II R filte r A(z) = 1/C(z). Whe n r{n) passe s throug h th e equalize r a(n) , the outpu t wil l b e x(n) = x(n) + (q * a)(n); there i s n o ISI . Suc h a n equalize r i s sai d t o b e zero-forcin g (ZF ) an d th e system i s calle d a n ISI-fre e system . Defin e th e outpu t erro r e{n) a s e(n) = x(n) — x(n — no),
2.3. Digita l modulatio n 1
7 q(n)
x(ri)
c(n)
±
a(n)
> x(n)
Figure 2.9 . LT I equalizer .
where th e syste m dela y n o = 0 i n thi s case . T h e n th e erro r e(n ) = (q * a)(n) consists onl y o f noise . I n practice , th e II R zero-forcin g equalize r 1/C(z) i s not frequentl y use d becaus e th e equalize r 1/C(z) wil l b e unstabl e whe n C(z) has zero s outsid e th e uni t circle . T o avoi d thi s problem , w e ca n us e a n F I R equalizer a(n). T h e outpu t o f th e equalize r i s x(n) =
(a * c* x)(n) + ( a * q)(n).
There ar e tw o commo n way s o f designin g a(n). On e i s t o choos e a(n) suc h t h a t th e IS I i s smal l i n som e sense . T h a t is , w e woul d lik e th e convolutio n (c * a)(n) t o b e a s clos e t o a dela y 5{n — no) a s possible . Anothe r wa y o f designing a(n) i s t o includ e th e effec t o f b o t h channe l nois e an d ISI . Not e t h a t whe n C(z) ha s a zer o zo ^ 0 , th e produc t A(z)C(z) canno t b e a dela y z~n° fo r an y F I R equalize r A(z) becaus e A(z)C(z) wil l hav e a zer o a t ZQ. In particular , whe n a n F I R channe l ha s mor e t h a n on e nonzer o t a p , a n F I R equalizer ca n neve r b e zero-forcing ; th e outpu t erro r wil l contai n b o t h th e channel nois e an d ISI . W e wil l stud y thes e solution s o f F I R equalizer s i n mor e detail i n Chapte r 3 . Signal t o nois e rati o (SNR ) I n a digita l communicatio n system , w e ofte n measure th e performanc e b y evaluatin g th e rati o o f signa l powe r £ x t o th e mean square d erro r a 2e = E[\x(n) — x(n — no)|2 ]. Thi s rati o i s know n a s th e signal t o nois e rati o (SNR ) an d i t i s give n b y
W h e n th e equalize r i s no t zero-forcing , th e erro r e(n ) = x(n) — x(n — no) contains no t onl y noise , bu t als o IS I terms . I n thi s case , /3 is als o know n a s the signa l t o nois e interferenc e rati o (SINR) , bu t w e shal l refe r t o i t simpl y as SNR .
2.3 Digita
l modulatio n
In digita l communicatio n systems , th e transmitte d bi t strea m consistin g o f "zeros" an d "ones, " i s ofte n partitione d int o segment s o f length , say , b. Eac h segment (codeword ) i s the n mappe d t o on e membe r i n a se t o f 2 b rea l o r complex numbers . Thi s proces s i s know n a s digita l modulation . T h e rea l or comple x number s representin g th e codeword s ar e know n a s modulatio n
18
2. Preliminarie s o f digita l communication s
symbols. A t th e receiver , base d o n th e receive d informatio n a decisio n wil l be mad e o n th e symbo l transmitted . Thi s proces s i s calle d symbo l detection . T h e resultin g symbo l i s the n decode d t o a 6-bi t codewor d (symbol-to-bit s mapping). Man y type s o f digita l modulation s hav e bee n developed . I n th e following, w e wil l describ e tw o widel y use d digita l modulatio n scheme s know n as th e puls e amplitud e modulatio n (PAM ) an d q u a d r a t u r e amplitud e mod ulation (QAM) . W e wil l analyz e thei r performanc e fo r transmissio n ove r a n AWGN channel . Not e t h a t i n a baseban d communicatio n system , wher e th e channel i s real , PA M i s ofte n employed , wherea s i n a passban d syste m th e channel i s complex , an d QA M i s usuall y employed .
2.3.1 Puls
e amplitud e modulatio n (PAM )
In 6-bi t puls e amplitud e modulatio n (PAM) , a codewor d o f b bits i s m a p p e d to a rea l numbe r G { 0 , 1 , . . ., 2 b~1 - 1} . (2.8
ek
s = ±(2f c + l ) A , wher
)
Figure 2.1 0 show s th e possibl e value s o f a PA M symbo l fo r b = 2 an d 6 = 3 , respectively [120] . Th e Gra y cod e indicate d i n th e figure wil l b e explaine d later. Suc h figures ar e calle d signa l constellations . Th e minimu m distanc e between tw o constellatio n point s i s 2A . Assum e t h a t al l constellatio n point s are equiprobable . The n th e signa l powe r o f a 6-bi t PA M constellatio n i s give n by gs = E[s 2] =
^-(2 2b-l). (2.9
)
T h e signa l powe r i s proportiona l t o th e squar e o f th e minimu m distanc e 2A . W h e n th e minimu m distanc e i s fixed, th e signa l powe r wil l increas e b y roughl y 6 d B fo r ever y additiona l bit .
(a)
(b)
-3A -
AA
-7A -5
3
A -3 A -
A
AA
3
A5
A7
A
Figure 2 . 1 0 . PA M constellations : (a ) 2-bi t PAM ; (b ) 3-bi t PAM .
Suppose t h a t a 6-bi t PA M symbo l s o f th e for m i n (2.8 ) i s t r a n s m i t t e d through a rea l zero-mea n AWG N channe l wit h nois e varianc e A/o . Th e re ceived signa l i s r = s + q. Assum e t h a t s an d q ar e independent . T h e conditional pd f o f th e receive d signa l r give n t h a t s i s t r a n s m i t t e d i s fr\s(r\s)=
f r\3(s +
q\s) = f
q(q),
2.3. Digita
l modulatio n
19
where f q(q) i s th e Gaussia n pd f give n i n (2.4) . Fro m th e receive d signa l r , w e make a decisio n o n th e transmitte d symbol . T h e commonl y adopte d decisio n rule i s th e neares t neighbo r decisio n rul e ( N N D R ) . I n a n NNDR , w e mak e a decisio n s = (2 z + l ) A , i
f \r- (2 z + 1)A | < \r - (2 j + 1)A | fo r al l j . (2.10
)
T h e decisio n J i s th e constellatio n poin t t h a t i s closes t t o th e receive d sig nal r . Fo r al l th e interio r constellatio n points , th e symbo l wil l b e detecte d erroneously i f th e channe l nois e ha s \q\ > A . Fo r th e tw o exterio r constella tion point s s = (2 b — 1)A an d s = (—2 6 + l ) A , w e wil l mak e a n erro r whe n q < — A an d q > A , respectively . Therefor e th e probabilit y t h a t th e detectio n is erroneou s i s give n b y P(s + s\s) =
( P(q > A ) , fo r s = ( - 2 6 + 1)A ; I P(q< - A ) , fo r s = (2 6 - 1)A ; [ P(\q\ > A ) , otherwise .
Using th e formul a fo r th e Gaussia n pd f give n i n (2.4) , th e conditiona l proba bility o f symbo l erro r i s give n b y
P(s^s\s)
\ Q ^W^W^ for* =
± (2>-l)A;
1 2Q i)l {2»-W 0)>0thCTWiSe '
where th e functio n Q(x) i s th e are a unde r a Gaussia n tail , define d a s
i r°°
Q{x)=
^mL e
~r2/2dT- (2
-n)
For equiprobabl e PA M symbols , th e symbo l erro r rat e (SER ) i s give n b y
SERpam(b) =
\ h E P&*
s
\g) = 2(1 " 2- b )Q(J ( 2 2 b 3 _ g s 1 ) A / - o ). (2.12
)
As th e functio n Q(x) decay s rapidl y wit h respec t t o x , fo r a moderat e SN R value mos t symbo l error s happe n betwee n adjacen t constellatio n points . W e can us e a mappin g i n whic h th e codeword s o f adjacen t constellatio n point s differ b y onl y on e bit . A n erro r betwee n adjacen t constellatio n point s result s only i n on e bi t error . T h e widel y use d Gra y cod e i s a mappin g schem e t h a t possesses thi s property . Figure s 2.11(a ) an d (b ) sho w a Gra y cod e mappin g for 2-bi t PA M an d 3-bi t PAM , respectively . W h e n th e Gra y cod e i s employe d in a 6-bi t PA M modulation , th e bi t erro r rat e (BER ) an d SE R ar e relate d approximately b y [120 ] BERpam(b) «
-SER
pam(b).
(2.13
)
From th e abov e formulas , w e se e t h a t th e bi t erro r rat e depend s o n th e signa l power S s an d th e nois e powe r A/o - T h e B E R curve s ar e ofte n plotte d agains t the SNR . Fo r AWG N channels , th e SN R i s simpl y th e rati o £ 8/N0. I n orde r
2. Preliminarie s o f digital communication s
20
to evaluat e th e accurac y o f th e BE R formul a derive d above , w e comput e th e BER curve s throug h Mont e Carl o simulatio n i n Fig. 2.12 . In the Monte Carlo simulation , sufficientl y man y round s o f simulation ar e carrie d ou t an d the result s ar e average d t o give a n accurate estimat e o f the actua l BER . As a rul e o f thumb, fo r a BER o f 10—z, w e nee d t o generate a t least 10 0 * 10z bits i n th e simulation s to obtain a n accurat e estimate . I n Fig . 2.12 , th e BE R approximations obtaine d fro m (2.13 ) ar e plotte d i n the dotte d curve s an d th e BER curve s obtained fro m th e Mont e Carl o simulation ar e plotted i n the soli d curves (a s the dotte d curve s almos t overla p with th e soli d curves , we see onl y the soli d curve s i n the figure) . Sinc e th e tw o curve s matc h almos t perfectly , the formul a i n (2.13 ) give s a very good approximatio n o f the tru e BE R values . Comparing th e PA M o f different constellatio n sizes , w e se e tha t fo r a BE R of 10~ 4 , w e nee d a n SNR o f around 11. 7 dB , 18. 2 dB , an d 24. 2 d B fo r 1-bit, 2-bit, an d 3-bi t PAM , respectively . T o achiev e th e sam e BER , th e require d SNR increase s roughl y b y 6 dB fo r ever y additiona l bit . (a) "
[V)
00
01
11
10
-3A
-A
A
3A
000
001
011
010
110
-7A -5 A -3 A - A A
3
111
101
A5
100
A7 A
Figure 2 . 1 1 . Gra y cod e mapping : (a ) 2-bit PAM ; (b ) 3-bit PAM .
Equation (2.12 ) relate s th e erro r rat e t o the SN R £ s/Afo. I t can b e used to obtai n th e numbe r o f bits tha t ca n b e transmitted fo r a given SN R an d target erro r rate . B y rearrangin g th e term s i n (2.12), w e ge t >=ilo&(l+
2v ^\
where
r para
L
r
^, ( 2 . 4
)
para
^ J-J -ftpam
2(1-2-^ If on e compare s th e formul a fo r b with th e channe l capacity , whic h i s given by 0.5log2 ( 1 + Ss/Afo) (bit s pe r use) , the quantit y T pam represents th e differenc e i n the require d SN R betwee n th e PAM schem e an d th e channe l capacity . Therefor e T parn is also know n a s th e SNR gap . Fo r a moderate erro r rate , th e inverse Q function i s relatively flat. Therefor e th e SN R ga p is well approximate d b y Tpam « \ [Q- 1 (SER pam/2)]2 .
(2.15
)
The SN R ga p i s a quantity tha t depend s onl y o n th e erro r rate . I n Table 2.1, we list th e value s of Tpam for som e typica l SER parn.
21
2.3. Digita l modulatio n
10 — B — 2-bi t PA M 0 3-bi t PA M 10"
10"' DC LU CO
10"'
10
10
10 1
52 SNR(dB)
0
25
30
Figure 2 . 1 2 . BE R performanc e o f PA M i n a real zero-mea n AWG N channel . Th e soli d curves ar e th e experimenta l value s obtaine d fro m th e Mont e Carl o simulatio n an d th e dotted curve s (almos t indistinguishabl e fro m th e soli d curves ) ar e th e theoretica l value s obtained fro m th e formul a i n (2.13) .
O Hi -LLpam
r L
para
Fpam i n d B
10"
2
2.21
3.44
10"
3
3.61
5.57
10" 4
5.05
7.03
icr icr 6
6.50
8.13
7.98
9.02
7
9.46
9.76
5
lO"
Table 2.1 . Th e SN R ga p T parn in (2.15 )
Binary phas e shif t keyin g ( B P S K ) modulatio n Fo r th e specia l cas e o f PAM wit h 6 = 1 , ther e ar e onl y tw o constellatio n points , + A an d —A , an d thi s is mor e commonl y know n a s binar y phas e shif t keyin g (BPSK ) modulation . For B P S K symbols , th e bi t erro r rat e an d symbo l erro r rat e ar e th e same .
22
2. Preliminarie s o f digita l communication s
T h e formul a (2.12 ) reduce s t o BERbpSk=
SERbp Sk=
Q
Unlike (2.13) , th e abov e B E R formul a fo r B P S K i s exact ; n o approximatio n is made . Example 2. 2 Suppos e th e transmitte r i s t o sen d th e followin g bi t stream : 000 01 0 11 1 11 0 01 0 10 0 10 1 110 . Assume t h a t th e modulatio n schem e i s a 3-bi t PA M wit h Gra y cod e mappin g as i n Fig . 2.10 . Th e "bits-to-symbo l mapping " bloc k take s ever y thre e inpu t bits an d map s t h e m t o a 3-bi t PA M symbol . Th e first thre e bit s ar e "000 " an d thus fro m Fig . 2.1 0 w e kno w t h a t th e first PA M symbo l i s — 7A. The n th e next thre e bit s ar e "010 " an d fro m th e figure w e hav e th e nex t PA M symbo l as —A . Continuin g thi s process , w e find t h a t th e abov e sequenc e o f 2 4 bit s i s m a p p e d t o th e followin g PA M symbols : s(n) :
- 7 A , - A , 3A , A , - A , 7A , 5A , A .
Now suppos e t h a t th e abov e PA M symbol s ar e t r a n s m i t t e d ove r a n AWG N channel an d du e t o channe l nois e th e receive d sequenc e i s r(n) :
- 9 A , - 0 . 7 A , 1.9A , 1.1A , - 1 . 3 A , 6.6A , 4.8A , 2.1A .
Assume t h a t a t th e receive r ther e i s n o additiona l signa l processin g an d t h a t N N D R i s applie d directl y t o r(n). Th e outpu t o f th e symbo l detecto r wil l b e s(n) :
- 7 A , - A , A , A , - A , 7A , 5A , 3 A .
Comparing s~(n) wit h s(n) , w e find t h a t w e hav e mad e tw o symbo l error s out o f eigh t t r a n s m i t t e d symbol s (indicate d b y boldface d symbols) . I n thi s experiment, th e symbo l erro r rat e i s given b y SER = 0.25 . Afte r th e "symbol to-bits mapping " bloc k usin g Gra y code , w e obtai n th e followin g sequence : 000 01 0 11 0 11 0 01 0 10 0 10 1 111 . Comparing th e decode d sequenc e wit h th e t r a n s m i t t e d sequence , tw o bit s (indicated b y boldface d numbers ) ar e receive d erroneously . Th e bi t erro r rat e is BER = 1/12 , whic h i s equa l t o SER/3 i n thi s exampl e becaus e th e tw o erroneously detecte d symbol s ar e adjacen t t o th e actua l t r a n s m i t t e d symbols .
2.3.2 Quadratur
e amplitud e modulatio n ( Q A M )
Unlike PA M symbols , QA M symbol s ar e comple x numbers . Fo r 26-bi t QAM , a codewor d o f 2b bit s i s m a p p e d t o a symbo l o f th e form 2 s = ±(2f c + l ) A ± j ( 2 Z + l ) A , w
h e r e k,
I G { 0 , 1 , . . ., 2 6 _ 1 - 1} . (2.16 )
2 Unless mentione d otherwise , th e QA M symbol s i n this boo k hav e a square constellation . Hence eac h QA M symbo l carrie s a n eve n numbe r o f bits .
23
2.3. Digita l modulatio n
Figure 2.13(a ) an d (b ) show , respectively , th e signa l constellation s fo r 2-bi t and 4-bi t QA M wit h th e correspondin g Gra y codes . T h e specia l cas e o f 2 bit QA M i s als o know n a s q u a d r a t u r e phas e shif t keyin g ( Q P S K ) . Al l th e four constellatio n point s i n Q P S K hav e th e sam e magnitude . Fro m (2.8 ) an d (2.16), w e se e t h a t th e rea l an d imaginar y part s o f a 26-bi t QA M symbo l ca n be viewe d a s tw o 6-bi t PA M symbols . Usin g thi s relation , man y result s fo r the QA M symbo l ca n b e obtaine d b y modifyin g thos e o f th e PA M symbols . For example , th e signa l powe r o f th e 26-bi t QA M symbo l i s
£s=E[\Sf]=
2
-^(2*b-l).
It i s twic e t h a t o f a 6-bi t PA M symbol .
•
01
-A 00
•
• A
• -A
A •
11
10
•
•
0110
1110
0011
•
• A 0111
1111
1011
0001
• -A 0101
1101
1001
•
.-3A
^3A •
0000
(a) (b
#3A_
0010
• I
^A
A •
•
0100
1100
1010
•
3A •
•
1000
)
Figure 2 . 1 3 . QA M constellatio n an d it s Gra y cod e mapping : (a ) 2-bi t QA M (als o known a s QPSK) ; (b ) 4-bi t QAM .
Suppose t h a t a 26-bi t QA M symbo l wit h powe r S s i s transmitted throug h a zero-mean comple x AWG N channe l wit h nois e varianc e A/o - Suppos e t h a t th e noise i s circularl y symmetri c s o t h a t it s pd f i s a s give n i n (2.5) . T h e n th e rea l part an d th e imaginar y par t ar e b o t h Gaussia n wit h varianc e A/o/2. Therefor e the transmissio n o f a 26-bi t QA M symbo l throug h a comple x AWG N channe l with nois e varianc e A/ o ca n b e viewe d a s th e transmissio n o f tw o 6-bi t PA M symbols, eac h wit h powe r £ s / 2 , throug h tw o rea l AWG N channels , eac h wit h noise varianc e A/o/2 . Fro m earlie r discussions , w e kno w t h a t whe n a 6-bi t PAM symbo l wit h powe r £ s/2 i s transmitte d throug h a rea l AWG N channe l
24
2. Preliminarie s of digital communication s
with nois e varianc e A/o/2, th e SE R i s given b y
5£fip„.„(» = 2(l-l)0( v / ( 2 2 t 3 _^ / 2 ). A QA M symbo l i s correctly decode d whe n bot h th e rea l an d imaginar y part s are correctl y decoded . Th e probabilit y fo r this i s (1 — S E Rparn{b))2. Thu s the SE R o f a 26-bit QA M symbo l i s given b y SERqam(2b) =
2SERpam(b) -
2
SER
pam(b).
When th e erro r rat e i s small, w e ca n ignor e th e second-orde r ter m an d th e SER i s well approximate d b y
SERqam{2b) « 2SERpam(b) = 4(l - ^ ) Q ( J^[^J- (
2 17
- )
Similar t o th e PA M case , if we use Gra y code s to ma p th e rea l an d imaginar y parts, respectively , a s shown i n Fig. 2.13 , the n an y QA M symbo l an d its nearest neighbo r wil l diffe r onl y b y on e bit . I n thi s case , th e BE R o f a 26-bit QAM ca n b e approximate d b y 1 -SER qam(2b). (2.18 26~
BERqam(2b) «
)
Figure 2.1 4 show s th e BE R performanc e o f 26-bit QA M fo r differen t b using Monte Carl o simulatio n an d th e formul a i n (2.18). Agai n w e se e tha t the experimental and theoretical BER curves match almos t perfectly . Th e formul a in (2.18 ) give s a very goo d approximatio n o f th e actua l BER . Comparison o f 6-bit PA M an d 26-bi t QA M Usin g the formulas i n (2.12) , (2.13), (2.17) , an d (2.18) , fo r th e sam e SN R £ 8/Af0 we hav e BERqarn(2b) «
BER
parn(b);
the BER s o f a 26-bi t QA M an d a 6-bi t PA M ar e approximatel y th e same , but th e bi t rat e o f QA M i s twice tha t o f PAM. However , w e should not e tha t the compariso n i s based o n differen t channe l settings . Fo r PAM , th e symbol s are rea l an d th e channe l nois e i s also rea l wit h varianc e A/o . Fo r QAM , th e symbols ar e comple x an d th e channe l nois e is complex wit h th e variance s o f both th e rea l and imaginar y part s equa l to A/o/2. I n other words , in passban d communication i f we choose a QAM ove r a PAM o f the sam e bi t rate , w e will have a gain i n SNR. Fo r example , fo r a BER o f 1 0- 4 , w e se e fro m Fig . 2.1 2 and Fig . 2.1 4 tha t a 2-bit QA M need s a n SN R o f aroun d 11. 7 d B wherea s a 2-bit PA M need s a n SNR o f around 18. 2 dB ; w e hav e a saving of 6.5 d B b y using QAM . For QA M symbol s w e can als o expres s 2 6 in term s o f the SNR , £ s/A/o, as we di d fo r PA M symbol s in (2.14). B y rearrangin g (2.17) , w e obtai n 2b = log2 (l + \-
£
-^) ,
L qam J
(2.19
)
25
2.3. Digita l modulatio n
* — 2-bi t QA M e — 4-bi t QA M 3 — 6-bi t QA M
10 1
52 SNR(dB)
0
Figure 2.14 . BE R performanc e o f QA M i n zero-mea n AWG N channels . Th e soli d curves ar e th e experimenta l value s obtaine d fro m th e Mont e Carl o simulatio n an d th e dotted curve s (almos t indistinguishabl e fro m th e soli d curves ) ar e th e theoretica l value s obtained fro m th e formul a i n (2.18) .
where T qarn i s the SN R ga p give n b y SERa 4(1 -2~
b
)
(2.20)
Again fo r moderat e erro r rates , th e followin g expressio n give s a very accurat e approximation o f th e SN R gap : ^qam ~ ~ \Q~ (S E Rqarn / 4)]
(2.21)
In Tabl e 2.2 , w e list th e value s o f Tqarn fo r som e typica l SER qarn. Althoug h the formul a i n (2.19 ) i s derive d fo r even-bi t QA M symbols , th e right-han d side i s als o use d fo r estimatin g th e numbe r o f bit s tha t ca n b e transmitte d when ther e i s n o even-bi t constrain t [38] . Quadrature phase shift keyin g (QPSK ) modulatio n Whe n a QAM sym bol ha s onl y tw o bits , i t i s als o commonl y know n a s a QPS K symbol . Th e constellation an d a Gra y cod e mappin g fo r QPS K ar e show n i n Fig . 2.13(a) . Suppose a QPS K symbo l wit h signa l powe r £ s i s transmitte d throug h a n AWGN channe l wit h nois e varianc e A/o - Fo r equiprobabl e QPS K symbols , we ca n comput e th e BE R b y computin g th e BE R fo r an y constellatio n poin t because o f th e symmetry . Suppos e tha t "11 " is transmitted . Le t q r an d qi
26 2
. Preliminarie s o f digita l communication s 3 -t^ -t^qam 2
ioicr 3 io- 4 io~5 io~6 io- 7
-1- qam
2.63
4.04 5.48
J- qam m
4.19
6.06 7.39
6.95
8.42
9.91
9.96
8.42
9.25
Table 2.2 . Th e SNR gap Tqarn i n (2.21)
be, respectively , th e rea l an d imaginar y part s o f nois e q. The n th e firs t bi t i s in erro r whe n q r < — A an d th e secon d bi t i s i n erro r whe n qi < —A . Thu s the BE R o f QPS K i s given b y BERqpsk=
0.5P(q r < - A ) + 0.5P(f t < - A ) = Q
The abov e BER formula fo r QPS K i s exact an d i t i s identical to that o f BPSK . Other modulatio n scheme s PA M an d QA M ar e th e mos t commonl y used modulatio n scheme s du e t o thei r simplicity . Ther e ar e man y othe r mod ulation schemes , for exampl e phas e shif t keyin g (PSK) , frequenc y shif t keyin g (FSK), differentia l phas e shif t keyin g (DPSK) , minimu m shif t keyin g (MSK) , and s o forth. I n practice , w e may choos e on e modulatio n schem e ove r others , depending o n th e application . Fo r example , i n som e application s i t migh t be desirabl e t o hav e modulatio n symbol s wit h a constan t magnitude , tha t is \sk\ = £ s for al l k. I n thi s case , w e ca n us e PS K modulatio n (show n i n Fig. 2.15) . I n a PS K modulatio n scheme , al l th e constellatio n point s ar e uniformly distribute d o n a circl e an d th e radiu s o f th e circl e determine s th e symbol power . Fo r a mor e detaile d an d complet e coverag e o f variou s digita l modulations, th e reader s ar e referre d t o [120 ] an d [137] . Example 2. 3 Suppos e w e wan t t o sen d th e 24-bi t sequenc e i n Exampl e 2. 2 using 4-bi t QA M symbols . Le t th e constellatio n an d Gra y cod e mappin g b e as show n i n Fig . 2.13(b) . A s eac h symbo l carrie s fou r bits , w e grou p th e 2 4 bits int o codeword s o f fou r bits : 0000 101 1 111 0 010 1 0010 1110 . From Fig . 2.13(b) , w e fin d th e correspondin g si x QA M symbol s s(n). Thes e QAM symbol s ar e sen t ove r a n AWG N channel . Suppos e th e receive d signal s r(n) ar e a s give n i n Tabl e 2.3 . Th e erro r probabilitie s i n thi s exampl e ar e fo r the sak e o f demonstration . Th e actua l error s ar e usuall y muc h smaller , suc h as 1 0 - 2 , 1 0 - 4 , etc . Assum e tha t a t th e receive r ther e i s n o additiona l signa l
2.3. Digita l modulatio n
27
(a)
(b)
Figure 2 . 1 5 . Phas e shif t keyin g modulation : (a ) 8-PSK ; (b ) 16-PSK .
n
s{n)
0 -3A - 3Aj 3A + Aj 1 2 A + 3Aj -A - Aj 3 4 -3A + 3Aj A + 3Aj 5
r(n)
-4.lA-2.6Aj 3.7A + 2.1AJ 0.9A + 2.8Aj -1.1A-Aj -4A + 1.7AJ 2.1A + 1.1AJ
s(n)
-3A - 3Aj 3A + 3Aj A + 3Aj -A - Aj -3A + Aj 3A + Aj
Table 2.3 . Transmitte d QA M symbol s s(n), receive d signal s r(n), an d detecte d sym bols 7i{n).
processing an d th e NND R i s applie d directl y t o r(n). Afte r symbo l detectio n we ge t s"(n) . Comparin g s(n ) wit h s(n) , w e find tha t w e hav e mad e thre e symbol error s (s"(l) , s"(4) , s"(5) ) ou t o f si x transmitte d symbols . Th e symbo l error rat e i s SER = 1 / 2 . Afte r symbol-to-bit s mappin g usin g th e Gra y cod e provided i n Fig . 2.13(b) , w e obtain th e followin g sequence : 0000 101 0 111 0 0101 0011 1011. Comparing th e decode d sequenc e wit h th e transmitte d sequence , fou r bit s (indicated b y boldface d numbers ) ar e receive d erroneously . Th e bi t erro r rat e is BER = 4/2 4 = 1/6 , whic h i s large r tha n SER/A = 1/8 . Thi s i s becaus e s(5) i s not adjacen t t o s(5) , whic h cause s a n erro r o f two bit s rathe r tha n on e
bit. ■
2. Preliminarie s o f digita l communication s
28
+s o
so
+s i
> -s ,
equalizer
channel
Figure 2 . 1 6 . A se t of M paralle l channel s and the corresponding zero-forcing equalizer .
2.4 Paralle
l subchannel s
In man y wideban d communicatio n systems , a wideban d channe l i s divide d into a se t o f subchannels , eac h wit h a smalle r bandwidth . Example s includ e the widel y use d O F D M an d D M T systems , whic h wil l b e studie d i n detai l in Chapte r 6 . I n thes e systems , a n F I R channe l i s converte d t o a se t o f parallel ISI-fre e channel s a s show n i n Fig . 2.16 . Th e receive d signa l o f th e i t h subchannel i s where Si i s th e symbo l t r a n s m i t t e d ove r th e i t h subchannel . Th e quantitie s di an d qi are , respectively , th e i t h subchanne l gai n an d noise . Becaus e eac h subchannel ha s onl y a singl e t a p , zero-forcin g equalizatio n ca n b e don e b y using simpl e scala r multiplier s 1/a ^ (i f ai ^ 0 ) a s indicate d i n th e figure . Le t s an d s " be, respectively , th e inpu t an d outpu t vectors . Defin e th e outpu t erro r vector a s e = s — s. T h e n w e ca n redra w th e paralle l channel s a s i n Fig . 2.17 . I t i s clea r t h a t th e output erro r o f th e i t h subchanne l i s e ^ = qi/ai. Th e erro r varianc e fo r th e i t h subchanne l i s
2_ M
where Mi i s th e varianc e o f q^. Fo r man y applications , th e subchannel s hav e the sam e nois e variance s Mi = A/o , bu t th e subchanne l gain s ca n b e ver y different. Thu s th e erro r variance s a^ ca n b e ver y differen t fo r differen t sub channels. Signal s t r a n s m i t t e d ove r differen t subchannel s encounte r differen t levels o f distortion . Fo r subchannel s wit h larg e erro r variances , th e bi t erro r rate wil l b e hig h an d th e overal l performanc e o f th e paralle l channel s wil l be limite d b y thes e bad subchannels . T o se e thi s effect , le t u s conside r th e following exampl e wit h onl y tw o subchannels .
29
2.4. Paralle l subchannel s
so
S\
£.
-► * o
->A l
S
M-\
Figure 2 . 1 7 . Equivalen t paralle l channel s o f Fig . 2.16 .
Example 2. 4 Suppos e t h a t ther e ar e onl y tw o subchannel s an d th e gain s ar e ao = 1 and a\ = 0.1 , respectively. T h e transmissio n powe r i s fixed a t 10 . T h e subchannel noise s qi ar e AWG N wit h varianc e J\fi = 1 . A quic k calculatio n shows t h a t th e outpu t erro r variance s ar e
Suppose t h a t th e maximu m signa l powe r allowe d o n eac h subchanne l i s £ m a x = 10. Le t th e transmitte d signal s Si b e B P S K symbol s wit h si = ± \ / l 0 s o t h a t the powe r i s Ei = 10 . Fo r B P S K modulation , B E R i s equa l t o SER . Fro m (2.12), th e B E R o f th e zerot h subchanne l i s Q ( \ / l 0 ) = 7.8 3 x 10~ 4 , wherea s the B E R o f th e first subchanne l i s Q ( \ / o T ) = 0.38 . T h e averag e B E R o f th e parallel channel s i s approximatel y 0.5*Q(VoT) = 0.19 . T h e performanc e o f th e syste m i s severel y limite d b y th e first subchannel . Instead o f transmittin g B P S K symbol s o n b o t h th e subchannels , suppos e t h a t we no w transmi t a 2-bi t PA M symbo l o n th e zerot h subchanne l an d th e first subchannel i s no t utilize d fo r transmission ; th e zerot h an d first subchannel s are allocate d tw o bit s an d zer o bits , respectively , s o t h a t th e tota l numbe r o f bits transmitte d i s stil l two . Usin g th e B E R formul a fo r 2-bi t PAM , w e find t h a t BER « 3/ 4 * Q{y/lQ/5)= 0.059 , whic h i s muc h smalle r t h a n 0.19 . ■ From th e abov e example , w e se e t h a t b y properl y loadin g th e bit s amon g the subchannels , w e ca n significantl y improv e th e B E R performanc e fo r th e same transmissio n rate . I n wha t follows , w e wil l sho w ho w t o achiev e this . Bit loadin g W h e n th e power s o n th e subchannel s ar e fixed a t £ m a x an d the subchanne l erro r variance s o 2e. ar e differen t fo r differen t z , the bit s assigne d to th e subchannel s bi can b e adjuste d t o improv e th e erro r rate . Thi s i s calle d bit loadin g (als o know n a s bi t allocation) . Below , w e wil l first conside r bi t loading fo r th e PA M cas e unde r th e pea k powe r constraint . Suppos e t h a t the erro r rate s o f al l th e subchannel s ar e th e sam e an d thei r SER s ar e equa l
2. Preliminarie s o f digital communication s
30
to SERQ. Fo r P A M symbols, w e kno w t h a t t h e numbe r o f bit s i s relate d t o the SN R by (2.14) . Therefor e t h e numbe r o f bit s t h a t ca n be t r a n s m i t t e d o n the i t h subchanne l i s
fc = i l o g 2 ( l +
% ^Y (2.22
)
where o\. i s t h e nois e powe r o f t h e i t h subchanne l output . T h e averag e bi t rate i s give n b y M-l
M^ T h e bit s compute d i n (2.22 ) ar e not intege r i n general . W h e n t h e constrain t of intege r bi t i s applied , t h e averag e bi t rat e i s b = ( 1 / M ) ^2i=^ \pi\, w n e r e [x\ denote s t h e larges t intege r les s t h a n o r equa l t o x. For Q A M symbols, suppos e t h e i th subchannel carrie s 2b i bits . Then , fro m (2.19), w e get t he number o f bits t h a t ca n be sent throug h t h e i th subchanne l as 2bi = log 2 1
\-
+—
%
L qam J
-.
T h e averag e bi t rat e i s give n b y M-l i=0
For t h e cas e o f intege r bi t loading , t h e averag e bi t rat e i s 1 / M ^ - = Q 2 [ ^ J . Example 2. 5 Conside r t h e paralle l channel s i n Exampl e 2.4 . Suppos e t h a t PAM symbol s ar e sent , t h e desire d S E R is SERQ= 1 0 - 7 , and t he m a x i m u m transmission powe r allowe d i s Smax = 1000 . Using Tabl e 2.1 , we have T pam = 9.46. T h e subchanne l SNR s ar e S W ? o = 1000 , SNRi =
10.
T h e maximu m achievabl e bit s fo r t h e two subchannels ar e b0 = 3.37 , 6
i = 0.52 .
T h u s t h e overal l maximu m achievabl e bi t rat e i s b = 1.95 . I f a n intege r bit allocatio n i s desired , thi s valu e become s b = 1.5 . Not e t h a t althoug h b\ = 0.52 , we roun d i t dow n t o zer o s o t h a t t h e desire d quality-of-servic e o f SER0= 1 0 - 7 is not violated . ■ In t h e abov e discussions , w e assum e t h a t t h e pea k signa l powe r o f eac h subchannel i s limite d b y S max. I n som e applications , w e ma y b e concerne d about t h e averag e signa l powe r rathe r t h a n t h e peak signa l power . T h e prob lem o f bit loadin g fo r thi s cas e wil l b e studie d i n Chapte r 8 .
2.5. Furthe r readin g
2.5 Furthe
31
r readin g
Some basi c concept s fo r digita l communicatio n system s wer e briefl y reviewe d in thi s chapter . Ther e ar e man y textbook s t h a t provid e a mor e detaile d an d comprehensive t r e a t m e n t o f thes e topics . T h e intereste d reader s ar e referre d to [50 , 67 , 120] , t o nam e jus t a few .
2.6 Problem
s
2.1 Suppos e w e hav e a communicatio n syste m wit h th e wavefor m (jpi * ca * p2)(t) give n b y ( \u\ f ( P i * ^ **»)(' ) =
-0.5|t-2 |+ l , for0<£<4 { ( , , otherwise ?
;
Find th e discrete-tim e equivalen t channe l c(n) whe n th e samplin g perio d is T = 1 . W h a t woul d c(n) b e i f w e increas e th e samplin g perio d t o T = 21 2.2 Multipath channels. have th e for m
I
n man y applications , th e transmissio n channel s L
k=l
where S a(t) i s a continuous-tim e impuls e (whic h i s als o know n a s a Dirac function) . Thes e channel s ar e know n a s multipat h channels . T h e parameters a^ an d r ^ are , respectively , calle d th e gai n an d dela y o f th e fcth path . T h e expressio n abov e show s t h a t c a(t) ha s L paths . Suppos e t h a t w e hav e a two-pat h channe l wit h a\ = 1 , a 2 = —0.5 , T\ = 0 , an d r2 = 3 . Assum e t h a t th e convolutio n o f th e transmittin g an d receivin g pulses yield s
{
£, i
f0 < t < 1 ;
2- t i f 1
.
Suppose t h a t T = 1 an d th e transmitte d symbol s ar e x(n) = (—l) n fo r 0 < n < 5 and zer o otherwise . I n th e absenc e o f noise : (a ) plo t x a(t) an d wa(t)\ (b ) find r(n) = w a(nT); (c ) verif y th e relatio n r(n) = ( c * x)(n), where c(n) i s th e discrete-tim e equivalen t channel .
2. Preliminarie s o f digital communication s 5 Repea t Proble m 2. 4 for T = 0.5 . 6 Le t t he continuous-time channe l nois e q a(t) b e a white rea l W S S process with powe r spectra l densit y (psd ) S qa(jfi)= Af a fo r al l ft. (a) Suppos e t h a t t h e receiving puls e p2 (t) i s an ideal lowpas s filte r wit h the passban d regio n [—0.5/T , 0.5/T] , wher e T i s t h e underlyin g symbol spacing . I s t he equivalen t discrete-tim e channe l nois e q(n) also a whit e W S S process? W h a t i s t he nois e power ? (b) No w let t h e passban d o f P2(t) b e [—1/T , 1/T] . Repea t par t (a) . 7 Conside r a syste m wher e t h e 2-bi t P A M symbol s s(n) ar e transmit ted ove r a hypothetica l noiseles s channe l wit h impuls e respons e c(n) = aS{n), wher e 0 < a < 1 . T h e received signa l r(n) = as(n) i s an attenu ated versio n o f t he t r a n s m i t t e d symbo l s(n). Suppos e t h e receive r doe s not kno w abou t t h e attenuation facto r a an d it make s it s decisions s~(n) using t h e N N D R base d o n t h e origina l constellatio n o f s(n). C o m p u t e the symbo l erro r rat e (SER) . Expres s you r answe r i n term s o f a. 8 Conside r a set of parallel channel s wit h fou r subchannel s a s in Fig. 2.16. Let t h e noise b e zero-mean AWG N wit h varianc e J\fk = 1 0 and t he gain s dk b e give n b y t he followin g table : A; 0
1
2
ak 5
1
1
3 0 \/4 0
Suppose t h a t t h e zero-forcin g equalize r i s employed . (a) W h e n B P S K symbol s wit h signa l powe r Si = 1 0 are transmitted , compute t h e subchanne l SNR s an d t he averag e B E R. (b) Suppos e t h a t w e do no t sen d an y bit s ove r t h e firs t tw o subchan nels an d transmi t 2-bi t P A M over t h e las t tw o subchannels . T h e maximum signa l powe r allowe d o n eac h subchanne l i s S max = 10 . Compute t h e averag e B E R assumin g t h a t t h e Gra y cod e i s use d for t h e mapping . 9 Suppos e P A M symbols ar e t r a n s m i t t ed ove r t h e set o f parallel channel s in Proble m 2.8 . Le t t h e desire d S E R be SERQ= 1 0 - 6 , and t h e maxi m u m transmissio n powe r b e S max = 1000 . Compute t h e number o f bit s bk (integer ) t h a t ca n be sen t throug h t h e kth subchanne l fo r 0 < k < 3. W h a t i s t he averag e bi t rat e b?
3 FIR equalizer s In wideban d communicatio n systems , th e transmissio n channel s ar e usuall y frequency-selective an d thu s wil l introduc e som e degre e o f intersymbo l inter ference (ISI ) i n additio n t o th e noise . A t th e receiver , som e signa l processin g is often carrie d ou t t o alleviat e the effec t o f these distortions . Suc h processin g is i n genera l know n a s channe l equalization . Figur e 3. 1 show s th e bloc k diagram o f a transmissio n syste m wit h a linea r equalize r a(n). I n man y ap plications, th e channe l ca n b e modele d a s a n FI R LT I filte r b y makin g th e order L c sufficientl y large : C(z) =
Y,c(n)2 n=0
The channe l nois e q(n) i s a zero-mea n wid e sens e stationar y (WSS ) process . In thi s chapter , w e shal l conside r FI R equalization . Th e equalize r A(z) ha s transfer functio n A{z) = J2a{n)2 n=0
noise q(n) x(n)
y(n)
channel c(n)
equalizer a(n)
x(n)
Figure 3 . 1 . Transmissio n syste m wit h a n LT I equalize r a(n)
Letting x(n) b e th e transmitte d signal , th e equalize r outpu t i s give n b y x(n) =
(a * c * x)(n) + ( a * q)(n), (3-1
)
where "* " denote s convolution. Th e goa l of channel equalization i s to produc e an outpu t signa l x(n) tha t i s "close " t o th e transmitte d signa l x(n) i n som e sense. Som e equalizatio n technique s wil l b e introduce d below . 33
34
3. FI R equalizer s
The zero-mea n assumptio n Unles s mentione d otherwise , i n thi s boo k we assum e t h a t al l th e rando m variable s hav e zero-mean . On e consequenc e of thi s i s t h a t th e mea n square d valu e i s equa l t o th e variance . T h a t is ,
E[\v\2] = vl for th e zero-mea n rando m variabl e v. Th e mea n square d valu e E[\ • | 2 ] an d the varianc e a 2 ar e use d interchangeabl y i n th e book .
3-1 Zero-forcin
g equalizer s
Consider Fig . 3.1 . Th e transfe r functio n fro m th e t r a n s m i t t e d signa l x(n) t o the equalize r outpu t x(n) i s give n b y T{z) =
A{z)C{z). z~ n°. Th
T h e equalize r A(z) i s zero-forcin g i f T(z) = forcing equalize r ca n b e expresse d a s x(n) =
e outpu t o f a zero -
x(n - n 0) + qo(n),
where qo(n) = (q * a)(n) i s th e equalize r outpu t noise . Th e intege r n o i s th e system delay . A s th e channe l nois e q(n) doe s no t affec t th e desig n o f a zero forcing equalize r (thoug h i t affect s th e syste m performance) , w e wil l assum e q{n) = 0 i n thi s section . For a n F I R channe l C(z) an d a n F I R equalize r A(z), thei r produc t A(z)C(z) is als o FIR . Th e transfe r functio n T(z) ca n b e a dela y i f an d onl y i f b o t h C(z) and A(z) ar e delays . W h e n th e channe l i s frequency-selective , C(z) ha s mor e t h a n on e t a p . Therefor e ther e doe s no t exis t a n F I R zero-forcin g equalize r fo r frequency-selective channels . A n alternativ e solutio n woul d b e t o fin d A(z) so t h a t A(z)C(z) i s "close " t o a dela y z~ n°. T o d o this , le t u s defin e d(n) =
(a * c)(n) — 5{n — no).
W h e n d(n) = 0 , th e equalize r A(z) i s zero-forcing . However , thi s i s no t possible unles s th e channe l c(n) ha s onl y on e nonzer o t a p . S o w e desig n th e equalizer a{n) t o minimiz e Y,n\d(n)\2' Thi s proble m ca n b e formulate d using ; matrix notatio n a s follows . Le t th e vector s t an d a b e give n b y
r*(o)i t(i)
" «(o)" and a =
t(L)_
«(1)
a{La)_
where L = L c + L a i s th e orde r o f T(z). The n the y ar e relate d b y t — C/oxt; a ?
(3.3)
35
3 . 1 . Zero-forcin g equalizer s
where th e ( L + 1 ) x (L a + 1 ) matri x Ci ow i s a lowe r triangula r Toeplit z matri x given b y
c(0) c(l)
0 c(0)
0
c(0)
C(ic)
0
Qo
(3.4)
0
c(L 0
0
0
c(0) c(l)
=(ic)
C(ic)
T h e subscrip t "low" serve s a s a reminde r t h a t th e matri x i s a "lowe r trian gular" matrix . Defin e a n ( L + 1 ) x 1 vecto r n0 1
(3.5)
which ha s onl y on e nonzer o entr y i n th e not h location . T h e n w e ca n writ e £d(no)
Z^
\d(n)\2 =
\\t
(3.6)
where | | • | | denote s th e Euclidea n nor m o f th e vector . T h e quantit y £d(no) is a measur e o f th e closenes s o f th e transfe r functio n T(z) t o th e dela y z~ n°. Thus th e proble m o f finding th e optima l a(n) become s a least-square s proble m of finding a t o minimiz e | | C / o w a — ln o | | 2 - T h e produc t C / o w a i s a linea r combination o f th e column s o f C/ ( We woul d lik e t o find th e vecto r i n the colum n space 1 o f Ci ow t h a t i s th e closes t t o l n o . Fro m linea r algebr a theory, w e kno w t h a t th e closes t vecto r i s th e orthogona l projectio n o f l n o onto th e colum n spac e o f Ci ow. A s th e matri x Qi ow ha s ful l colum n ran k unless C(z) = 0 (Proble m 3.1) , th e least-square s solutio n i s uniqu e an d i t i s given b y [52 ] d-ls — {Clow^low) C
/ow
lno,
(3.7)
where t denote s transpos e conjugation . T h e correspondin g equalize r ai s(n)= [aj s ] n wil l b e calle d th e least-square s equalizer . T h e invers e ( C | o w C / o w ) _ 1 always exist s a s Ci ow ha s ful l colum n rank . Substitutin g th e expressio n i n (3.7) int o (3.6) , w e obtai n £d,is(no) =
| | B / s l n o — ln o | | ,
(3.8)
where B / s i s th e positiv e semidefinit e matri x give n b y Ciow(Clow-tCiow) C
int
The colum n spac e o f a matri x i s th e subspac e spanne d b y it s colum n vectors .
(3.9)
3. FI R equalizer s
36 Expanding th e right-han d sid e o f (3.8 ) w e ge t £d,u(no)=
lJHBj 8B,8lno - 4
0
Bisl„0- 4
Using th e fact s t h a t BJ" S = B / s an d B j s B / s = expression t o £d,u(no) =
1 " [
B
0
Bj8lno+ 1 .
B / s , w e ca n simplif y th e abov e
^ „ „ , „ „ , (3-10
)
where [B/ s ] n o i s th e not h diagona l entr y o f B / s . On e ca n choos e th e sys t e m dela y n o t o minimiz e £d,is(no)- I t follow s fro m (3.10 ) t h a t th e smalles t £d,is{n$) i s achieve d whe n n o i s chose n suc h t h a t th e not h diagona l entr y o f B / s i s the largest . W h e n th e syste m dela y n o i s chose n optimally , th e equalize r is calle d a delay-optimized least-square s equalizer . I t i s lef t a s a n exercis e t o show t h a t al l th e diagona l entrie s o f th e matri x B / s satisf y 0 < [ B ^ ] / ^ < 1 . T h e maximu m o f [B/ S]fcfc ca n b e 0 o r 1 only fo r som e ver y specia l case s (Prob lems 3. 4 an d 3.5) . W e summariz e th e desig n procedur e a s follows : • comput e th e matri x A =
(CJ^C^)
- 1
^^;
• find n o s o t h a t th e not h diagona l entr y o f th e matri x B / s = Ci the largest .
owA
i
s
From (3.7) , th e delay-optimize d least-square s equalize r i s give n b y th e not h column o f A . Example 3. 1 Le t u s conside r th e followin g tw o F I R channels : C0(z)=l + d(z)=
2z-
1
l + 0.95z
, _1
.
For th e channe l Co(z), a causa l stabl e zero-forcin g equalize r a zf(n) doe s no t exist becaus e l/Co(z) ha s a pol e a t z = —2 , which i s outsid e th e uni t circle . On th e othe r hand , th e invers e \jC\(z) i s causa l an d stable . No w w e conside r F I R least-square s equalizer s wit h differen t orders . Th e syste m dela y n o i s chosen optimally . Figur e 3. 2 show s th e plo t o f £^i s versu s L a, an d Tabl e 3. 1 shows th e smalles t L a neede d t o achiev e th e liste d targe t value s o f £^i s for Co(z) an d Ci(z), respectively . Observ e t h a t fo r th e channe l Co(z), wit h a relatively smal l orde r L ai w e ar e abl e t o mak e th e overal l impuls e respons e very clos e t o a n impuls e 8{n — no) . O n th e othe r hand , i t i s no t a s eas y to equaliz e th e channe l C\(z). T o achiev e th e sam e targe t £d,isi w e nee d a much longe r equalize r fo r C\{z). Fo r example , whe n th e targe t £^i s = 10 -3, the smalles t equalize r orde r neede d i s L a = 4 fo r Co(z) an d L a = 4 4 fo r C\(z). Thi s ca n b e explaine d a s follows . T o ge t a smal l £&, th e equalize r a(n) shoul d approximat e th e impuls e respons e o f z~ n° jC\(z\ whic h ha s th e form (—0.95) n _ n °, a quantit y decayin g slowl y wit h n . Thu s w e nee d a lon g equalizer a{n). O n th e othe r hand , fo r th e channe l Co(z), i f w e choos e th e equalizer a s A(z) =
0.5z~ La ( 1 - 0.5z + 0 . 5 V +
(-0.5)
La La
z ),
then th e transfe r functio n wil l b e T(z) =
A(z)C 0(z)=
0 . 5 ( - 0 . 5 ) L a + z~
La
~\
37
3 . 1 . Zero-forcin g equalizer s
If th e syste m dela y i s chose n a s n o = L a + 1 , the n th e quantit y £d(no) i s equa l to 0 . 5 2 L a + 2 , whic h decay s a t a muc h faste r rate . Thi s explain s wh y w e ar e able t o achiev e th e targe t value s o f £^i s liste d i n Tabl e 3. 1 wit h a relativel y small L a. Not e t h a t th e fac t t h a t 1/CQ(Z) i s unstabl e doe s no t m a t t e r whe n we desig n th e F I R least-square s equalizers . ■
■C0(z)
C^z)
x x x x x x x x x x x x x x x x x 51 01 52 0 L
Figure 3.2 . Se e Exampl e 3.1 . A plo t o f £ d,i
Target £ dj8 0.1
La forQ (*) 1
La for CiO ) 6
0.05
1
10
0.01
3
23
0.005
3
29
0.001
4
44
Table 3 . 1 . Se e Exampl e 3.1 . Smalles t equalize r orde r L a neede d fo r Co(z) an d Ci(z) to achiev e th e targe t Sd,is-
Effect o f channe l nois e T h e channe l nois e ha s bee n ignore d i n th e abov e analysis. Suppos e no w ther e i s channe l nois e q(n). T h e nois y outpu t x(n) i s
3. FI R equalizer s
38 given b y (3.1) . Defin e th e outpu t erro r e(n) the outpu t erro r i s give n b y e(n) =
x(n) — x(n — no). I n thi s case ,
(a* c* x)(n) — x(n — no) + (a * q)(n) .
(3.11)
eq(n)
G-sig yTl)
W h e n th e convolutio n ( a * c)(n) i s no t a n impuls e 8{n — no), th e quantit y £sig{n) i s no t zero . Th e outpu t erro r consist s o f tw o terms : th e signal dependent t e r m e sig(n) an d th e noise-dependen t t e r m e q(n). A least-square s equalizer ai s(n) minimize s onl y th e powe r o f e sig(n). Th e filtered nois e (ais * q)(n) ma y hav e a larg e power , a s demonstrate d i n Exampl e 3.2 . Example 3. 2 I n thi s example , th e channe l i s C\(z) = 1 + 0.952: - 1 , wit h a n AWGN q(n) o f variance A/o = 0.3 . Th e signa l x{n) i s assumed t o b e zero-mea n WSS, uncorrelate d wit h th e noise , an d it s spectru m i s assume d whit e wit h variance £ x = 1 . Thu s th e SN R i s 10/ 3 (5.2 3 dB) . Th e equalize r i s designe d using th e delay-optimize d least-square s approach . I n Fig . 3.3 , w e plo t th e following thre e erro r variance s versu s th e equalize r orde r L a:
E[\e{nf
E[\eslg(n)\\ at
=
E[\e
2
q(n)\
}.
Figure 3.3 . Se e Exampl e 3.2 . Th e variance s o f e(n) , e; s ;(n), an d e q(n) define d i n (3.11) versu s th e equalize r order s L a.
From th e plot , w e se e t h a t th e varianc e o f th e signal-dependen t t e r m e sig(n) decreases monotonicall y wit h respec t t o L a, bu t th e varianc e o f th e noise dependent t e r m e q(n) increase s a s L a increases . A s a result , whe n L a i s
39
3.2. Orthogonalit y principl e an d linea r estimatio n
large the outpu t erro r varianc e o 2e is dominated b y the nois e power o 2e , which increases a s L a increases . Increasin g th e lengt h o f the least-square s equalize r leads t o a large r outpu t error ! The nois e amplificatio n ca n als o b e understoo d fro m a frequenc y domai n viewpoint. Whe n L a increases , th e approximatio n ci(n ) * ai s(n) ~ S(n — no) become s mor e accurate . I n th e frequenc y domain , Ai s(e^UJ) approache s l/Ci(ejuJ) whe n L a i s large . A s Ci(e juJ) i s a lowpas s filter wit h Ci(e j7r) = 0.05, th e equalize r Ai s{e^) i s a highpas s filter wit h Ai s(e^) ~ 20 ; the high frequency componen t o f th e channe l nois e q(n) i s severel y amplifie d b y th e least-squares equalizer . ■ As w e ca n se e fro m th e abov e example , thoug h a least-square s equalize r ais(n) ca n effectivel y reduc e th e signal-dependen t ter m e sig(n), it s abilit y t o combat th e channe l nois e i s limited . A bette r desig n o f th e equalize r shoul d take bot h e sig(n) an d e q(n) int o consideration . B y designin g a n equalize r that minimize s th e outpu t erro r varianc e ^[|e(n)| 2 ], w e ar e abl e t o minimiz e the combine d effec t o f th e signal-dependen t ter m an d th e nois e term . Befor e we proceed t o stud y suc h equalizers , w e introduce i n Sectio n 3. 2 a usefu l too l called th e orthogonalit y principle . Thi s principl e play s a n importan t rol e i n many area s o f signa l processin g an d communications .
3.2 Orthogonalit
y principl e an d linea r estimatio n
In digital communications, w e frequently fac e the problem of estimating a random variabl e x fro m som e noisy observation s yo , 2/i ? • • • ? VK-I- Th e estimat e x i s ofte n obtaine d b y takin g a linea r combinatio n o f th e observe d samples ; that is , x = a 0y0 + aiy i H h a K-\VK-\. This i s known a s a linear estimator . W e would lik e to find ai s o that x i s close to x i n som e sense . Defin e th e estimatio n erro r e a s e = x — x. One commonl y use d criterio n i s th e mea n square d erro r (MSE ) ^[|e| 2 ]. Th e optimal estimat e x tha t minimize s th e MS E i s known a s the minimu m mea n squared erro r (MMSE ) solution . A powerfu l too l fo r finding th e MMS E solution i s the orthogonalit y principle , outline d i n Theore m 3.1 . Theorem 3. 1 Th e linea r estimat e x minimize s th e MS E ^[|e| 2 ] i f an d onl y if E[ey?]=0, fo
r i = 0 , 1 , . . . , K - 1 ; (3.12
)
that is , i f an d onl y i f th e estimatio n erro r e i s orthogona l t o ever y observe d sample yi. ■ Proof. Suppose x± i s the estimat e tha t satisfie s (3.12 ) an d e± = x± — x. Let x b e anothe r linea r estimat e o f x. Le t u s rewrit e th e MS E fo r x a s E[\x - x\ 2} = E[\x± -x + x- x
2
±\
}.
3. FI R equalizer s
40
Expanding th e right-han d sid e o f th e abov e equation , w e hav e E[\e±\2} +
E[\x - x ±\2} + E[e ±(x -
x ±)*} + E[e* ±(x - x
±)].
Note t h a t x — x± i s a linea r combinatio n o f yi. Becaus e e± i s orthogona l to ever y yi , i t i s orthogona l t o x — x±. Thu s w e hav e 2 E[\x-x\= }
E[\e ±\2} +
E[\x - x ±\2} >
E[\e
2
±\
}.
T h e inequalit y become s a n equalit y i f an d onl y if x = x±. ■
T h e orthogonalit y principl e finds man y applications . Fo r example , i t ha s been successfull y applie d t o th e topi c o f linea r predictio n theor y [164] . W e will sho w ho w t o appl y th e principl e t o th e desig n o f MMS E equalizer s i n th e next section .
Figure 3.4 . Geometrica l interpretatio n o f th e MMS E estimate .
T h e orthogonalit y principl e als o ha s a nic e geometrica l interpretation . Note t h a t th e estimat e x i s a linea r combinatio n o f yi an d thu s belong s t o th e subspace spanne d b y yi. W e wis h t o find th e elemen t i n thi s subspac e t h a t is "closest " t o x. Fro m th e theor y o f linea r algebra , thi s closes t elemen t x± is th e orthogona l projectio n o f x ont o th e subspac e spanne d b y yi. Thi s i s precisely wha t i s state d i n th e orthogonalit y principle : th e MMS E estimat e x± i s suc h t h a t th e erro r e± i s orthogona l t o ever y yi an d henc e al l o f thei r linear combinations . Therefor e th e thre e quantitie s x, x±, an d e± for m a right triangl e wit h x, x±, an d e± bein g th e hypotenuse , th e base , an d th e height, respectively . Figur e 3. 4 illustrate s thi s geometrica l relationship . I n particular, i t implie s t h a t E[x±e*±] = 0; t h a t is , th e MMS E estimat e an d it s estimatio n erro r ar e orthogonal . Fro m the P y t h a g o r e a n theorem , w e kno w t h a t the y satisf y
E[\x\2] =
E[\e
2
±\
]+E[\x±\%
a relatio n t h a t ca n als o b e verifie d directl y b y usin g th e orthogonalit y princi ple. I t follow s fro m th e abov e equatio n t h a t th e powe r o f th e MMS E estimat e x± i s alway s les s t h a n t h a t o f x, unles s th e estimatio n erro r i s zero .
3.2. Orthogonalit y principl e an d linea r estimatio n
3.2.1 Biase
41
d an d unbiase d linea r estimate s
A linea r estimat e x o f x i s sai d t o b e a n unbiase d estimat e i f
otherwise i t i s sai d t o b e biased . I n man y digita l communicatio n systems , the linea r estimat e x i s usuall y th e outpu t o f a n equalizer . I t i s ofte n possibl e to writ e th e estimat e i n th e for m x = ax + T , (3.13
)
where r i s a rando m variabl e relate d t o th e channe l nois e an d interferin g symbols. Usuall y w e ca n mak e th e followin g assumptions 2 abou t r : (i) r ha s zero-mean , E[T]
=
0;
(ii) r i s statisticall y independen t o f x. From (3.13) , w e ca n writ e ^ [ x | x ] = ax +
^[T|X] .
Using th e abov e tw o assumptions , ^ [ r | x ] = E[r] = 0 an d w e hav e ^ [ x | x ] = ax. Therefore w e conclud e t h a t th e estimat e x expresse d i n th e for m i n (3.13 ) i s biased i f a ^ 1 . Not e t h a t th e fac t t h a t r i s statisticall y independen t o f x implies t h a t x an d r ar e uncorrelated , bu t th e convers e ma y no t b e true . As w e wil l se e i n Sectio n 3.3 , th e outpu t o f a n MMS E equalize r i s a bi ased estimat e o f th e desire d signal . I f th e bia s i s no t remove d befor e applyin g symbol detectio n a t th e receiver , th e bi t erro r rat e (BER ) wil l increas e (Sec tion 3.4) . T h e bia s ca n b e remove d easil y b y dividin g x b y a; th e resul t wil l be a n unbiase d estimat e X
T
—= £ + a a
(3.14)
Biased an d unbiase d SNR s Le t x b e a n estimat e o f x an d assum e t h a t they ar e relate d b y (3.13) . Defin e th e erro r a s e = x — x. W e sa y th e SNR , 2
^biased = - f , (3.15
)
is a biase d SN R i f x i s biased . W h e n th e bia s i s remove d a s i n (3.14) , w e T h e unbiase d SN R i s give n b y
=
J ^ . (3.16
)
2 In mos t practica l communicatio n systems , thes e assumption s ar e ofte n vali d becaus e (i) th e channe l nois e i s usuall y zero-mea n an d independen t o f th e desire d signa l x, an d (ii ) the transmitte d symbol s ar e usuall y ii d wit h zer o mean , whic h implie s tha t th e interferin g symbols ar e als o independen t o f x an d o f zero-mean .
42
3. FI R equalizer s
Note t h a t w e hav e use d variance s i n th e abov e definition s o f th e biase d an d unbiased SNRs . Becaus e al l rando m variable s ar e assume d t o hav e zero-mean , their variance s ar e th e sam e a s thei r mea n square d errors . As w e wil l se e below , th e MMS E estimat e x± i s alway s biase d unles s it s estimation erro r e± = 0 , whic h i s impossibl e whe n th e observe d sample s yi contain noise . Ther e i s a simpl e relatio n betwee n th e biase d an d unbiase d SNRs o f a n MMS E estimate , a s state d i n th e followin g lemm a [28] . Lemma 3. 1 Suppos e t h a t th e MMS E estimat e x± ca n b e expresse d a s x± = ax + r , wher e r satisfie s Assumption s (i ) an d (ii) . The n th e biase d SN R fibiased an d th e unbiase d SN R /3 are relate d b y ^biased = P + I Moreover th e constan t a i s real , satisfyin g 0 < a < 1 , an d th e biase d an d unbiased SNR s ca n b e respectivel y expresse d a s ^biased = " , (3-17 1— a P= - .a (3.18 I— a
) )
Proof. Le t x± b e th e MMS E estimat e o f x. Usin g (3.13) , w e ca n writ e the estimatio n erro r a s e± = x± — x = (a — l)x + r . Note t h a t th e unconditiona l mea n i£[ej_ ] i s zer o becaus e th e expectatio n operation i s ove r al l rando m variables . Th e MS E an d varianc e o f e± ar e the same . A s x an d r ar e uncorrelated , th e varianc e (o r MSE ) o f e± can b e writte n a s a2± = \a-l\ 2a2x+a2. (3.19 ) On th e othe r hand , w e hav e i?[e^e^ ] = E[e±(x* ± —#*)] , whic h i s equa l to E[—e±x*] becaus e e± i s orthogona l t o x±. S o w e ca n writ e a2e± = E[-e ±x*}=
E[-((a -
l)x + r)x*] =
( 1 - a)a
2
x.
(3.20
)
One direc t implicatio n o f th e abov e expressio n i s t h a t , whe n w e writ e an MMS E estimat e a s x± = ax + r , th e constan t a i s a rea l numbe r < 1. Usin g th e fac t t h a t G\ < o^ , Eq . (3.20 ) implie s t h a t a > 0 . Substituting (3.20 ) int o (3.15) , w e immediatel y ge t th e expressio n o f biased SN R give n i n (3.17) . Usin g (3.19 ) an d (3.20) , w e find t h a t o~T = a(l — a)a x. Substituting th e abov e expressio n int o (3.16) , w e ge t 22
13 a(l — a)cr ' x which simplifie s t o (3.18) . I t follow s fro m th e tw o expression s (3.18 ) an d (3.17) t h a t biased = /
3+ l .■
3.2. Orthogonalit y principl e an d linea r estimatio n
43
Note t h a t th e expressio n (3.20 ) implie s t h a t a = 1 if an d onl y i f the estimatio n error e± = 0 . I n othe r words , an MMSE estimate is always biased unless x± = x. Becaus e a n MMS E estimato r minimize s th e mea n square d erro r ^ [ | e | 2 ] , i t maximize s th e biase d SN R biased = ^ / ^ [ | e | 2 ] - Ou r nex t lemm a shows t h a t i f w e remov e th e bia s o f th e MMS E estimat e a s i n (3.14) , thi s bias removed MMS E estimat e i s als o th e bes t linea r unbiase d estimate . Therefor e to ge t a n optima l unbiase d estimate , w e ca n firs t find a n MMS E estimat e an d then remov e th e bia s o f th e MMS E estimat e befor e symbo l detection . A s bia s is invariabl y remove d befor e symbo l detection , th e bia s remova l operatio n i s implicit i n al l MMS E receiver s an d i t i s usuall y no t show n i n th e syste m bloc k diagram. Lemma 3. 2 [28 ] Suppos e t h a t th e MMS E estimat e ca n b e expresse d i n th e form o f (3.13) . Defin e th e unbiase d estimat e %unb,A. —
X_\_/OL.
T h e n an y othe r linea r unbiase d estimat e o f x canno t hav e a large r SN R t h a n %unb,.L' ^
Proof. Le t u s assum e th e contrary . Suppos e t h a t ther e i s a n unbiase d linear estimat e wit h a highe r SNR . Le t
be suc h a n unbiase d linea r estimate . It s unbiase d SN R i s give n b y
which i s assume d t o b e large r t h a n /3 , the SN R o f x Mn 6,±- Usin g th e fac t t h a t ^biased = / 3 + 1 , w e hav e ^biased < ft' + 1 , whic h ca n b e rewritte n as 22
ox
Hbiased —
where a\ i
^
o
"
^_x_
^o
\
I i
±i
s th e MS E o f x±. Rearrangin g th e terms , on e get s
< > ra-
(3-2
Now construc t a ne w estimat e a s 22 X =
_
9i
_ 2 X=
ZJ2
I
_ 2y
X
~^~
22
J
-cri at u
x^
eu
x^ue
) '
D
44
3. FI R equalizer s As th e estimat e x' i s unbiased , w e hav e i£[0|a; ] = 0 , whic h implie s t h a t the zero-mea n rando m variable s x an d 0 ar e uncorrelated . S o th e ne w MSEis
Simplifying th e abov e expression , w e ge t 22
2 _ °l°0
o\
2i
n- 2 *
Using (3.21) , w e wil l hav e o 2e,, < a^ ±, a contradiction ! Thu s w e conclud e t h a t ther e doe s no t exis t an y unbiase d linea r estimat e wit h a highe r SN R t h a n x,. nh. i • ■
3.2.2 Estimatio
n o f multipl e rando m variable s
T h e orthogonalit y principl e ca n als o b e applie d t o solv e th e proble m o f esti mating mor e t h a n on e rando m variable . Suppos e w e ar e t o estimat e M ran dom variable s Xi, 0 < i < M — 1, fro m K observe d sample s yj , 0 < j < K — 1. A linea r estimat e o f xi ha s th e for m K-l Xi =
/
j
dijVj-
3=0
Defining th e vector s x = [xo . . . % _ i ] T , y = [yo . . . yK-i] T', an d x = [xo . . . XM-I] T•, th e proble m become s on e o f estimatin g th e rando m vecto r x from th e observe d vecto r y : x = Ay , where A i s th e M x K matri x give n b y [A]^ - = a^- . Defin e th e erro r vecto r e = x — x , wher e th e kth entr y i s th e kth estimatio n erro r e ^ = Xk — XkOur ai m i s t o find A suc h t h a t i^e^e ] i s minimized . Not e t h a t minimizin g i^e^e] i s equivalen t t o minimizin g ^[|e/e| 2 ] fo r ever y k. B y th e orthogonalit y principle, th e estimat e Xi i s optima l i n th e MMS E sens e i f an d onl y i f ever y estimation erro r e ^ = Xi — Xi i s orthogona l t o ever y yj. T h a t is , E[eiVj] = 0 , fo
r0 < i < M - 1 ,0 < j < K - 1 ,
or equivalentl y i n matri x for m £?[eyt] = 0 . (3.22
)
T h e MMS E estimat e x ^ i s suc h t h a t th e erro r vecto r e ^ i s orthogona l t o th e observed vecto r y . A s th e MMS E estimat e ha s th e for m x ^ = A y , th e erro r vector i s orthogona l t o x ^ a s well :
E[e±^}=
0.
In othe r words , ever y e^± i s orthogona l t o ever y % , j _ . Usin g th e fac t t h a t i ? [ e ^ x j= 0 (whic h follow s directl y fro m th e abov e expression) , i t ca n
45
3.3. M M S E equalizer s
be show n t h a t th e length s o f th e thre e vector s xj_ , ej_ , an d x satisf y th e P y t h a g o r e a n theorem : # [ x t x ] = E[JC\JC ±] +
E[e{e
±};
the sam e righ t triangl e relatio n continue s t o hold . A s i n the cas e o f one rando m variable, th e MMS E estimat e x ^ i s a biase d estimat e unles s th e estimatio n error e ^ = 0 . Thu s th e followin g SN R fo r th e kth estimat e i s biased : G
o
xh ,
n «^/c,_
Pk,biased —
o
L
•
W h e n th e kth estimat e ca n b e expresse d a s Xk,± = otk^k + r^ fo r som e zero mean rando m variabl e r ^ t h a t i s statisticall y independen t o f £& , the n i t ca n also b e show n t h a t th e biase d an d unbiase d SNR s are , respectively , give n b y fik,biased = an
1 - Oi
k
t
d j3k
=.
- a
k
T h e relatio n (3k,biased = flk + 1 continue s t o hol d fo r al l k. Moreove r th e bias-removed MMS E estimat e Xk,±/otk als o maximize s th e unbiase d SN R /3k among al l unbiase d estimat e o f x^.
3.3 M M S
E equalizer s
We ar e no w read y t o deriv e minimu m mea n square d erro r (MMSE ) equal izers. Conside r th e transmissio n syste m i n Fig . 3.1 . A n equalize r i s calle d a n MMSE equalize r i f i t minimize s th e quantit y E[\e(n)\2}=E[\x(n)-x(n-n0)\2}, where th e intege r n o i s th e syste m delay . I n thi s section , w e wil l illustrat e ho w the orthogonalit y principl e i s used t o deriv e MMS E equalizer s fo r F I R channel s and MIM O (multi-inpu t multi-output ) frequenc y nonselectiv e channels . W e consider onl y F I R equalizers . Reader s ca n find mor e detail s i n [64] .
3.3.1 FI
R channel s
For a n L c t h orde r channe l c(n) wit h nois e q(n), th e receive d signa l i n Fig . 3. 1 is Lc
y(n) =
2_. c(k)x(n — k=o
k) + q(n).
It i s assume d t h a t th e transmitte d signa l x(n) i s a zero-mea n wid e sens e stationary (WSS ) rando m proces s an d th e nois e q(n) i s a zero-mea n WS S process uncorrelate d wit h x(n). T h e outpu t o f a n L a t h orde r equalize r a(n) is give n b y x(n) =
^ a(k)y(n /c=0
-
k). (3.23
)
3. FI R equalizer s
46
Define th e outpu t erro r e(n) = x(n) — x(n — no) , wher e n o i s th e syste m delay. The n ou r goa l i s to find a(n) s o that i?[|e(n)| 2 ] i s minimized . Tha t is , we wis h t o find th e MMS E equalizer . Fro m (3.23) , w e se e tha t finding th e MMSE equalize r i s equivalen t t o th e proble m o f finding th e MMS E estimat e of x(n — no) fro m th e (L 0 + l ) receive d samples y(n), y(n — l), . . . , y(n — L a). The orthogonalit y principl e say s tha t th e estimat e i s optima l i f an d onl y i f the outpu t erro r e(n) satisfie s E[e(n)y*(n-k)]=0, fo
r 0 < k < L a.
(3.24)
We have (L a + 1 ) equations an d w e can solv e for th e (L a + 1 ) unknowns a(n), 0 < n < L a. A matri x approac h t o solvin g suc h a proble m i s give n below .
Matrix formulatio n Define th e followin g vectors : ■ a(0 ) "
a(l)
a(La)
y(n) y(n- 1 )
q(n) q(n - 1 )
y(n - L a)
q(n - L a)_
x
x(n) x(n — 1 )
(3.25)
x(n — L)
where L = L a-\- L c. The n w e ca n writ e th e erro r a s e(n) = y T a — x(n — no). The orthogonalit y conditio n i n (3.24 ) become s E[y*(yTa-x(n-n = 0 0))]
,
where th e superscrip t * indicate s comple x conjugation . Le t u s denot e th e autocorrelation matrice s o f y, q , an d x a s R^ , R g , an d R^ , respectively , an d define th e cros s correlation vector r xy(no) = E[x(n — no)y*]. The n th e MMS E equalizer i s give n by 3 a
± = [ R^rlr^(™o).
Next w e will expres s a ^ i n term s o f the channe l tap s c{n). Usin g th e relatio n that y{n) = ( c * x)(n) + q(n), w e ca n writ e th e vecto r y a s
y = cz L x + ^ where Ci ow i s th e ( L + 1 ) x (L a + 1 ) Toeplit z matri x i n (3.4) . Unde r th e assumption tha t th e nois e q(n) an d transmitte d signa l x(n) ar e uncorrelated , we hav e R 7/ C T T D fi* Rq. Similarly w e ca n expres s th e cros s correlatio n vecto r r xy(no) a s i-xy
(no)
^lowrXjxJ-n0i
where l n o i s th e uni t vecto r define d i n (3.5) . Th e produc t R * l n o i s simpl y the not h colum n o f R* . Substitutin g th e expression s o f R ^ an d r xy(no) int o the expressio n o f a^ , w e obtai n a±
C
lowIlxClow +
R q
^lowrXjxJ-n0-
(3.26)
3 Because eac h entr y o f y i s a mixtur e o f channe l nois e an d transmitte d signal , th e autocorrelation matri x H y i s i n genera l invertible , excep t fo r som e ver y specia l cases .
3.3. MMS E equalizer s
47
T h e outpu t signa l o f th e MMS E equalize r i s y Ta±=
x±(n)=
x T Q o w a ± + q T a ± . (3.27
)
Next w e procee d t o deriv e th e minimize d mea n square d error . Becaus e th e error e±(n) = x±(n) — x(n — no) i s orthogona l t o y , i t i s als o orthogona l t o x±(n). Therefor e w e ca n writ e S±(TIQ) = ^[|e^(n)| 2] a s £±(no) =
E [ - (x±(n) -
x(n - n 0)) x*(n - n 0)] .
Substituting (3.27 ) int o th e abov e expressio n an d simplifyin g th e result , w e have £±{no) = where S x =
£ x- l n
0
R
x c iow \C
r x (0 ) = E[\x(n)\ 2] i B
^=
^T
R
lowIl*xCiow
+
R* J C
ZowR*lno,
(3.28
)
s th e signa l power . Le t u s defin e th e matri x
^ C l o w [C
lowRlCiow
R* J C
+
lowYCx.
T h e n th e minimize d mea n square d erro r ca n b e rewritte n a s £±N)=
£*(l-[Bj
no
, n o ) . (3.29
)
One direc t consequenc e o f th e abov e expressio n i s t h a t al l th e diagona l entrie s of th e matri x B ^ satisf y 0 < [ B _!_]&& < 1 (th e non-negativit y o f [ B _!_]&& follow s from th e fac t t h a t B ^ i s positiv e semidefinite) . Therefor e th e optima l syste m delay n o i s suc h t h a t th e not h diagona l entr y o f th e matri x B ^ i s th e largest . One ca n follo w a procedur e simila r t o t h a t o f th e least-square s equalize r t o obtain a delay-optimize d MMS E equalizer .
Zero-mean ii d input s In man y systems , th e transmitte d sample s x(n) ar e zero-mea n an d ii d (in dependent an d identicall y distributed) . This , i n particular , implie s t h a t th e autocorrelation matri x R x = £ XI. I n thi s specia l case , th e MMS E equalize r and th e matri x B ^ become , respectively , a±
( C L C I O .+
B_L = C
iow
^ R ; ) cLlno
I C lowCiow +
^ - R g J Clow
, (3.30 '^
) 3 31
* ^
It follow s fro m th e abov e expression s t h a t a ^ an d B ^ ar e relate d b y Ci owa.±= B ^ l n o . Substitutin g thi s resul t int o th e expressio n o f x±(n) i n (3.27) , w e ca n find t h a t th e MMS E estimat e ca n b e expresse d a s x±(n)=
ax(n —
where a = [ B J Jn
0,n0
>
no) + r ( n ) ,
3. FI R equalizer s
48
Note t h a t th e quantit y r{n) ha s zero-mea n an d i t i s statisticall y independen t of th e desire d signa l x(n — no). Becaus e [Bj_] n 0 j n o < 1 (excep t fo r th e unre alistic cas e t h a t q(n) = 0 an d th e channe l c(n) ha s onl y on e nonzer o t a p , se e Problem 3.15) , th e MMS E estimat e i s a biase d estimate . W e ca n conclud e from Lemm a 3. 1 t h a t th e unbiase d SN R (3 and th e biase d SN R Phased a r e > respectively, give n b y [Bjn o.n o
T h e relatio n fiuased =
1
d pbiased
a n
1 - [ B Jn o , n o 1
- [ B Jn 0 ) n o
/3 + 1 continue s t o hold .
Uncorrelated nois e W h e n th e nois e q(n) i s als o uncorrelated , i.e . H A/oI, th e expression s o f a ^ an d B ^ can , respectively , b e simplifie d a s a
^ = [C
lowCiow
+
-I J C
B_L ( Cj„Ci ow"I-\ ow \^low^low _L =— Ci ^low
q
Zowlno,
"—I - J -C1 ^lowi
where 7 i s th e SN R £ X/NQ. I t i s wel l know n t h a t whe n th e SN R approache s infinity, a n MMS E equalize r reduce s t o a least-square s equalizer . On e ca n see t h a t a s th e quantit y I / 7 goe s t o zero , a ^ an d B ^ i n th e abov e formula s reduce, respectively , t o a/ s an d B / s i n (3.7 ) an d (3.9) . Usin g th e matri x identity give n i n Proble m 3.18(b) , th e equalize r an d th e matri x ca n als o b e expressed a s
ax = Cj olu (c lowC\ow +
V) l
no , (3.32
Bi. = Q o t u CL ( C ^ C L + ±1) .
(3.33
) )
These alternativ e formula s ar e sometime s usefu l fo r simplifyin g expressions .
3.3.2 M I M
O frequency-nonselectiv e channel s
As w e wil l se e i n late r chapters , MIM O frequency-nonselectiv e channel s aris e in man y wideban d communicatio n systems . Conside r a transmissio n schem e where th e t r a n s m i t t e d signa l i s a n M x 1 vector x an d th e receive d signa l i s a K x 1 vector y . Fo r frequency-nonselectiv e channels , thes e vector s ar e relate d by y = C x + q , (3.34 ) where C i s a K x M channe l matri x an d q i s a K x 1 nois e vector , whic h is independen t o f x . Le t th e equalize r b e a n M x K matri x A . The n th e equalizer outpu t i s x = Ay . T h e outpu t erro r vecto r i s give n b y e = x — x. B y th e orthogonalit y principle , the MMS E equalize r satisfie s E[ey^} =
0.
3.3. MMS E equalizer s
49
Substituting th e expressio n e = A y — x int o th e abov e equation , on e ca n sho w t h a t t h e MMSE equalize r i s given by ■A-_L = K'xyK'y
i
where th e auto - an d cros s correlatio n matrice s are , respectively , R ^ = ^ [ y y ^ ] and H xy = E ^ x y t ] . Usin g t h e fact t h a t x a n d q ar e uncorrelated, on e ca n express thes e correlatio n matrice s i n terms o f the channe l matri x C an d arriv e at A ± = R x C t [ C R x C t + R g ] _ 1 . (3.35 ) It ca n b e verified t h a t t h e autocorrelation matri x o f the correspondin g outpu t error vecto r e ^ i s given by Rex =
E[e ±e{]=
K X- K
x&
[CR
xCt
+ Rg ] _ 1 CR X.
If b o t h R x a n d Hq ar e invertible, w e can apply t h e matrix inversio n lemm a in Appendi x A to rewrite t h e above expressio n mor e compactl y a s Rex =
(R- 1 + CtR7 1C)"1.
T h e minimize d mea n square d erro r fo r t he kth inpu t signa l Xk is give n b y ^[|e/c,±| 2 ] = [Rejjfcfc . T h e minimize d tota l mea n square d erro r i s trace (R e j _). Special cas e o f scalar frequency-selectiv e channel s Observ e t h a t whe n C = [c(0 ) c ( l ) . . . c(L c )], x = [x(n) x(n-l) . . . x(N - L C ) ] T , y = y(n) , a nd q = q(n), t h e MIMO channe l mode l i n (3.34 ) reduce s t o t he special cas e of scalar channel s discusse d i n t he previous subsection . Therefor e al l t he results of th e scala r cas e ca n b e obtained fro m t h e formulas fo r t he MIMO case . For example, t h e MMSE equalize r i n (3.26 ) ca n be derive d fro m t h e formula i n (3.35) b y making som e appropriat e substitution s (se e Proble m 3.16) .
Zero-mean ii d inputs W h e n t h e input signal s Xi are zero-mean a n d iid, t h e autocorrelation matri x reduces t o R x = £ XI. T h e MMSE equalize r A ^ a n d t he autocorrelatio n matrix R e ± becom e A±=
Cf
CCf + -^R £
R e ± = S X[I- B
Q
J , (3.36
)
where C f ( C Cf + - ^ R 0 ) C
.
Comparing t h e matrix B ^ wit h t h a t o f the F I R channel cas e i n (3.31) , one finds t h a t the y ar e identical i f we se t C = Cf ow. Therefor e man y result s are similar t o those o f the F I R channe l case . Fo r example , al l th e diagona l entrie s
3. FI R equalizer s
50
of B ^ ar e smalle r t h a n on e an d th e MMS E equalize r i s alway s biased . T h e biased SN R o f th e kth estimat e Xk,± i s give n b y Pk,biased
1 - [Bjfcf c
T h e unbiase d SN R i s give n b y /3 k = /3k,biased — 1 . I f th e nois e vecto r q i s als o uncorrelated, on e ca n furthe r simplif y th e result s b y substitutin g H q = A/o I into th e expression s o f A ^ an d B_L .
3.3.3 Example
s
Example 3. 3 Frequency-nonselective scalar channel. Le t u s conside r th e trans mission o f th e signa l x ove r a frequency-nonselectiv e channel . Th e receive d signal i s y = ex + q, where th e channe l nois e q i s assume d t o b e zero-mea n an d independen t o f x. Suppos e w e ar e t o us e th e linea r estimat e x = ay. B y th e orthogonalit y principle, th e MMS E solutio n i s suc h t h a t th e estimatio n erro r e = x — x i s orthogonal t o th e observatio n y, i.e . E[(x — x)y*] = 0 . Thi s mean s E[xy*] = E[xy*]. Observ e t h a t becaus e x an d q ar e orthogonal , E[xq*\ = 0 ; w e hav e E[xy*] = c*a 2x an d E[xy*} = a(\c\ 2al + a 2). W e arriv e a t c*<72 7c
a±=
where 7 = (J^jai. T h
| c | ^ + af
*
=
^ T T ' (3
'37)
e MMS E estimat e i s 2
^ 7|c|
ic*
T h e outpu t erro r e± = x± — x i s e±
- 1 7c 7|c|
2
+ l
* 2
|c| + l
7
which contain s contribution s fro m b o t h q and x. Th e minimize d MS E i s give n by
E
^n = ^frr- ( 2
,2i °-
3 38
-)
T h e biase d SN R i s therefor e give n b y 2 Pbiased= ^ [ |
e
J2] = 7|c|
2
+ l .
To remov e th e bias , w e ca n multipl y x b y 1/a t o obtai n th e bias-remove d M M S E outpu t a s x un\> = x + q/c. I t follow s t h a t th e unbiase d SN R i s P = l\c\
2
,
3.3. MMS E equalizer s
51
which i s equa l t o ^biased — 1 , a s expecte d fro m th e result s o f Lemm a 3.1 . I f one i s t o us e a zero-forcin g equalizer , th e outpu t wil l b e xzf=
y/c =
x + q/c ;
it i s identica l t o th e unbiase d estimat e x unb o f th e MMS E equalizer . I n thi s case, th e MMS E equalize r ha s th e sam e unbiase d SN R a s th e zero-forcin g equalizer. ■ In th e abov e example , w e se e t h a t whe n th e channe l ha s onl y on e nonzer o t a p , the unbiase d SN R o f a n MMS E equalize r i s th e sam e a s t h a t o f a zero-forcin g receiver. W h e n th e bia s o f a n MMS E equalize r i s removed , th e resultin g o u t p u t i s th e sam e a s a zero-forcin g equalize r an d henc e the y hav e th e sam e performance. Fo r th e mor e genera l cas e o f F I R channels , th e situatio n i s different. A n MMS E equalize r ca n yiel d a highe r unbiase d SN R t h a n a zero forcing equalize r (i f i t exists ) a s demonstrate d i n Exampl e 3.4 . Example 3. 4 Frequency-selective scalar channel I n thi s example , th e channe l is C(z) = 1+0.952: - 1 . Assum e t h a t b o t h th e signa l x(n) an d th e nois e q(n) ar e zero-mean, white , an d independen t o f each other . Firs t w e fix th e signa l powe r at S x = 1 an d th e nois e powe r a t A/ o = 0.3 . W e plo t th e variance s o f e(n) , esig(n), an d e q(n) (se e Eq . (3.11) ) fo r th e delay-optimize d MMS E equalize r versus th e equalize r orde r L a. T h e result s ar e show n i n Fig . 3.5 . Comparin g these result s wit h thos e o f th e least-square s equalize r ai s(n) i n Exampl e 3.2 , we se e t h a t ther e i s n o nois e amplificatio n fo r th e MMS E equalizer . T h e o u t p u t erro r varianc e o\ i s n o longe r dominate d b y th e nois e powe r a 2e whe n La i s large .
52
3. FI R equalizer s
H e(n
) e . (n ) sigv '
-x e
HI
q(n)
I | | | | || | | | | | | | | |
*X
) ( )( X X X ) ( X X X X X X X X X X
10 1
52
L
02
5
Figure 3 . 5 . Exampl e 3.4 . Th e variances o f e(n) , e Sig(n)} an d e q(n) (define d i n (3.11)) o f an MMSE equalize r plotte d agains t th e equalizer order s L a.
DQ ■D
cc 1 0 CO
55 — zero-forcin g — — — least-square s MMSE
10
15
20 Y(dB)
25
30
Figure 3 . 6 . Exampl e 3.4 . Th e unbiased SNRs of the M R zero-forcing equalizer 1/C(z), the least-square s equalize r Ai s(z), an d the MMSE equalize r A±(z). Th e orders o f bot h Ais(z) an d A±(z) ar e L a = 16 . Her e 7 = 6 X/MQ.
53
3.3. M M S E equalizer s
S (e JC0)
-30
0.2 0.
4 0. 6 0. Frequency normalize d by7t
Figure 3.7 . Exampl e 3.4 . Th e powe r spectr a \C(e
8
juJ 2
)\ Sx(ejuJ) an
d ^(e -
Next w e compar e th e unbiase d SNR s o f th e MMS E equalizer , th e least squares equalizer , an d th e II R zero-forcin g equalize r 1/C(z). Th e result s ar e shown i n Fig . 3.6 , wher e 7 = £ x/Afo an d th e order s o f A±(z) an d Ai s(z) ar e La = 16 . W e se e tha t th e unbiase d SN R o f th e MMS E equalize r i s alway s higher tha n tha t o f th e least-square s equalizer , confirmin g th e fac t state d i n Lemma 3. 2 tha t th e MMS E equalize r ha s th e larges t unbiase d SN R amon g all unbiase d estimate s base d o n th e sam e se t o f observations . Th e differenc e is mor e prominen t whe n 7 i s small . Thi s i s becaus e fo r a larg e 7 , th e MMS E equalizer reduce s t o th e least-square s equalizer . Not e tha t i n th e SN R regio n of 7 < 2 4 dB , wit h a relativ e smal l orde r o f L a = 16 , th e unbiase d SNR s of bot h th e MMS E an d least-square s equalizer s ar e highe r tha n tha t o f th e IIR zero-forcin g equalize r eve n thoug h a n II R equalize r employ s al l th e pas t samples fo r symbo l recovery . Whe n 7 i s large , th e IS I ter m become s mor e significant. Th e MMS E an d least-square s equalizer s suffe r fro m th e IS I effec t and thu s ar e inferio r t o th e zero-forcin g II R equalizer . ■ To understan d ho w a n MMS E equalize r avoid s th e nois e amplificatio n problem, w e examin e th e equalizer s i n Exampl e 3. 4 fro m th e frequenc y do main viewpoint . I n Fig . 3.7 , w e plot th e powe r spectr a \C(e^ UJ)\2Sx(e^UJ) (th e desired signa l componen t i n the receive d nois y signa l y(n)) an d S q(eJUJ) = A/o (the nois e component ) fo r S x = 1 an d A/ o = 0.3 . I t i s see n tha t th e low frequency conten t o f th e receive d sample s contain s mostl y th e signa l compo nent, wherea s it s high-frequenc y conten t i s dominate d b y th e nois e compo nent. Figur e 3. 8 show s th e magnitud e response s o f the zero-forcin g equalize r 1/(7(2:), th e least-square s equalize r Ai s(z) an d th e MMS E equalize r A±(z),
54
3. FI R equalizer s
where th e order s o f b o t h Ai s(z) an d A±(z) ar e L a = 16 . Fro m th e figure, we ca n se e t h a t b o t h th e II R zero-forcin g equalize r an d th e F I R least-square s equalizer t r y t o recove r al l th e frequenc y component s o f th e signa l x(n). I n the proces s o f recoverin g x(n), the y als o greatl y amplif y th e nois e componen t in th e high-frequenc y region . O n th e othe r hand , th e MMS E equalize r A±(z) also trie s t o recove r x(n) i n th e low-frequenc y regio n becaus e th e receive d signal contain s mostl y th e desire d signa l i n thi s region . I n th e high-frequenc y region, wher e th e receive d sample s contain s mostl y noise , th e MMS E equalize r avoids th e nois e amplificatio n proble m b y havin g a smalle r gain .
15
CD
10
— zero-forcin g — least-square s - ■ MMSE
CD C/) £=
o
Q. "D
E
-5
-10
0.2 0.
4 0. 6 0. Frequency normalize d by7t
8
Figure 3.8 . Exampl e 3.4 . Th e magnitud e response s o f th e II R zero-forcin g equalize r 1/C(z), th e least-square s equalize r Ai s(z), an d the MMS E equalize r A±(z). Th e order s of bot h Ai s(z) an d A±(z) ar e L a = 16 .
Example 3. 5 MIMO channel Le t th e t r a n s m i t t e d signa l x an d th e receive d signal y b e b o t h 2 x 1 vector s an d le t th e channe l matri x b e ~1 c c1 where c i s real . Suppos e t h a t c ^ ± l s o t h a t C i s invertible . I n thi s cas e th e zero-forcing receive r exist s an d it s outpu t i s x= x + C
_1
q.
Assume t h a t R ^ = £ xl an d H q = A/oI . Th e autocorrelatio n matri x o f the out put erro r i s the n give n b y R e = A/oC~"l"C -1 . On e ca n calculat e th e unbiase d
55
3.3. M M S E equalizer s
Figure 3.9 . Exampl e 3.5 . Compariso n o f th e unbiase d SNR s o f th e zero-forcin g an d MMSE receiver s fo r c = 0.5 .
SNR o f th e zero-forcin g receiver , whic h i s give n b y Pk,zf =
(1-c2)2 \ ^ 2 1+ c
z
7 , for k = 0 , 1 ,
where th e quantit y 7 = S x/Afo. Nex t conside r th e MMS E receiver . Substi tuting C int o th e expressio n o f th e autocorrelatio n matri x H e± i n (3.36) , w e find th e tw o outpu t erro r variance s ar e th e same , equa l t o
MKl + T^ + c 2)
J [Re,l] fcfc crj, k,mmse ~ L"e,_LjK K — Q _^ _ ~-l _| _ c 2 \ 2 _ 4 C 2 '
for k = 0 , 1 .
T h e biase d SN R o f th e MMS E receive r i s ~2\
Pk,biased = ( 7 + 1 + 7 C )
47C 2
I + 7 - 1 + c ,2: *
T h e unbiase d SN R o f th e MMS E receive r ca n b e compute d usin g th e formul a Pk = Pk,biased ~ 1 , an d i t i s give n b y Pk = 7 + 7
c2
47c 2
l+ 7"
1
+c2'
for / c = 0 , 1 . Computin g th e differenc e betwee n th e unbiase d SNR s o f th e MMSE an d zero-forcin g receivers , w e ge t Pk — Pk,zf =
T
47C 2
47C 2
I + 7 - 1 + c ,:2 '
which i s alway s greate r t h a n zero . Unlik e th e scala r cas e i n Exampl e 3.3 , the MMS E receive r ha s a highe r unbiase d SN R t h a n th e zero-forcin g receiver . T h e tw o unbiase d SNR s fo r c = 0. 5 ar e plotte d i n Fig . 3.9 . I t i s see n t h a t th e difference ca n b e significan t fo r a moderat e SNR . ■
3. FI R equalizer s
w W
MMSE receiver
A
Ma
rp 1
A / I X rp
P1 ^
,
symbol detection
Figure 3 . 1 0 . Symbo l detectio n base d o n th e unbiase d estimat e x
3-4 Symbo
w W
unb.
l detectio n fo r M M S E receiver s
From earlie r discussions , w e kno w t h a t a n MMS E estimat e i s biased . A s the equalize r outpu t i s x = ax + r fo r som e a < 1 , th e constellatio n point s are contracte d b y a facto r o f a an d th e boundarie s fo r th e nearest-neighbo r decision rul e (NNDR ) hav e bee n changed . Therefor e i f th e bia s i s no t re moved befor e makin g a symbo l decision , th e origina l decisio n boundarie s i n the N N D R wil l n o longe r b e optimal . Thi s wil l increas e th e symbo l erro r rat e (SER) a s demonstrate d late r b y Exampl e 3.6 . Thu s fo r MMS E equalizers , i t is importan t t o remov e th e bia s an d mak e th e symbo l detectio n base d o n th e unbiased estimat e Xunb = x/a =
x + r/a.
This operatio n i s demonstrate d i n Fig . 3.10 . Th e outpu t erro r o f th e unbiase d estimate i s therefor e give n b y r/a; i t i s amplifie d b y a facto r o f 1/a. Fro m (3.18), w e kno w t h a t a i s relate d t o th e unbiase d SN R (3 as
One ca n se e fro m th e abov e formul a t h a t a wil l approac h on e a s f3 increases . For example , a = 0. 9 i f (3 = 9 , an d a = 0.9 9 i f (3 = 99 . W h e n th e SN R i s ver y high, th e bia s become s negligible . Computing SE R usin g formula s I n Chapte r 2 , th e SE R formula s fo r PAM an d QA M symbol s ar e derive d (se e (2.12 ) an d (2.17)) . Thes e formula s are derive d unde r th e assumptio n t h a t th e nois e i s Gaussian . Fro m earlie r discussion, w e kno w t h a t th e quantit y r consist s o f b o th nois e an d ISI . Thoug h the channe l nois e is , i n general , Gaussian , th e IS I t e r m i s a linea r combinatio n of th e t r a n s m i t t e d symbols , whic h ar e no t Gaussian . However , i t i s know n [119] t h a t r ha s a n approximatel y Gaussia n distributio n whe n th e numbe r of t r a n s m i t t e d symbol s contributin g t o th e IS I t e r m i s sufficientl y large , e.g . 32. Th e Gaussia n tai l render s a ver y nic e approximatio n o f bi t erro r rat e [119, 189 , 190] . A s w e will se e i n Exampl e 3.7 , thi s approximatio n i s extremel y good, an d th e formula s derive d i n Sectio n 2. 3 giv e a ver y accurat e estimat e of SER . Example 3. 6 Suppos e th e modulatio n symbo l s i s a n equiprobabl e 2-bi t PA M symbol wit h s G { ± 1, ± 3 } . Suppos e t h a t ther e i s n o signa l processin g a t th e transmitter s o t h a t th e t r a n s m i t t e d signa l i s x = s. On e ca n calculat e th e signal powe r an d i t i s give n b y a^ = £ x = 5 . Assum e t h a t th e channe l
3.4. Symbo l detectio n fo r MMS E receiver s
57
is AWGN . S o th e receive d signa l i s y = x + g , wher e th e nois e powe r i s (Tq = A/ o = 1 . T h e quantit y 7 i s S x/Afo= 5 . W e conside r tw o differen t receivers. (a) Zero-forcing equalizer T h e outpu t o f a zero-forcin g equalize r i s x zf= y = x + q. T h e MS E i s give n b y 2 E[\ezf\2]=E[\q\ = ]
l.
From (2.12) , th e symbo l erro r probabilit y i s give n b y SER= zf
3/2Q(l ) =
0.238 .
(b) FIR MMSE equalizer Fro m (3.37) , w e hav e a± = the equalize r outpu t i s x± =
a±y =
7 / ( 7 + 1 ) = 5/ 6 an d
5/6 x + 5/6g .
T h e MMS E estimat e x± i s biased . T h e MS E i s give n b y E[\e J 2 = ]= ^
5/6 ,
which i s smalle r t h a n ^ [ | e ^ / | 2 ] . Thoug h i t ha s a smalle r erro r variance , the MMS E equalize r doe s no t necessaril y hav e a bette r SE R performanc e t h a n th e zero-forcin g equalizer . I n fac t i f th e bia s i s no t remove d befor e making a symbo l decision , th e receive r t h a t employ s a n MMS E equalize r can hav e a wors e SE R performance . Suppos e th e receive r use s N N D R for makin g decision s withou t takin g th e bia s int o consideration . Le t us comput e th e SER . Give n t h a t s i s transmitted , x = 5/6 s + 5/6q i s a Gaussia n rando m variabl e wit h mea n m = 5/6 s an d varianc e a 2 = ( 5 / 6 ) 2 , t h a t is , p(£|s) =
_^
= e
-(x-m)2/(2^)B
Below w e calculat e th e probabilit y o f symbo l erro r whe n s = 1 is trans mitted. T h e derivation s fo r othe r case s ar e similar . Accordin g t o NNDR , the decisio n J i s correc t i f 0 < x < 2 . A symbo l erro r occur s i f eithe r x < 0 o r x > 2 , t h a t is , P ( s ^ s\s = 1 ) = P(x < 0\s = 1 ) + P(x > 2\s = 1) . Using th e Q function , w e ca n writ e P(x < 0\s = 1 ) = Q ( l ) an d P(x > 2\s = 1 ) = Q ( 7 / 5 ) . Therefor e P(s + s\s = 1 ) = Q ( l ) + Q ( 7 / 5 ) = 0.2394 . Using a simila r procedure , on e ca n verif y t h a t P(s ^ s\s = —1 ) = P(s ^ s\s = 1 ) an d P ( s ^ s\s = ±3) =
P(x < 2\s = 3 ) = Q ( 3 / 5 ) = 0.2743 .
As a result , w e hav e a symbo l erro r rat e o f SER^^^ = 0.2569 , whic h is large r t h a n SER zf i n P a r t (a) . T h e MMS E receive r ha s a wors e SE R performance i f th e bia s i s no t corrected !
58
3. FI R equalizer s
Suppose w e remov e th e bia s b y dividin g x± b y 5/6 , th e resul t wil l b e x/a = x + q = x zf, i t i s identica l t o t h a t o f a zero-forcin g equalizer . Thu s w e conclude t h a t i n thi s specia l cas e th e MMS E an d zero-forcin g equalizer s hav e the sam e B E R performance . Thi s exampl e demonstrate s th e importanc e o f bias remova l befor e th e symbo l detectio n a t th e receiver . ■
10
LU 1 0
MMSE, Monte-Carl o * MMSE
10 10
, formula
— — — IIR , zero-forcing
10
15
20 Y(dB)
25
30
Figure 3 . 1 1 . Exampl e 3.7 . Symbo l erro r rat e versu s SN R 7 = £
x/J\fo.
Example 3. 7 Conside r th e channe l i n Exampl e 3.4 . Suppos e t h a t x(n) i s an equiprobabl e 2-bi t PA M symbol . Tw o differen t equalizers , th e II R zero forcing equalize r an d a 16t h orde r MMS E equalizer , ar e considered . T o obtai n the SE R o f th e system , w e us e Mont e Carl o simulation . Figur e 3.1 1 show s the plot s o f SE R versu s th e SN R 7 = £ x/Afo. Not e t h a t whe n th e channe l is frequency-selective , a n F I R MMS E equalize r ca n hav e a smalle r SE R t h a n the II R zero-forcin g equalize r fo r a moderat e SN R 7 . I n th e sam e figure, w e also plo t th e SE R value s obtaine d b y substitutin g th e unbiase d SN R o f th e M M S E equalize r int o th e SE R formul a i n (2.12) . I t ca n b e see n t h a t th e SE R values obtaine d fro m th e Mont e Carl o experimen t ar e indistinguishabl e fro m those obtaine d fro m th e formula ; th e IS I t e r m i s wel l modele d b y a Gaussia n distribution. ■ Example 3. 8 I n thi s example , w e stud y th e effec t o f channe l zero s o n th e performance o f th e MMS E equalizers . Th e orde r o f th e equalize r i s L a = 32 . T h e t r a n s m i t t e d sequenc e x{n) i s a 3-bi t PA M symbo l an d th e nois e q(n) i s
3.5. Channel-shortenin g equalizer s 5
10 1
52
02
9
53 SNR (dB)
03
54
0
Figure 3 . 1 2 . Exampl e 3.8 . Effec t o f channe l zer o o n th e SE R performanc e o f th e MMSE equalizers .
white an d Gaussian . Th e channe l i s give n b y
Note tha t th e channe l i s normalize d s o tha t i t ha s uni t energ y fo r al l p. The channe l ha s a zer o a t z = p. Figur e 3.1 2 show s th e SE R o f th e delay optimized MMS E equalize r fo r variou s position s o f th e channe l zeros . Whe n p = 0 , th e channe l become s a n AWG N channe l an d th e SE R i s th e smallest . It i s see n fro m th e figur e tha t whe n th e zer o move s close r t o th e uni t circle , SER become s worse . Th e syste m ha s th e wors t SE R performanc e whe n th e channel zer o i s o n th e uni t circle . Moreover , not e tha t whe n th e channe l zero i s replace d wit h it s reciprocal , th e SE R performanc e i s unchanged. Thi s phenomenon ca n b e explaine d usin g th e resul t i n Proble m 3.12 . ■
3.5 Channel-shortenin
g equalizer s
Example 3.8 demonstrates that th e performance o f equalizers depends strongl y on the locatio n o f channel zeros . Whe n th e channe l ha s a zero on the uni t cir cle, th e zero-forcin g equalize r i s unstabl e an d th e performanc e o f th e MMS E equalizer become s unsatisfactory . A s w e wil l se e i n late r chapters , w e ca n greatly alleviat e thi s proble m b y usin g transmissio n scheme s tha t inser t som e redundant sample s known as the guard interva l to the transmitted signal . Th e
3. FI R equalizer s
60
number o f redundan t sample s require d i s usuall y equa l t o th e channe l orde r Lc. Fo r transmissio n environment s suc h a s digita l subscribe r loop s (DSL) , the channe l ca n b e ver y long ; havin g redundan t sample s a s lon g a s th e chan nel orde r become s impractical , a s i t wil l significantl y reduc e th e transmissio n rate. Thu s fo r thes e applications , w e ofte n emplo y a n equalize r a t th e receive r to shorten the channel. Suc h a channel-shortenin g equalize r i s ofte n calle d th e time domai n equalize r (TEQ) . I t i s s o name d becaus e th e ai m i s t o equaliz e the channe l i n th e tim e domai n t o a shorte r length , an d a T E Q i s ofte n use d together wit h anothe r se t o f equalizer s calle d frequenc y domai n equalizers . In thi s section , w e shal l stud y th e desig n o f F I R T E Q fo r F I R channels . Let th e channe l b e a n L c th-order F I R filter C{z). Th e receive d signa l i s give n by y(n) =
(c*x)(n) +
q(n),
where q(n) i s th e channe l noise , whic h i s assume d t o b e uncorrelate d wit h x(n). Le t th e T E Q a(n) b e a n L a t h - o r d e r F I R filter. The n w e ca n writ e it s output a s x(n) =
(t * x)(n) + (a * q)(n),
where th e "effectiv e impuls e response " t(n) i s give n b y t(n) =
La
(a * c)(n) =
N , a(k)c(n — k=o
k).
Unlike th e previousl y studie d equalizers , th e goa l o f havin g a T E Q a(n) i s no t to produc e a n outpu t x(n) t h a t i s clos e t o x(n — no) fo r som e no. It s goa l is t o shorten th e effectiv e channe l t(n) t o som e predetermine d targe t length , say (y + 1) , whic h i s usuall y a n intege r muc h smalle r t h a n L c. Fo r a n Lcth order F I R channe l an d a n L a t h - o r d e r F I R T E Q , th e lengt h o f ( a * c){n) i s {La + L c + 1) ; th e channe l i s i n fac t lengthened rathe r t h a n shortened . T h e cascade (a * c)(n) ca n neve r b e a filter o f lengt h (y + 1) , bu t i t ca n b e a goo d approximation i n th e sens e t h a t th e coefficient s o f (a * c) (n) t h a t fal l outsid e a window o f length (z^+1 ) ar e insignificant . B y doin g thi s w e effectively shorte n the lon g impuls e respons e c(n) t o a "short " impuls e respons e ( a * c)(n). W e now describ e th e mathematica l formulatio n o f th e problem . Suppos e w e ar e to desig n a(n) s o t h a t mos t o f th e energ y o f t(n) lie s i n a prescribe d windo w of lengt h (y + 1) , sa y n o < n < no + v. Le t u s rewrit e th e equalize r outpu t a s n0 + v
x(n) =
2_j t(k)x(n — k=n0 kg[n vv / Xd(n) x
k) + V vj
^ t(k)x(n 0,n0+v]
_ 1
,
^/ u(n)
—
k) +(a* q)(n). Qo(n)
.
T h e thre e quantitie s Xd(n), x u(n), an d q 0{n) are , respectively , th e desire d signal du e t o th e impuls e respons e withi n th e window , th e undesire d signa l (i.e . the IS I term ) du e t o th e impuls e respons e outsid e th e window , an d th e nois e term. Assum e t h a t th e t r a n s m i t t e d signa l x(n) i s whit e wit h autocorrelatio n coefficients r x(k)= S xS(k) an d t h a t th e nois e q(n) i s uncorrelate d wit h x(n). T h e n w e ca n writ e th e desire d signa l power , th e IS I power , an d th e nois e
3.5. Channel-shortenin g equalizer s
61
power, respectively , a s no + i/
Pd = £ x Y, ^
fc
)| 2 > p isi= £* Yl
fc
l^
k=no fc0[no,no +
)! 2 ' F 9 = ^[l
^]
T h e coefficient s o f t{n) t h a t fal l insid e th e windo w contribut e t o th e desire d signal powe r wherea s thos e t h a t fal l outsid e th e windo w contribut e t o th e IS I power. I n wha t follows , w e wil l expres s thes e thre e quantitie s i n term s o f the T E Q coefficient s a(n). Defin e L = L a + L c an d le t th e vector s t an d a , respectively, b e define d a s i n (3.2) . T h e n thes e vector s ar e relate d b y (3.3) , which w e reproduc e below : where Ci ow i s th e ( L + 1 ) x (L a + 1 ) Toeplit z matri x i n (3.4) . T o captur e th e desired impuls e respons e i n th e prescribe d window , w e defin e a n (L + 1 ) x (L + 1 ) diagona l matri x D n o whos e diagona l entrie s ar e r,
1 noJ
_ f 1 , n 0 < n < n 0 + is; "\ 0 , otherwise .
T h e n th e coefficient s t h a t li e insid e an d outsid e th e windo w are , respectively , given b y (I-Dno)t=
(I-D„
0
)CJowa.
From th e abov e expressions , w e ca n writ e th e thre e quantitie s i n (3.39 ) a s Pd Pisi =
£ xa^CljI -
D
no)Qowa,
(3.40
)
±q = a Ixqa , where H q i s the (L a + 1 ) x (L a + 1 ) autocorrelatio n matri x o f th e channe l nois e q(n). Not e t h a t w e hav e use d D j l o D n o = D n o an d ( I — D n o ) t ( I — D n o ) = (I —Dn o ) i n th e abov e equations . W i t h th e abov e expressions , w e no w procee d to th e desig n o f th e T E Q a{n). Ther e ar e man y desig n criteri a fo r T E Q . Below w e conside r th e T E Q t h a t maximize s th e SI R o r SIN R dependin g o n the availabilit y o f th e nois e statistics . (1) Maximizatio n o f SI R T h e signa l t o interferenc e rati o (SIR ) i s define d as
Pd
SIR:
P■ 1 ISI
Substituting th e expression s fo r Pd an d P ^ int o th e abov e equation , th e T E Q t h a t maximize s th e SI R i s give n b y a
^Imi^riQ
Cl
ow3.
&ovt = ar g niax r — -^ . a a tcL(I-Dno)Qowa Note t h a t th e proble m o f finding a t o maximiz e th e rati o Pd/Pisi i s equivalen t to t h a t o f finding a t o minimiz e P ^ unde r th e constrain t Pd = 1 . T h e latte r criteria i s use d t o desig n T E Q i n [90 ] an d th e optima l T E Q i s calle d th e MSSNR (maximum-shortenin g SNR ) T E Q . Thu s th e SIR-maximize d T E Q i s the sam e a s th e MSSN R T E Q .
62
3. FI R equalizer s
(2) Maximizatio n o f SIN R W h e n th e nois e statistic s ar e available , w e can exploi t t h e m i n th e desig n o f T E Q . Defin e th e signa l t o interferenc e an d noise rati o (SINR ) a s
SINR
=-ET3hr-
T h e n th e optima l T E Q a(n) become s a
3.opt =
ar g m ax ■
^
owDn0Qowa-
LtcL(I-Dno)Clowa+^atR,a
Note t h a t scalin g th e vecto r a ha s n o effec t o n th e SI R an d th e SIN R values. W e ca n conside r a unit-nor m vecto r a . The n th e proble m o f finding the optima l a(n) ca n b e expresse d a s aopt =
ajQ0a( ar g m ax - — - — , (3.41 a aTQx a
.
)
for som e positiv e semidefinit e matri x Q o an d positiv e definit e m a t r i x 4 Q i . T h e solutio n t o th e abov e proble m ca n b e obtaine d b y applyin g th e Rayleigh Ritz principl e (Appendi x A) . Le t Q 2 b e a positiv e definit e matri x suc h t h a t Q ^ 1 = Q2Q 2 a n d defin e th e vecto r b = Q ^ X a . The n (3.41 ) ca n b e rewritte n as
b+Q+Q0Q2b
Q22 ar *opt = v^ a i ggmma ax A■ —p
-.
From th e Rayleigh-Rit z principle , th e optima l T E Q i s give n b y a. opt = Q2V , where v i s th e eigenvecto r o f th e matri x Q2Q0Q 2 associate d wit h it s larges t eigenvalue. A s w e shal l se e i n th e numerica l exampl e below , th e optima l T E Q based o n eithe r objectiv e functio n ca n effectivel y shorte n th e channel . The Choic e o f n o Th e startin g poin t n o o f the windo w affect s th e optima l T E Q solution . T o ge t th e bes t no , on e ha s t o solv e th e optima l T E Q fo r 0 < n o < L — v an d choos e th e valu e o f n o t h a t maximize s th e objectiv e function. I n practice , i t i s foun d t h a t th e optima l n o i s usuall y a n intege r much smalle r t h a n L c. Example 3. 9 Le t th e channe l c{n) b e a n F I R filter o f lengt h 100 : 0.9 n co s ( — 0.8 n co s — 0.7 n si n — ) + 0.6 n si n , V40 / V60 / V80 / VIOO / for 0 < n < 100 . Thi s impuls e respons e i s plotte d i n Fig . 3.13 . Suppos e t h a t we ar e t o desig n a sixth-orde r T E Q a(n) wit h v = 10 . T E Q s optimize d fo r the tw o criteri a describe d earlie r ar e considered . Fo r th e cas e o f SINR , w e assume t h a t th e nois e i s white an d th e SN R 7 = £ x/Afo = 3 0 d B. Th e resultin g impulse response s (a*c)(n) ar e show n i n Fig . 3.14 . I t i s see n t h a t b o t h T E Q s can effectivel y shorte n th e channe l wit h lengt h 10 0 t o th e prescribe d windo w of lengt h 11 . Th e SI R i s 94. 3 d B fo r th e SIR-maximize d T E Q an d th e SIN R is 28. 9 d B fo r th e SINR-maximize d T E Q . ■ c(n) = yJ
4 In fact , i t ca n b e showe d (se e Proble m 3.20 ) tha t th e matri x C^ ow; (IQ i ) i s positiv e definit e excep t fo r th e degenerat e cas e o f L c < z^ , whic h implie s tha t n o channel shortenin g i s needed .
3.5. Channel-shortenin g equalizer s
63
100
Figure 3 . 1 3 . Exampl e 3.9 . Channe l impuls e response .
64
3. FI R equalizer s
shortened — — — origina l
40 6
n
0
80
100
100
Figure 3.14 . Exampl e 3.9 . Shortene d impuls e respons e wher e th e TE Q a(n) i s designed t o maximiz e (a ) SIR, (b ) SINR.
3.6. Concludin g Remark s
3.6 Concludin
65
g Remark s
In thi s chapter , w e studie d th e proble m o f channe l equalizatio n fo r F I R chan nels. T h e F I R zero-forcin g equalize r an d th e MMS E equalize r wer e derived . Both equalizer s wer e obtaine d b y simpl e matri x computations . A powerfu l tool fo r solvin g man y communicatio n problems , th e orthogonalit y principle , was als o studied . Fo r mor e detaile d t r e a t m e n t s o f estimatio n an d detectio n problems, se e [64 , 65] . T h e ide a o f channel-shortenin g equalizer s wa s introduce d b y Mels a i n [90] . Since then , man y desig n approache s hav e bee n proposed . T h e SI R an d SIN R criteria give n i n thi s chapte r fo r designin g T E Q wer e propose d i n [37 , 152] . One ca n als o desig n th e T E Q b y incorporatin g th e specifi c structur e o f th e transmitter an d receive r int o th e T E Q optimization . Thes e approache s ar e often mor e comple x bu t ca n resul t i n bette r syste m performance . Fo r furthe r details, th e reade r i s referre d t o [6 , 10 , 42 , 82 , 87 , 165] .
3.7 Problem
s
3.1 Prov e t h a t th e matri x Ci order channe l C(z) ^ 0 .
i n (3.4 ) ha s ful l ran k wheneve r th e L
cth-
3.2 Le t C(z) = 1 + z - 1 . Fin d th e second-orde r least-square s equalize r ai W h a t i s th e optima l syste m dela y no ?
s(n).
ow
La 3.3 Le t A(z) = a(0)+a(l)2: _ 1 H \-a(L b e th e delay-optimize d least a)z~ Lc squares equalize r o f th e channe l C(z) = c(0 ) + c(l)2: _ 1 H Vc(L c)z~ , and le t th e correspondin g optima l syste m dela y b e no.
(a) Prov e t h a t th e delay-optimize d L a t h - o r d e r least-square s equalize for th e channe l C\z) = c{L c) + c(L c - l ) z _ 1 -\ \c(0)z~Lc i _1 La given b y A'(z) = a(L a) + a(L a - l ) z -\ \a(0)z~ . W h a is th e optima l syste m dela y n 0 ? Ho w ar e th e minimize d MSE s fo these tw o case s related ?
r s t r
(b) W h a t i s the L a t h - o r d e r delay-optimize d least-square s equalize r whe n the channe l i s C"{z) = c(0)-c(l)z- 1+c(2)z-2 + (-l) Lcc(Lc)z-Lc? W h a t i s th e optima l syste m delay ? I s th e minimize d MS E fo r th e case o f C ,,(z) th e sam e a s t h a t o f C(z)l Justif y you r answer . (c) Prov e t h a t A(z 2) i s th e 2L a th-order delay-optimize d least-square s equalizer fo r C(z 2). 3.4 (a ) Le t U A V b e th e singula r valu e decompositio n o f th e matri x Ci in (3.4) . Sho w t h a t th e matri x B / s i n (3.9 ) ca n b e expresse d a s B,S=U
00
ow
ut.
(b) Expres s th e kth diagona l entr y [ B ^ ] / ^ o f B / s i n term s o f the entrie s in th e fcth ro w vecto r o f U . (c) Prov e t h a t 0 < [Bi s]kk <
1 fo r 0 < k < L.
3. FI R equalizer s
66
3.5 Fro m th e previou s problem , w e kno w t h a t 0 < [B/ S]fcfc < 1 fo r al l k. (a) Fin d th e condition s o n th e channe l c(n) suc h t h a t maxfc[B| s ]^ = 0 . (b) Fin d th e condition s o n th e channe l c{n) suc h t h a t [ B ^ ] / ^ = some k.
1 fo r
3.6 Suppos e t h a t th e t r a n s m i t t e d signa l an d th e channe l nois e ar e b o t h white an d t h a t the y ar e uncorrelated . Le t e/ s (n) = x(n) — x(n — no) be th e outpu t erro r o f a n L a t h - o r d e r least-square s equalizer . C o m p u t e the outpu t erro r varianc e £I S{JIQ)= -E[|e/ S (n)| 2 ]. Expres s you r answe r in term s o f Ci ow, £ x an d A/o . 3.7 Orthogonality principle for the vector case On e ca n generaliz e th e or thogonality principl e t o th e vecto r case . Suppos e w e ar e t o estimat e an M x 1 rando m vecto r x fro m a se t o f M x 1 observe d vector s yo, y i , . . . , YK-I- Th e linea r estimat e i s x=
A 0y0 + Aiy iH h
AK-IYK-I .
Show t h a t th e estimat e x ^ i s optima l i f an d onl y i f it s erro r vecto r e ^ = x ^ — x satisfie s ^ [ e ^ y j ] = 0 , fo r i = 0 , 1 , . . . , K — 1. 3.8 Suppos e w e ar e t o estimat e xi usin g th e observatio n variable s yi = aiXi + qi. Assum e t h a t x>i are uncorrelate d wit h qi fo r 0 < i , j < K — 1, and th e autocorrelatio n matrice s o f x an d q ar e b o t h diagona l (no t nec essarily equa l t o a scale d identit y matrix) . Sho w t h a t th e vecto r MMS E estimation proble m reduce s t o K scala r MMS E estimatio n problems . 3.9 Le t th e M x N matri x A b e th e optima l linea r estimato r o f a n M x 1 vector s give n a n N x 1 observatio n vecto r r . Suppos e no w w e ar e t o estimate s usin g B r , wher e B i s a n invertibl e matrix . Prov e t h a t th e op timal estimato r i n thi s cas e i s A B _ 1 an d th e mea n squar e erro r remain s the same . (Thi s show s t h a t , withou t los s o f generality , applyin g a n in vertible matri x t o th e observatio n vecto r doe s no t chang e th e minimize d MSE o f a n MMS E equalizer. ) 3.10 Verif y t h a t th e minimize d MS E fo r a n MMS E equalize r ca n b e expresse d as i n (3.28) . 3.11 Le t C(z) = 1 + z~ x. Suppos e b o t h th e t r a n s m i t t e d signa l an d channe l noise ar e whit e wit h power s £ x an d A/o , an d the y ar e uncorrelated . Fin d the second-orde r MMS E equalize r a±(n). I f th e channe l i s change d t o C'{z)= 1 — z _ 1 , ho w i s th e ne w MMS E equalize r a'j_(n) relate d t o a±(n)? 3.12 Suppos e b o t h th e t r a n s m i t t e d signa l an d channe l nois e ar e whit e wit h powers £ x an d A/o , and the y ar e uncorrelated . Le t th e channe l b e C(z) = S n = o c (n)z~n a n d th e L a t h - o r d e r delay-optimize d MMS E equalize r b e a±(n) wit h th e optima l decisio n dela y give n b y no (a) Expres s a\ (th e outpu t erro r varianc e du e t o th e channe l noise ) i n terms o f A/ o an d a± (n) .
67
3.7. Problem s
(b) Prov e t h a t th e delay-optimize d L a t h - o r d e r MMS E equalize r fo r th e reversed channe l C'(z) = c(L c) + c(L c — l ) ; ?- 1 + • • • + c(0)z~ Lc i s f given b y a j_(n) = a±(L a — n). W h a t i s th e optima l decisio n dela y n 0 ? HO W ar e th e minimize d MSE s fo r thes e tw o case s related ? (c) W h a t i s th e delay-optimize d L a t h - o r d e r MMS E equalize r a![_(n) when th e channe l i s C'\z)= c(0 ) - c ( l ) ^ - 1 + c(2)z~ 2 - •• • + Lc Lc (—l) c(Lc)z~ 7 W h a t i s th e optima l syste m delay ? I s th e mini mized MS E fo r th e cas e o f C" {z) th e sam e a s t h a t o f C(z)7 Justif y your answer . (d) W h a t i s th e 2L a th-order delay-optimize d equalize r whe n th e chan nel i s C(z 2)7 Justif y you r answer . 3.13 Explai n wh y th e quantitie s min£j_(no) an d mm n0 n
Ed,is (^o) 0
given i n (3.10 ) an d (3.29 ) canno t increas e wit h L equalizer.
ai
th e orde r o f th e
3.14 Suppos e t h a t w e hav e a two-inpu t one-outpu t channe l whos e transfe r matrix i s c = [ 1 1] . T h e inpu t vecto r x i s 2 x 1 with R x = E xl2 an d th e noise q ha s varianc e A/o - Fin d th e MMS E equalize r an d th e minimize d MSE t r a c e ( R e ) . 3.15 Conside r th e matri x B ^ give n i n (3.31) . Sho w t h a t [Bj^f c = 1 for som e k onl y i f th e channe l c(n) ha s onl y on e nonzer o t a p an d th e nois e q(n) is zero . (Hint : [ B _!_]&& = 1 fo r som e k i f an d onl y i f E±(k) = 0 fo r som e k.) 3.16 Le t C = [c(0 ) c(l ) . . . c(L c )], x = [x(n) x(n - 1 ) . . . x(N - L C ) ] T , y = 2/(n) , an d q = q(n). Sho w t h a t th e MMS E equalize r i n (3.35 ) reduces t o t h a t i n (3.26) . 3.17 MMSE equalizers for linear phase channels A matri x J i s sai d t o b e a reversal matri x i f al l it s entrie s ar e zer o excep t th e anti-diagona l entries . For example , a 3 x 3 reversa l matri x i s give n b y
"0 0 01 10
1" 0 0
Let th e inpu t signa l x(n), th e channe l c(n) , an d th e nois e q(n) b e rea l in Fig . 3.1 . (a) Le t x an d q b e th e vector s define d i n (3.25) . Sho w t h a t thei r autocorrelation matrice s satisf y J R X J = R x an d J R g J = H q. (b) Sho w t h a t i f c(n) i s symmetric, i.e . c(L c—n)= c(n) , the n JCi ow= J Clow, wher e C\ ow i s th e ( L + 1 ) x (L a + 1 ) Toeplit z matri x define d in (3.4) . (c) Suppos e t h a t th e orde r L a o f th e equalize r i s suc h t h a t L = L a-\-Lc is even . Le t th e syste m dela y b e n o = L/2. Sho w t h a t th e M M S E equalizer a±(n) i s symmetri c whe n c(n) i s symmetric .
3. FI R equalizer s
68
(d) W h a t woul d a±(n) b e i f c(n) i s anti-symmetri c (i.e . c(L c — n) = -c(n))? 3.18 Le t Q b e a n m x m positiv e definit e matri x s o t h a t Q = SS" ^ fo r som e invertible S . Conside r th e followin g matrix : P=
A t [AA t + Q j ^ A ,
where A i s a n m x n matrix . B y followin g th e procedur e describe d below , we ca n prov e t h a t al l th e diagona l entrie s o f P ar e strictl y smalle r t h a n one. (a) Sho w t h a t P ca n b e expresse d a s B ^ [BB ^ + I m ] B
, wher e B =
(b) Fo r an y m x n matri x B , verif y th e identit y B [ B t B + I n] =
[BBt+I
m
]B,
which prove s t h a t [ B B t + I m ] _ 1 B = B [B+ B + 1 „ ]
_1
for an y B . (c) Verif y th e followin g identity : B + B [B+ B + I n ] _ 1 + [B+ B + I n ] _ 1 = I . (d) Usin g th e result s i n (a) , (b) , an d (c) , an d th e fac t t h a t th e matri x [B^B + I n ] i s positiv e definite , sho w t h a t th e diagona l entrie s of P ar e strictl y smalle r t h a n one . 3.19 Single-input multi-output channel:
channel
(SIMO) Conside r th e followin g SIM O
y = ex + q ,
where x i s th e scala r t r a n s m i t t e d signal , y i s th e i V x l receive d vector , and q i s th e nois e vecto r whic h i s assume d t o b e uncorrelate d wit h x. Let th e equalize r outpu t b e x = a Tcx + a
T
q.
Define th e unbiase d SN R
|aTc|Vg aTRga*
Assume t h a t th e nois e autocorrelatio n matri x H q i s positive definit e an d let R q = QQ+ . (a) Fin d th e equalize r a^l t t h a t maximize s th e unbiase d SN R /3. Ex press you r answe r i n term s o f c an d Q . (b) Le t a ^ b e th e MMS E equalizer . Ho w i s 3^ pt relate d t o a ^ ?
69
3.7. Problem s
3.20 Sho w t h a t th e matri x CJ ow(I — ~D no)Ciow i n (3.40 ) ca n b e singula r onl y if th e channe l orde r L c < v. Thi s implie s t h a t th e IS I ca n neve r b e completely remove d b y th e T E Q unles s th e origina l channe l c(n) i s al ready shorte r t h a n o r equa l t o th e targe t channe l length . (Hint : W h e n Lc > z/ , ca n w e find a nonzer o T E Q a(n) suc h t h a t ( I — D n o ) t = 0? ) 3.21 Le t th e channe l b e C(z) = l + z _ 1 + z " 2 + z " 3 + z " 4 . Le t v = 2 and n 0 = 2. Assum e t h a t b o t h signa l x(n) an d nois e q(n) ar e whit e wit h powe r Sx an d A/o , respectively . Fin d a first-order T E Q A(z) = a(0 ) + a{\)z~ 1 t h a t maximize s (a ) SIR, (b ) SINK 3.22 An MMSE criterion for designing TEQs [6 ] Conside r th e bloc k diagra m shown i n Fig . P3.22 . A differen t wa y o f designin g th e T E Q a{n) i s t o find a(n) an d td(n) s o t h a t th e mea n square d erro r
E[\w(n) - y(n)f
(3.42)
is minimized . T h e sequenc e td(n) i s calle d th e targe t impuls e respons e (TIR). Thi s desig n proble m i s spli t int o thre e step s a s describe d below . (Assume t h a t th e signa l x(n) an d nois e q(n) ar e uncorrelated , an d th e signal x(n) i s whit e wit h signa l powe r £ x.) q(n) x(n)
w W
c(n)
wr
\w 9W
a(n)
e(n)
V-
y(n)
^
td(n)
Figure P 3 . 2 2 . Th e TE Q tha t minimize s th e outpu t mea n squared error .
(a) Le t td(n) b e F I R wit h nonzer o coefficient s fo r n o < n < UQ + v. Given td(n), find a(n) s o t h a t th e mea n square d erro r £(td) =
E[\w(n) -
(x *t
2
d)(n)\
}
is minimized . Expres s you r answe r i n term s o f th e T I R vecto r td = [td(no) Ufao + 1 ) • • • td(no + ^ ) ] T , the matri x C iow, an d th e noise autocorrelatio n matri x H q. (b) Le t £o(td) b e th e minimize d mea n square d error . Expres s £o(td) i n terms o f t ^ an d Ci ow. (c) Fin d a unit-nor m t ^ s o t h a t £o(td) i s minimized . W h a t i s th e corresponding optima l T E Q a ( n ) ? 3.23 Le t th e syste m parameter s b e th e sam e a s thos e i n Proble m 3.21 . Us e the metho d describe d i n Proble m 3.2 2 t o desig n a n optima l first-order T E Q A(z) = a(0 ) + a ( l ) z ~ 1 . I s th e resultan t T E Q identica l t o th e T E Q designed i n Proble m 3.21(b) ?
70
3. FI R equalizer s
3.24 I n thi s problem , w e woul d lik e t o prov e t h a t th e optima l T E Q obtaine d by usin g th e MMS E criteri a describe d i n Proble m 3.2 2 i s a scale d versio n of th e optima l T E Q t h a t maximize s th e SINR . Thi s ca n b e don e b y following th e step s give n below . (a) Le t a±(n) an d td,±(n) be , respectively , th e optima l T E Q an d de sired respons e obtaine d i n Proble m 3.22(c) . Prov e t h a t td,±(n) = (a_i_ * c)(n) fo r n o < n < no + v. (Hint : Sho w t h a t i f td,±(n) ^ (a± * c)(n) fo r som e n o < n < no + z/ , the n w e ca n find anothe r td(n) wit h a smalle r mea n square d error. ) (b) Usin g th e resul t fro m (a) , sho w t h a t finding th e a{n) an d td(n) t h a t minimize th e mea n square d erro r define d i n (3.42 ) i s equivalen t t o finding th e a(n) t h a t minimize s P ^ + P q. (c) Prov e t h a t th e optima l a(n) t h a t minimize s P ^ + P q, unde r th e constraint t h a t Pd = 1 , differ s fro m th e T E Q t h a t maximize s th e SINR onl y b y a scal e factor . As th e T E Q performanc e i s no t change d b y scaling , thi s prove s t h a t th e M M S E T E Q an d SINR-maximize d T E Q hav e th e sam e performance .
4 Fundamentals o f multirate signa l processing Multirate signa l processing has bee n foun d t o b e very usefu l i n modern com munication systems . I n thi s chapter , w e introduce basi c multirat e concept s which ar e fundamenta l t o man y o f our discussion s i n futur e chapters .
4.1 Multirat
e buildin g block s
The decimator an d expander are two fundamental buildin g blocks in multirate signal processin g systems . Th e M-fol d decimator , show n i n Fig . 4.1(a) , i s defined b y the input-outpu t relatio n yd(n)=x(Mn), (4.1
)
which means that ever y Mt h sampl e of the inpu t i s retained. Thi s i s demonstrated i n the figure for M = 3 . Note that th e samples are renumbered b y the decimator suc h tha t 2/d(0) = x(0) , y
d(l)
=
x(3), y
d{2)
= x(6),
and s o forth. Th e decimato r i s als o know n a s the downsampler . I n general , decimation result s in a loss of information. Bu t i f a signal is bandlimited i n a certain wa y then i t ca n be decimated withou t los s of information, a s we shall see in Sectio n 4.2. The M-fol d expander , o r upsampler , show n i n Fig . 4.2(a) , i s define d i n the tim e domai n b y the input-outpu t relatio n , . _ ( x(n/M), n = multiple o f M; ( ~ \ 0 , otherwise . ^'
Ve[n)
.
Z)
Thus the expander merely inserts zero-valued samples in a systematic manner, namely M — 1 zeros ar e inserte d betwee n adjacen t sample s o f x(n). Thi s i s demonstrated i n Figs . 4.2(b ) an d (c ) fo r M = 3 . Evidentl y ther e i s n o los s of informatio n du e t o th e expandin g operation . Fro m th e definitio n o f th e expander w e see tha t ye(Mn)= x(n), 71
72
4. Fundamental s o f multirat e signa l processin g |M
(a) x(n)
n
-+ yd(
)
decimator
x(n)
L_L
(b) -
3
Ir
-
2
-
1
0
1
,
2
3
4
5
6
7
x(0) x(6)
*(-3)
yd(n)
x(3)
_J_
(c) - 1 0
1
2
Figure 4 . 1 . (a ) The M-fold decimato r o r downsampler. (b),(c ) Example s of input and output signal s wit h M = 3 .
\M
(a) x{ri)
n
-+ y yee([ )
expander
x(n) (b) - 1 0
1
2
x(0)
42)
i i
n
ye( )
x(l) (c)
•
—1 • -
3
-
2
-
1
1
• -^ 1
^
0
3
1
2
•• •
—• 4
5
•►
6
7
Figure 4 . 2 . (a ) Th e M-fold expande r o r upsampler . (b),(c ) Example s o f inpu t an d output signal s wit h M = 3 .
73
4.1. Multirat e buildin g block s
which show s tha t w e ca n recove r th e inpu t x(n) fro m y e(n) simpl y b y Mfold decimation . Th e expande r i s usually followed b y a digital filter calle d a n interpolation filte r whic h makes it mor e interesting (Sectio n 4.3).
4.1.1 Transfor m domai n formula s In the z-domai n the input-output relatio n for th e expander ca n be written a s X(zM) (expander)
Ye(z) =
. (4.3
)
To see this not e that y e (n) = 0 when n i s not a multiple o f M, an d w e get oo o
Ye(z)= Y,
o Mn
ye(Mn)z-
=
Y,
x{n)z-
Mn
=
X{z
M
),
n = —o o n = —oo
where we have used the fac t tha t y e(Mn) = x(n). Next , fo r th e decimato r we claim that th e input-outpu t relatio n i n the z-domai n ca n be written a s Yd(z) =
~^Y1
x
(zl/MW£) (decimator)
, (4.4
)
£=0
where the quantit y W i s defined a s W =
e
-j2*/Mm (
45)
M
Note tha t W i s a n Mt h roo t o f unity , tha t i s W = 1 . W e shall defe r th e proof o f the abov e expression to Sectio n 4.4. It i s importan t t o understan d (4.3 ) an d (4.4 ) i n term s o f th e frequenc y variable UJ. Substitutin g z = e JUJ w e have y
e
X(e JujM) (expander
(0=
and
) (4.6
)
M-l
y
d
(0=
— Yl X(e J^-27ri^M) (decimator)
. (4.7
)
£=0
Thus, fo r th e expander , Y e{e^) i s obtaine d b y squeezing or shrinkin g th e plot o f X{e^) b y a facto r o f M (Fig . 4.3) . Ther e ar e M squeeze d copie s of X(e^UJ) in a period o f 2TT. Th e extr a M — 1 copies are called images . Fo r th e decimator, the output Yd(e^) i s a sum of M terms . Th e zeroth term contains X{e?^lM\ whic h i s simply a n M-fol d stretched version of X{e^). Th e firs t term is X(e i ( >" 2 7 0 / M ) which i s th e stretche d versio n shifte d b y 2n. Mor e generally , th e ^t h ter m X{e^-2^IM) i s th e stretche d versio n shifte d b y 2n£. Figure 4. 4 demon strates thi s fo r M = 3. The shifted version s in general overlap with the original stretched version . This is called the aliasing effect, an d it is similar to aliasing created by undersampling a bandhmited signa l [105] . Sinc e the transform domai n expressions for decimatio n ar e rather complicated , w e often abbreviat e the m a s follows: M-l
X(eH o
IM
r \x(e
JUJ
= )]
= *£ 0
— V x(e^-
2
^^M); (4.8
)
74
4. Fundamental s o f multirat e signa l processin g
X{eia)
(a)
Y<en
(b)
/3
0 rc
Figure 4 . 3 . Fourie r transform s o f (a ) inpu t t o th e expande r an d (b ) outpu t o f th e expander fo r M = 3 . I n genera l ther e ar e M — 1 images .
nQ
X{eia)
c \
stretched version
shifted version
/^
V)
-27i \ o
-> (a
2TE
-2TC
x
)
shifted version
/
2j
(b)
l
overlap o r aliasing
Figure 4 . 4 . Fourie r transform s o f (a ) inpu t t o th e decimato r an d (b ) outpu t o f th e decimator fo r M = 3 . I n genera l ther e ar e M — 1 shifte d version s o f th e stretche d version.
X(z)
IM
or
X(z)
A ^M
M-l
^y M
X{Z
X M
I WI).
(4.9)
^=o
Similarly, th e notatio n [x(n)]^ M i s ofte n use d i n th e tim e domai n instea d o f x(Mn). Fo r th e expande r w e us e simila r notations , namel y [x(n)]^ M an d [ X ( Z ) ] ^ M - Thes e notation s ar e especiall y usefu l i n th e simplificatio n o f com plicated mathematica l expression s containin g multirat e operators . Remark I n man y signa l processin g applications , suc h a s subban d codin g and subban d adaptiv e filtering, i t i s ofte n desirabl e t o hav e a multirat e sys t e m t h a t i s fre e fro m th e aliasin g effect . Therefor e i n thes e applications , th e multirate syste m i s usuall y designe d i n suc h a wa y t h a t th e aliasin g effec t is eliminate d o r minimized . However , i n mos t moder n communicatio n sys tems, a n inpu t signa l ofte n passe s throug h a n M-fol d expande r befor e i t goe s through a n M-fol d decimator . A s w e wil l se e next , suc h a cascad e syste m o f
75
4 . 1 . Multirat e buildin g block s
an expande r followe d b y a decimato r doe s no t create aliasing .
4.1.2 Multirat
e identitie s
In man y applications , decimator s an d expander s ar e use d i n cascad e wit h linear tim e invarian t systems . Sinc e th e decimato r an d expande r ar e linear , but time-varyin g buildin g blocks , they canno t i n general be interchanged wit h other system s i n th e cascade . However , unde r som e condition s a restricte d amount o f movement i s possible. Fo r example, th e two systems show n in Fig. 4.5(a) ar e equivalent an d so are the two systems show n i n Fig. 4.5(b) . Thes e are know n a s noble identities . Fo r convenience w e shall refe r t o them a s the first an d the second nobl e identity , respectively . A proof o f these identitie s i s as follows .
(a)
(b)
x(n) W
\M
► H(z)
W
H(z)
► \M
x(n)
y(n)
x(n) w
W
x(n)
y(n) W
W
H(zM) —► \M
H(zM)
\M
y(n) W
y(n) W
Figure 4 . 5 . Nobl e identities , (a ) Th e firs t nobl e identity , an d (b ) th e secon d nobl e identity.
Proof. Th e filte r H(z M) o n th e righ t i n Fig . 4.5(a ) ha s th e outpu t X(z)H(zM). Th e output i s therefore th e decimated versio n M-l
Y(z) =
[X(z)H(z M)]iM =
M
Wk)H[(z^MWk)M),
^ E X(z^ k=0
where W = e~^l M. B
y using th e fact tha t W M = M-l
Y(z)
M
k Y,X(z1/MW = )
1 , we find
H(z)(x(z))
k=0
IM
The right-han d sid e is precisely the output o f the system o n the left sid e of Fig . 4.5(a) . Thi s prove s th e firs t identity . Fo r the secon d identit y note tha t th e output o f the system o n the left sid e o f Fig. 4.5(b ) i s Y(z) =
(x(z)H(z)) =
X(z
M
)H(zM),
which i s precisely th e outpu t o f the syste m o n the righ t side . Thi s establishes th e second identity . ■
4. Fundamental s o f multirat e signa l processin g
76
x(n)
\M
H(z)
|M
y(n) x(n) ►= H
[H(z)] M
l
y(n)
Figure 4 . 6 . Th e polyphase identity .
Note t h a t t h e first nobl e identit y ca n also b e written i n t he for m M (H(Z = )X(Z)) H(z)(x(zj)
(4.10
)
As thi s i s valid fo r any two ^-transforms X{z) a n d H(z), i t i s useful t o writ e this a s (X1(zM= )X2(z)) X^ix^z)) (4.11 ) Next conside r t h e cascad e show n o n t h e lef t i n Fig . 4.6 . Her e a transfe r function H(z) i s sandwiche d betwee n a n expande r a n d a decimator . T h e output ca n be written a s Y(z) =
(H{Z)X{Z M)) ^
=
X(z)(H(z))
' IM
(4.12)
where w e have use d (4.11) . Thu s t h e system i s equivalent t o a simpl e trans fer functio n [H(Z)]IM, whic h represent s a n LT I system wit h t h e decimate d impulse respons e h(Mn). Thi s prove s t h e identity i n Fig. 4.6 , whic h i s calle d the polyphas e identity . Not e t h a t i t i s simpl y t h e first nobl e identit y wit h the role s o f X(z) a n d H(z) interchanged . A s we will se e in futur e chapters , the polyphas e identit y i s ver y usefu l fo r t he applications o f multirat e signa l processing i n communicatio n systems . W e summarize t h e results i n t he fol lowing. Theorem 4. 1 Polyphas e Identity . Conside r t h e cascade syste m o n t he left i n Fig. 4.6 . T he system i s LTI with transfe r functio n [H{Z)]^M a s shown o n t he right o f t he figure. O r equivalentl y i n t he time domai n it s impulse respons e is h(Mn). ■
4.1.3 Blockin
g an d unblockin g
T h e multirat e buildin g block s ca n be used t o describe schematicall y tw o basi c operations calle d blockin g a n d unblockin g t h a t appea r frequentl y i n man y modern communicatio n systems . Conside r Fig . 4.7(a) . Her e t h e vector x ( n) is mad e fro m successiv e sample s o f x(n). I f t h e origina l sample s x(n) ar e spaced apar t b y T seconds , the n t h e samples o f x ( n) ar e spaced apar t b y MT seconds, a s show n i n t h e figure . Not e t h a t t h e vector s x ( n ) a nd x (n + 1 ) have n o commo n samples . Thu s t h e sample s o f t h e vecto r x ( n ) represen t successive nonoverlappin g block s o f t h e inpu t strea m x(n) (Fig . 4.8) . W e therefore sa y t h at x ( n ) is an M-blocke d versio n o r just a blocked version o f x{n). T h e integer M i s t he bloc k size . I n othe r words , a seria l inpu t d a t a stream i s converte d t o a se t o f paralle l d a t a streams . Fo r thi s reason , t h e
4 . 1 . Multirat
77
e buildin g block s
blocking operatio n i s als o know n a s seria l t o paralle l conversion , denote d b y S / P ( M ) . Wheneve r th e detail s o f blockin g ar e no t relevant , w e shal l us e th e schematic diagra m o f Fig . 4.7(b) . MT
■■ I ■
x(n)
>
|M
► x(Mn)
|M
► x{Mn+\)
-► x(Mn) x(ri)
S/P(M)
_^. x(Mn+\)
-+ x(Mn+M-\) advance chain
(b) Figure 4 . 7 . Th e blockin g operation o r serial to paralle l conversion : (a ) implementatio n with multirat e buildin g blocks ; (b ) schemati c diagram .
block 0 bloc
x(n)
k1
InlllllhliLi.Ji M
2M
Figure 4 . 8 . Th e blockin g operatio n demonstrate d i n th e tim e domain .
T h e invers e o f th e blockin g operatio n i s unblocking , whic h i s show n i n Fig . 4.9(a). Her e a se t o f M signal s i s interleaved b y th e combinatio n o f expander s and th e dela y chain . W e sa y t h a t x(n) i s th e unblocked versio n o f th e se t {xk(n)}. W e als o sa y t h a t th e signal s Xk(n) ar e time-multiplexed t o ge t x(n). T h e inpu t sequenc e Xk(n) i s relate d t o x(n) b y ^fc(^) = x(Mn +
k).
For obviou s reasons , th e unblockin g operatio n i s als o regarde d a s a paralle l t o seria l conversion , an d it s schemati c diagra m i s show n i n Fig . 4.9(b) . If th e blockin g an d unblockin g system s ar e connecte d i n cascad e (i n eithe r order) i t i s clea r t h a t w e ge t identity . Thi s i s show n i n Fig . 4.10 . Thi s trivial multirate identity i s surprisingl y useful , a s w e shal l se e later .
78
4. Fundamental s o f multirat e signa l processin g
x0(n) -
—►
fM
x{(n) -
—►
fM
x(n)
V") "
.r
fM
XM-i(n) H
►
W
x (n)
-l
w • • •
i--1
v (vt\
n
*M-\ \
)
P/S(AQ
> w x(ri)
w
delay chai n
(a)
(b)
Figure 4 . 9 . Th e unblockin g operation , o r paralle l t o seria l conversion : (a ) implemen tation wit h multirat e buildin g blocks ; (b ) schemati c diagram .
w
|M
w
z1 r
(a)
z
1
A
| M
W
w
f
w
(b)
1 A
•
\M
\M
\M
|M -1
|M A
•
i' J l
1 w
z^ r
|M
fM
w
-1
\M
•
Ak.Z
W
\M
z^ r
w w
1
•
|M
1
Figure 4 . 1 0 . Tw o system s whic h ar e equivalen t t o identit y systems , (a ) Blockin g followed b y unblocking ; (b ) unblockin g followe d b y blocking .
79
4.2. Decimatio n filter s
4.2 Decimatio
n filter s
Consider th e frequenc y domai n relatio n (4.7 ) fo r a decimator . Thi s wa s demonstrated i n Fig . 4.4 . Not e t h a t th e stretche d versio n X(e juj/M) i n genera l overlaps wit h som e o f th e shifte d versions . T h e origina l inpu t x(n) canno t i n general b e recovere d fro m th e decimate d outpu t yd(n) = x(Mn). Thi s overla p is commonl y referre d t o a s aliasing . However , whe n th e origina l signa l x(n) is ban d limited t o th e regio n -TT/M <
UJ < T T / M ,
then ther e i s n o aliasin g becaus e th e shifte d version s d o no t overlap . I n thi s case w e ca n recove r th e higher-rat e signa l x(n) fro m it s decimate d versio n Vdin). A signa l x(n) nee d no t b e a bandlimite d lowpas s signa l i n orde r t o b e recoverable fro m it s M-fol d decimate d version . Fo r example , conside r th e bandpass signal s wit h Fourie r transfor m show n i n Fig . 4.11 . T h e tota l band width i s 2TT/M. I t ca n b e verifie d (Proble m 4.2 ) t h a t i f th e signa l i n Fig . 4.11(a) i s decimated , the n th e stretche d versio n X{e?^l M) doe s no t overla p with th e shifte d version s x(e j^-27ri^M). Fo r th e signa l i n Fig . 4.11(b) , decimation b y M wil l no t generat e an y aliasin g i f an d onl y i f CJ Q = kn/M. In thes e cases , w e ca n therefor e recove r x(n) fro m x(Mn) b y usin g a n idea l bandpass filter wit h passban d regio n coincidin g wit h th e nonzer o par t o f th e signal. W h e n a signa l ca n b e decimate d b y M withou t creatin g an y aliasing , it i s sai d t o satisf y th e aliasfree(M ) condition . T h e mos t genera l aliasfree(M ) condition ca n b e foun d i n [159] .
(a) 2n/M
(b)
-7i - c o
0 - 7 i / M -oe>
0
0
ce>
0
CO 0 +TC/M n
Figure 4 . 1 1 . Bandpas s signal s wit h tota l bandwidt h 2TT/M. (a real signal .
ce
>
) Comple x signal ; (b )
Given a n arbitrar y signa l x(n) w e hav e t o pas s i t throug h a n appropriat e bandlimiting filter H(z) s o t h a t ther e i s n o aliasin g upo n decimatio n b y M. This i s show n i n Fig . 4.12(a) . T h e preconditionin g filter H(z) i s calle d th e decimation filter . I t ca n b e a lowpass, bandpass , o r highpass , a s demonstrate d in Figs . 4.12(b)-(d ) fo r M = 3 . T h e outpu t y(n) i s relate d t o th e inpu t x(n) by y(n)
=
Y^ h(k)x(Mn -
k) = ^ x(k)h(Mn
-
k).
80
4. Fundamental s o f multirat e signa l processin g
(a) x(n)
H(z)
s(n)
|M
■> y(n)
decimation decimato filter
r
H{J") lowpass (b)
-n/3 0
7i/
3
//(/*) bandpass (c)
-n/3 0
n/3
tf(^) highpass (d)
Figure 4 . 1 2 . (a sponses fo r H(z).
-n -27i/
3
27i/3 n
) Decimatio n filter ; (b)-(d ) example s o f aliasfree(3 ) frequenc y re -
In practic e o f cours e th e filte r canno t hav e th e idea l response s shown . Ther e can onl y b e finite attenuation i n th e stopbands . Bu t th e effec t o f aliasin g ca n be mad e arbitraril y smal l b y makin g thi s attenuatio n large .
4.3 Interpolatio
n filter s
An interpolatio n filte r i s a discrete-tim e filte r use d a t th e outpu t o f a n ex pander, a s show n i n Fig . 4.13(a) . Thi s combinatio n o f a n expande r followe d by a n interpolatio n filte r i s sai d t o b e a discrete-tim e interpolator . Fro m its definitio n w e kno w t h a t th e expande r insert s zero-value d sample s betwee n adjacent sample s o f th e input . Th e purpos e o f th e interpolatio n filte r i s t o replace thes e zero s wit h a weighte d averag e o f th e inpu t samples , a s w e shal l explain later . First conside r wha t happen s i n th e frequenc y domain . Recal l t h a t th e expander ha s M squeeze d copie s (images ) o f th e inpu t Fourie r transfor m
4.3. Interpolatio n filter s
81 s(n)
\M
(a) x(n)
H(z)
"► An)
expander interpolatio filter
n
(b)
Si^) (c)
image _ -71
i
^ -7i/3 0
mage
\
\JL^v_ 7C/3
► CO
lowpass filter
H(J") (d)
-TI/3
0
TT/
3
CO
final output
7(<>) (e) -71
-TI/3
0
TT/3
CO
Figure 4 . 1 3 . (a ) Discrete-tim e interpolator ; (b)-(e ) explanatio n o f it s operatio n i n the frequenc y domai n fo r M = 3 .
X ( e j a ; ) , a s show n i n Fig . 4.13(b) . T h e interpolatio n filte r retain s on e out of these M copie s o f t he Fourier transform . Thi s i s demonstrated i n Fig. 4.1 3 for the cas e whe n t h e filte r i s lowpas s an d M = 3 . Thus t h e interpolate d outpu t Y(e^) i s simpl y a squeeze d versio n o f X(e^ UJ) wit h t h e image s removed .
82
4. Fundamental s o f multirat e signa l processin g
4.3.1 Tim e domai n vie w o f interpolatio n filte r Returning t o th e tim e domain , Fig . 4.1 4 show s th e variou s signal s involved . T h e outpu t o f th e interpolatio n filte r i s th e convolutio n o f it s inpu t s(n) wit h the impuls e respons e h(n), t h a t i s oo o
y(n)= Y^
o
s(k)h(n-k)=
s(kM)h(n
^
= k= — oo k
-
JfcM) ,
— oo
where th e secon d equalit y follow s becaus e s(k) = 0 unles s A : is a multipl e o f M. Sinc e s(kM) = x{k) b y definition , w e therefor e hav e oo
y(n)= ^
x(k)h(n-kM).
(4.13
)
/e= —o o
Suppose H(e^) i function
s idea l lowpass . Th e impuls e respons e h{n) wil l b e th e sin e . , , sin(7rn/M ), Kn)= — ,' • (4.14
A^^
)
Figure 4.14(c ) show s h{n) fo r th e cas e M = 3 . Th e convolutio n ( s * h)(n) results i n th e replacemen t o f th e zero-value d sample s o f s(n) wit h interpolated values, a s demonstrate d i n Fig . 4.14(d) . Th e outpu t y{n) ha s a smoot h lowpass appearanc e becaus e o f th e filtering.
4.3.2 Th
e Nyquist(M ) propert y
Note t h a t th e sin e impuls e respons e satisfie s th e propert y h(Mn)=
S(n) (Nyquist(M
) property) . (4.15
)
T h a t is , i t ha s regula r zer o crossing s a t nonzer o multiple s o f M , an d moreove r h(0) = 1 . A filter H(z) satisfyin g thi s propert y i s calle d a N y q u i s t ( M ) filter, or jus t a Nyquis t filter . W e wil l sho w t h a t thi s propert y ensure s t h a t th e original inpu t sample s x(n) ar e retaine d b y th e interpolatio n filter withou t distortion, a s show n b y th e thic k line s i n Fig . 4.14(d) . Mor e precisely , w e wil l show t h a t y(Mn)= x(n). T h u s a n M-fol d decimate d versio n o f th e interpolate d signa l y{n) return s th e original signa l x(n) withou t distortion . Proof. Th e fac t t h a t th e Nyquis t ( M) conditio n ensure s t h a t y(Mn)= x{n) ca n b e verifie d directl y fro m Eq . (4.13) . Settin g n = £M, thi s equation ca n b e rewritte n a s follows : oo o
y(£M) =
^
x(k)h(£M
o
-
kM) =
^
x(k)g(£
-
fc),
/ e = — oo / e = — o o
where g(n) = h(Mn). W e hav e prove d t h a t th e sequenc e y(nM) i s equa l to th e convolutio n (x * g)(n). S o i t follow s t h a t y(nM) = x(n) fo r al l inputs x(n) i f an d onl y i f th e Nyquis t conditio n h(Mn) = 8{n) i s satis fied. ■
4.3. Interpolatio n filter s
83
x(n) (a)
s(n) (b) -
3
-
2
-
1
0
1
2
3
-
3
-
2
-
1
0
1
2
3
4
5
6
7
5
6
7
h(n) (c)
~^^ 4
1
y(n) t
(d)
tT
ft
1T t , , 1
1
•• •
►
-1 0 Figure 4 . 1 4 . Operatio n o f th e interpolatio n filte r i n th e tim e domain , (a ) Inpu t signal; (b ) outpu t o f th e expander ; (c ) impuls e respons e o f th e interpolatio n filter ; (d ) the fina l interpolate d output .
In practice , th e interpolatio n filte r i s a realizabl e syste m wit h finite attenuation i n stopbands . T h e image s ar e therefor e no t completel y suppressed . I t is, however , stil l possibl e t o satisf y th e Nyquis t propert y exactly , thu s ensur ing t h a t origina l sample s x(n) ar e preserved . Ther e ar e man y method s fo r the desig n o f Nyquis t filters . On e simpl e approac h i s th e windo w method . Starting wit h th e idea l sin e functio n i n (4.14) , on e ca n choos e an y windo w function w(n), suc h a s a Blackman , Hanning , o r Kaise r window . T h e n th e following i s a n F I R Nyquis t filter :
hyn) =
sin(7m/M) w[n). Tin
We ca n als o obtai n F I R Nyquis t filters b y incorporatin g th e Nyquis t constrain t into s t a n d a r d filter desig n methods , e.g . th e eigenfilte r metho d [114 , 151 , 156].
84
4. Fundamental s o f multirat e signa l processin g
4.4 Polyphas
e decompositio n
One ver y usefu l techniqu e i n th e analysi s an d desig n o f multirat e system s is th e polyphas e decomposition . Conside r a signa l x(n) wit h ^-transfor m X(z) = J^^L-o c x (n)z~n'• Thi s ca n alway s b e rewritte n i n th e for m M-lo
X ( z ) = J ^ J2
c Mx
l ex Mn
(
+
^)^"
(M +fc)
"
=
J2
= k=0 n= — oc /e
z
) k
~ Yl
x{Mn
+ k)z-
Mn
0 n = — oo
for an y positiv e intege r M. Equivalently , M-l
X(z)= Y^
k=0
z~ kXk(zM) (Typ
e 1 polyphase) (4.16
)
where X k (z) i s the ^-transfor m o f the kth subsequenc e xk(n)=x(Mn +
k), 0
< k < M - 1 . (4.17
)
Equation (4.16 ) i s calle d th e Typ e 1 polyphase decompositio n (o r represen tation) o f X{z) wit h respec t t o M. Th e M quantitie s X k{z) ar e sai d t o b e the polyphas e component s o f X{z) wit h respec t t o M (equivalentl y Xk(n) are th e M polyphas e component s o f x(n))} Not e tha t th e zerot h polyphas e component xo(n) i s simply th e decimate d versio n o f x(n), tha t i s xo(n) = x(Mn). It i s als o possibl e t o defin e th e polyphas e decompositio n M-l
X{z) =
^ z
k
Xk{zM) (Typ
e 2 polyphase), (4.18
)
k=0
where th e Typ e 2 polyphase component s X k (z) ar e ^-transform s o f x k (n) = x(Mn — k) instea d o f x(Mn + k). Figur e 4.1 5 show s ho w th e polyphas e com ponents ca n be represented i n terms o f delays and decimators . A time domai n example fo r Typ e 1 polyphase decompositio n i s shown in Fig. 4.1 6 for M = 3 ; note how the samples of the polyphase component s ar e numbered wit h consec utive integers . W e ca n als o appl y th e polyphas e decompositio n t o a transfe r function H(z). Th e Typ e 1 an d Typ e 2 polyphas e decomposition s o f H(z) are, respectively ,
f T,^ z~ kGk{zM), (Typel) ; H(z) = I (4.19 I Zk^o 1 * kSk(zM), (Typ e 2).
)
The usefulnes s o f polyphase decomposition s wil l become clea r later . Wit h th e polyphase representation , w e ar e no w read y t o prov e th e transfor m domai n formula o f the decimato r output . 1 Clearly X^{z) depend s o n M , s o a notatio n suc h a s Xj^ '(z) woul d hav e bee n mor e appropriate; bu t th e quantit y M i s almos t alway s clea r fro m th e context . S o w e shal l us e the simple r notatio n Xj^(z).
4.4. Polyphas e decompositio n
x(n)
85
type 1 polyphase components
A'
| M
►x
type 2 polyphase components
*(TI)
Q(n)
z
| M
►x
x{n)
| M
—► X
►x
Q(n)
| M
►x
x(n)
-..^ 1
'T
• • •
| M
'■ll M-IW
• • • | M
—► ^ M - I W
(b)
(a)
Figure 4 . 1 5 . Simpl e bloc k diagra m interpretatio n o f polyphase components , (a ) Type 1 polyphas e component s Xk(n); (b ) Type 2 polyphas e component s Xk(n).
x(ri)
w —Li 0 1
^0(«) (b)
t. . ■ 23
4
5
6
7
J_L
•• •
xx(n) (c)
* 2 0) (d)
Figure 4 . 1 6 . A signa l x ( n ) an d its three polyphas e component s Xk(n), fo r M = Note carefull y ho w the samples ar e renumbered.
3.
86 4
. Fundamental s o f multirat e signa l processin g Proof of (4-4)- Replacin g z wit h zW £ i n th e polyphas e representatio n (4.16) w e ge t M-l
X(zWe) = Y^ z-
k
W~MXk{zM).
k=0
Thus
M-l M-
l M-
Using th e fac t W = M-l
k
l
^2 x(zw ) = ^2 z - xk{z ) ^2 w ~M- (
= £=0 k 0
i
M
42
- °)
=£ 0 j27r M
e
/ , on e ca n verif y th e following :
M
E ™- = { f (7
0—C\ v
1J
'
if / c is a multipl e o f M ; otherwise.
Substituting th e abov e expressio n int o (4.20) , w e ge t M-l
^2x(z^MW£= )
MX 0(z).
£=0
Since Yd(z) =
XQ(Z) 1 th
e proo f i s complete .
Example 4. 1 Le t H(z) b e th e F I R filte r H(z) =
2-z~ 1+ z~
2
+ 4z~ 3 - 2z~
4
- 3z~
5
+ 5z "
■6
Let u s rewrit e i^(^ ) i n term s o f th e Typ e 1 an d Typ e 2 polyphas e decompo sitions wit h respec t t o M = 3 a s H(z) =
( 2 + 4 z ~ 3 + 5z~ 6 ) + z - \ - l - 2z~ 3) +
#(*) =
( 2 + 4 z ~ 3 + 5;T 6 ) + z 1(z~3 -
3z~ 6) +
z " 2 ( l - 3z~ 3), 2 2 ( - ; T 3 - 2;T
6
),
respectively. The n th e polyphas e component s are , respectively , give n b y G0(z) = Gi(z)= G2(z) =
2 + Az- 1 + 5 z ~ 2 , 5 - 1 - 22?- 1 , 5i(2? l-3z~\ S
= 2 + 4 * - 1 + 5z~ ) = z' 1 - 3z~ 2, 2(z) = 0 (^)
2
,
For th e II R case , conside r th e filte r 1 wx 1-2ZF(z) -1 * 1 +2 "
To ge t th e Typ e 1 polyphas e component s o f F(z) wit h respec t t o M = 2 , w e multiply b o t h th e numerato r an d th e denominato r b y th e t e r m ( 1 — z _1) s o t h a t th e resultan t denominato r i s a functio n o f z 2. Thu s F{z) ca n b e rewritte n as 1 + 2z- 2 l + 2z- 2 _ 3 w , l-3z~ x F z z ( )= i = = 2 1 ^ + i = 2 " 1— z ^ 1 —z^ 1 — z 2 From th e abov e equation , w e immediatel y ge t th e Typ e 1 polyphas e compo nents a s Z G0(z) = _ , G^z) = _z_1. 1
87
4.4. Polyphas e decompositio n
Similarly w e can ge t th e Typ e 2 polyphase component s o f F(z) a s S0(z)=1-±^, S
1(z)
"
3
1
"_1
Observe tha t whe n H(z) i s an FI R filte r havin g L coefficients , th e tota l num ber o f coefficients i n th e M polyphas e component s i s als o L. Th e sam e i s no t true fo r th e II R case . I f w e implemen t th e II R polyphas e component s Si(z) (or Gi(z)), th e tota l numbe r o f multiplier s i s large r tha n tha t i n F(z).
4.4.1 Decimatio
n an d interpolatio n filter s
First conside r th e decimatio n filte r reproduce d i n Fig . 4.17(a) . Not e tha t one ou t o f ever y M outpu t sample s ar e discarded . I t i s therefor e wastefu l t o compute al l th e output s o f H(z). Th e structur e fo r H(z) ca n b e rearrange d in suc h a wa y tha t non e o f th e discarde d sample s ar e compute d i n th e firs t place. Fo r thi s w e ca n us e th e polyphas e decomposition . Conside r th e Typ e 2 polyphase decompositio n H(z) =
M-l
£ z
k
Sk(zM).
k=0
We ca n redra w Fig . 4.17(a ) a s i n Fig . 4.17(b) . Th e firs t nobl e identit y (Fig . 4.5) ca n no w b e applie d t o redra w thi s i n th e for m o f Fig . 4.17(c) , whic h is calle d th e polyphas e implementatio n o f th e decimatio n filter . Not e tha t none of the compute d result s ar e discarded, an d furthermor e tha t al l the com putations (filterin g wit h Sk(z)) ar e implemente d a t th e lowe r rate , i.e . afte r the decimator . Th e polyphas e implementatio n i s therefor e computationall y very efficient . Not e fro m Fig . 4.1 7 tha t th e Typ e 2 polyphas e component s Sk(z) o f th e filte r ar e operatin g individuall y o n th e Typ e 1 polyphase com ponents Xk(n) o f the inpu t x(n) (compar e Fig . 4.15(a) ) befor e th e result s ar e added t o produc e y(n). Equivalentl y th e polyphas e component s Sk(z) operat e on th e blocke d versio n x(n ) o f th e inpu t x(n). Next conside r th e interpolatio n filte r show n i n Fig . 4.18(a) . Th e zero valued sample s inserte d b y the expande r ente r th e multiplier s f(k) an d resul t in waste d computation . Furthermor e thes e multiplier s operat e a t th e highe r rate (the y only have T/M second s per computation), wher e T i s the separatio n between inpu t samples) . T o avoi d suc h inefficien t computation s w e represen t the filte r i n it s Typ e 1 polyphase form : M-l
F(z) =
£ z-
k
Gk{zM).
k=0
Then the system ca n be redrawn a s in Fig. 4.18(b) . B y using the secon d nobl e identity (Fig . 4.5(b) ) thi s ca n b e simplifie d t o th e for m show n i n Fig. 4.18(c) . In thi s implementatio n ther e ar e n o zero-value d sample s enterin g th e filter s Gk(z). Furthermor e al l th e computation s ar e takin g plac e a t th e lowe r rat e (i.e. befor e th e expanders) . Observ e tha t th e outpu t y(n) i s th e interleaved
4. Fundamental s o f multirat e signa l processin g (a) x(n)
+\ H(z)
|M ►
W
y{n)
blocked inpu t vector x(n) x(n)
|M
M
—> | M
M
z\ r
w^
► S0(z )
Sx{z ) 1
• SuJzM)
•
x(/z) A"
zv
►
—►
|M
^
| M
ZY
T
Hi3-t
y(n)
► S o 00
xQ(n)
ii
5,(z) Xj(n)
• •
• • •
A
L-J | M | — J
M-lW
Figure 4 . 1 7 . (a ) Decimatio n filter ; (b ) Typ e 2 polyphas e representation ; (c ) simplifi cation wit h th e hel p o f th e firs t nobl e identity .
(a) x(n)
*\\M >]
F(z) ►
y(n) blocked outpu t vector y(n)
x(n)
(b) 1
fM
GizM)
fM
Gx(zM)
x(ri) Az J = kz
I ^iMl^jq^O l 1
—► i
i
L^
' -+ '
G0(z)
> |M
Gi(z)
► |M
•• •
y(n) A.-i
• •
*z~l
| G M_x(z) [_^|| M [—T
Figure 4 . 1 8 . (a ) Interpolatio n filter , (b ) Typ e 1 polyphas e representation , an d (c ) simplification wit h th e hel p o f th e secon d nobl e identity .
version o f th e o u t p u t s o f Gk(z). Thu s eac h Typ e 1 polyphas e componen t Gk (z) o f th e filte r compute s th e correspondin g Typ e 1 polyphas e componen t of th e outpu t 2/(n) , an d thes e result s ar e interleave d t o ge t y{n). In orde r t o maximiz e th e efficienc y o f computatio n i n multirat e systems , we move all decimators as far to the left as possible an d move all expanders as far to the right as possible. I n thi s way , th e computationa l buildin g block s operate a t th e lowes t possibl e rate .
4.4. Polyphas e decompositio n 8
4.4.2 Synthesi
9
s filte r bank s
We nex t conside r th e syste m show n i n Fig . 4.19 . I n thi s system , ther e ar e M interpolation filter s wit h input s Sfe(n) , an d thei r output s ar e added . I n general , the interpolatio n rati o N ca n b e differen t fro m th e numbe r o f branche s M (usually N > M i n communication s an d N < M i n othe r application s suc h a s signal compressio n an d adaptiv e filtering) . T h e syste m i s calle d a synthesi s filter ban k becaus e i t combine s a se t o f signal s Sk(n) int o a singl e signa l x{n). We no w sho w ho w thi s filte r ban k ca n b e expresse d i n polyphas e form , whic h leads t o a mor e efficien t implementation . sQ{n) - ► ■^(n) - ►
\N
—►
fJV
—►
x(n) *■<>(*)
4k
•
\J7—>
F
M-\(Z)
—>
\\N
Fy{z)
•
W n >-
[\7 —>
s0(n)
Jk *
• • •
\J
t*
i/-iW-
(b)
(a)
*0W
s0(n)
^
s,(n) -
(c)
G(z)
^
\N
-l
^z
-l
G(zN)
NJ
▲
X in)
\N
-1 ^Z
i
xx(n)
—>
x{n)
—►
-1
• • • L
**-iW ^
\
\N
1
Figure 4 . 1 9 . (a ) Synthesi s filte r bank ; (b ) polyphas e version ; (c ) simplificatio n wit h the us e o f th e secon d nobl e identity .
Assume t h a t eac h filte r F Tn(z) i s expresse d i n Typ e 1 polyphase for m wit h respect t o TV : N-l
Fm{z) =
k
Gkm(zN). (4.21
Y, z-
)
k=0
Expressing th e ban k o f M filter s a s a ro w vecto r w e therefor e ge t [ F 0(z) F
x(z)
..
. FM-I(Z)
]
=
[
1
-(JV-l) ]
G
( ^ (4.22)
4. Fundamental s o f multirat e signa l processin g
90
where G(z) i s th e N x M matri x o f polyphas e components :
G(z)
G01(z) Gn(z)
Goo(z) Gw(z)
Go,MGi,M-
(4.23)
G iV-l,M-l (*)
GW-1,0(2) GN-I,I(Z)
We sa y t h a t G(z) i s th e polyphas e matri x o f th e synthesi s filte r bank . Equa tion (4.22 ) i s sai d t o b e th e polyphas e representatio n o f th e filte r bank , an d can b e represente d a s i n Fig . 4.19(b) . W i t h th e hel p o f th e secon d no ble identit y w e ca n furthe r redra w i t a s i n Fig . 4.19(c) , whic h i s calle d th e polyphase implementatio n o f th e synthesi s filte r bank . I n thi s representatio n the inpu t vecto r s(n ) i s filtere d b y th e MIM O syste m G(z) t o produc e th e output vecto r x ( n ) , whos e component s ar e interleave d (unblocked ) t o produc e the scala r outpu t x(n).
4.4.3 Analysi
s filte r bank s
Now conside r th e syste m show n i n Fig . 4.20(a) . I n thi s system , ther e ar e M decimatio n filter s wit h a commo n inpu t x(n). Th e decimatio n rati o N is i n genera l differen t fro m M (agai n usuall y N > M i n communication s and N < M i n othe r application s suc h a s signa l compressio n an d adaptiv e filtering). Suc h a syste m i s calle d a n analysi s filte r ban k becaus e i t split s a signal x(n) int o M components . W e no w sho w ho w thi s filte r ban k ca n b e expressed i n polyphas e form . Assume t h a t eac h filte r H rn(z) i s expresse d i n Typ e 2 polyphas e form : N-l ^ Z
Hm{z) =
(4.24)
k
Smk(zN)-
fc=0
Expressing th e ban k o f M filter s a s a colum n vecto r w e therefor e ge t
H0(Z) ffi(z) I
i
_ HM-i(z) J
r
..
|
S(z»)
\
i
z
(4.25)
_ z (N-l)
where S(z) i s th e M x N matri x o f polyphas e components :
S(z)
Soo(z) Sio(z) SM-I,O(Z) SM-I,I(Z)
Soi(z) Su(z)
So,N- l(*) Sl,N- l(*) SM-
1,7V-1
(4.26)
(*)
We sa y t h a t S(z) i s th e polyphas e matri x o f th e analysi s filte r bank . Equa tion (4.25 ) i s sai d t o b e th e polyphas e representatio n o f th e filte r bank , an d can b e represente d a s i n Fig . 4.20(b) . W i t h th e hel p o f th e firs t nobl e identit y this polyphas e representatio n ca n b e redraw n a s show n i n Fig . 4.20(c) . I n this representatio n th e inpu t x(n) i s firs t blocke d int o it s Typ e 1 polyphas e components Xk(n) an d thes e ar e filtere d b y th e MIM O syste m S(z) t o produc e the outpu t vector .
91
4.5. Concludin g remark s
x(n) r
w
-+ r
H0(z) Hx(z)
k [17 IN
-+
x(n) w
z
—►
z'
\N
H HM-X®
f
• • •
S(zW)
>-
—W
\N
►
x(w)
AIL
r
1A f
z1 r
L
• •
^
• • • |JV
/
(c)
IN
W
(a)
Zi
—► r W
•
•
1
►
(b)
x0(n) ^
^ ^W ^ ^
S(z)
W")^
^/
Figure 4 . 2 0 . (a ) Analysi s filte r bank ; (b ) polyphas e version ; (c ) simplificatio n wit h the us e o f th e firs t nobl e identity .
4.5 Concludin
g remark s
Filter ban k system s hav e bee n successfull y applie d t o man y area s suc h a s sig nal compression , subban d adaptiv e filtering , an d s o forth . Reader s intereste d in a mor e detaile d t r e a t m e n t o f thes e system s ar e referre d t o [4 , 30 , 43 , 49 , 146, 159 , 169] . I n thi s book , w e focu s o n th e applicatio n o f multirat e signa l processing i n moder n communicatio n systems . I n th e nex t fe w chapter s w e shall fin d t h a t man y recen t design s o f communicatio n system s ca n b e effi ciently represente d i n term s o f th e multirat e filte r ban k languag e introduce d in thi s chapter .
4.6 Problem
s
4.1 Determin e whethe r th e decimato r an d expande r hav e on e o r mor e o f th e following properties : (a ) linearity ; (b ) causality ; (c ) tim e invariance . 4.2 Sho w t h a t th e signa l i n Fig . 4.11(a ) satisfie s th e aliasfree(M ) condition . Suggest a circui t t o recove r th e origina l signal s fro m thei r decimato r outputs.
92
4. Fundamental
s o f multirat e signa l processin g
4.3 Sho w t h a t whe n w e appl y a n M-fol d decimatio n t o th e signa l whos e Fourier transfor m i s show n i n Fig . 4.1 1 (b) , ther e i s n o aliasin g i f an d only i f c^ o i s a n intege r multipl e o f TT/M. Thi s show s t h a t decimat ing a signa l wit h a tota l bandwidt h o f 2TT/M ca n creat e aliasin g i f th e frequency band s ar e no t properl y located . 4.4 Le t H(z) =
E ^ 1 z~ kGk(zM), X(z)
=
l + z~\ an d Y(z) =
H(z)X(z).
(a) Identif y th e Typ e 1 polyphas e component s o f Y(z). (b) Identif y th e Typ e 2 polyphas e component s o f Y(z). 4.5 Fin d th e Typ e 1 an d Typ e 2 polyphas e representatio n (wit h respec t t o 3) o f th e filte r 1 wx l-z4.6 Conside r th e tw o multirat e system s show n i n Fig . P4.6 . Suppos e t h a t M = 3 , L = 2 , an d C(z) = 1 . Sho w t h a t th e tw o system s hav e th e sam e output. Thi s mean s t h a t w e ca n exchang e a two-fol d expande r an d a three-fold decimator .
\u
—►
> |M
C(z) — ► | M
► C(z)
►
n
►
—►
F i g u r e P 4 . 6 . Tw o cascade s o f multirat e buildin g blocks .
4.7 Suppos e t h a t L = 2 an d M = 3 , an d th e transfe r functio n C{z) i n Fig. P4. 6 i s a dela y z~ x. D o th e tw o system s hav e th e sam e o u t p u t ? Justify you r answer . 4.8 Le t M = N = 2 an d le t th e synthesi s filter s F 0(z) an d F 1(z) o f Fig . 4.19(a) b e relate d b y Fi(z) = FQ(—Z). HO W ar e th e Typ e 2 polyphas e components o f thes e filter s related ? Exploitin g thi s relation , find a n implementation (i n th e polyphas e form ) o f th e synthesi s filter ban k wit h the smalles t numbe r o f multipliers . 4.9 Le t M = N = 2 an d le t th e analysi s filters H 0(z) an d H^z) o f Fig . 4.20(a) b e relate d b y Hi(z) = HQ(—Z). HO W ar e th e Typ e 1 polyphas e components o f thes e filters related ? Exploitin g thi s relation , find a n implementation (i n th e polyphas e form ) o f th e analysi s filter ban k wit h the smalles t numbe r o f multipliers . 4.10 Conside r th e system s i n Fig . P4.10 . Le t Hk(z) an d F k(z) b e th e follow ing idea l filters: Hk(e*")=
F k(e*")=
i , M < M < —jir~ > 0, otherwise ,
4.6. Problem
93
s
for fc = 0 , l , . . . , M- 1 . (a) Determin e U k(ejui) i n Fig. P4.10(a) i n term s o f X(e jui
(b) Determin e Y(e ) i (a)
x(ri) w
n Fig. P4.10(b) i n term s o f X(e
(b)
1
r
-+ | M
|M
—►
**(*>
^oW
—► | M
► \M
—►
Fob)
^(z)
—► | A f
► \M
—►
Fx(z)
•
*J H M_x{z) \M\M \-^\
\M\-+\
F
).
uo(n) * » i
ux(n)
)
\'
i
•
•
•
).
ju;
%<
Hk(z)
x(«)
r
ju;
M_X{Z)
"uJn) '
Figure P 4 . 1 0 . Tw o multirat e systems .
4.11 DFT banks. Conside r t h e analysi s filte r ban k i n Fig . 4.20 . Suppos e t h a t N = M an d t h e analysi s filter s H m(z) ar e relate d b y H m(z)= H0(zWm), fo r m = 1 , 2 , . . . , M - 1 . Such a filte r ban k i s called a D FT bank. D F T bank s ca n be implemente d efficientl y usin g t h e polyphas e structure an d they ar e widely use d i n man y applications . (a) Suppos e t h a t HQ(Z) is a n idea l lowpas s filte r wit h t h e passban d [—7r/M, 7r/M] . Plo t t h e magnitud e response s o f H k(z) fo r 0 < k <M - 1 . (b) Le t t he Type 2 polyphase decompositio n o f HQ(Z) wit h respec t t o Mbe M-l
H0(z) = J2 z kSk(zM). k=0
Write t h e Typ e 2 polyphas e decompositio n o f H m(z) i n term s o f Sk(z). (c) Le t h(z) =
[H 0(z) H
±(z)
. . . H M-i(z)]T. Sho
w tha t
h(z)=WD(zM)e(z), where W i s t he M x M normalize d D F T matrix wit h (/c , Z)th entr y (l/y/M)e~i27rkl/M, e(z) = [lz . . . z 1^'1}7\ an d D ( z) is a diagona l matrix. W h a t ar e t he diagonal entrie s o f D ( z ) ? (d) Suppos e t h a t HQ(Z) is an Lth-orde r F I R filter. Usin g t h e result i n (c), sugges t a low-cos t implementatio n o f t h e analysi s filte r ban k shown i n Fig . 4.20(c) . Sho w t h a t t h e implementatio n cos t o f the analysi s filte r ban k i s equa l t o t h a t o f HQ(Z) plu s a n M x M
94
4. Fundamental s o f multirat e signa l processin g
DFT matrix , whic h ca n b e implemente d efficientl y usin g FF T al gorithms. Similarly, whe n th e synthesi s filters hav e th e frequenc y shif t property , the filter ban k i s calle d a DF T ban k an d i t ca n als o b e implemente d efficiently usin g th e polyphas e structure .
5 Multirate formulatio n o f communication system s In thi s chapter , w e giv e th e formulatio n o f som e moder n communicatio n sys tems usin g th e multirat e buildin g blocks . Unde r th e multirat e framework , we sho w ho w t o inser t redundanc y a t th e transmitte r sid e t o hel p th e tas k o f equalization. Thes e formulation s ar e fundamenta l t o man y o f ou r discussion s in futur e chapters . W e als o sho w t h a t th e multirat e framewor k ca n b e use d to simplif y th e stud y o f th e so-calle d fractionall y space d equalize r syste m i n which th e receive r tak e sample s a t a highe r rate .
5.1 Filte
r ban k transceiver s
Figure 5. 1 show s a syste m calle d th e filte r ban k transceive r (transmitte r an d receiver). Suc h a syste m i s als o know n a s a transmultiplexer . Thi s syste m and it s variation s wil l b e centra l t o man y o f our discussion s i n thi s book . Fro m the figure w e realiz e t h a t th e transmitte r i s a synthesi s filter ban k wherea s the receive r i s a n analysi s filter ban k (se e Sectio n 4.4) . T h e filters Fk(z) ar e known a s th e transmittin g filter s an d Hk(z) ar e calle d th e receivin g filters . W h e n al l thes e filters ar e FIR , w e sa y t h a t th e transceive r i s a n F I R filter bank transceiver ; otherwis e i t i s calle d a n II R filter ban k transceiver . T h e M inpu t signal s Sk(n) ar e usuall y symbo l stream s (suc h a s PA M o r QA M signals, se e Sectio n 2.2) . Thes e coul d b e symbol s generate d b y differen t user s t h a t shar e th e sam e communicatio n channel . O r the y coul d b e differen t part s of th e signal s generate d b y on e user . T h e numbe r M i s als o calle d th e numbe r of subchannel s o r bands . T h e symbo l stream s Sk(n) ar e passe d throug h th e transmitting filters Fk(z) an d th e result s ar e adde d t o for m th e t r a n s m i t t e d signal M-l
( ) ^2 ^2 s k(i)fk(n - iM).
x n =
k=0 i
Figure 5. 2 show s th e actio n o f th e transmittin g filter Fk(z) o n th e kth inpu signal Sfe(n) . I t take s eac h sampl e o f Sk(n) an d "put s a puls e fkiji) aroun it." Thu s th e filters Fk(z) ar e als o calle d pulse-shaping filters. Figur e 5.2(b shows a n exampl e o f th e shape d outpu t whe n Fk(z) i s a lowpas s filter. T h 95
t d ) e
5. Multirat
96
s0(n) - ► \M
—►
F0®
s^n) -+> \M
—►
F^iz)
noise q(n)
uQ(n) x{n)
C(z)
rw channel r
i iw
ux(n)
H0(z)
—> Hx(z) 1
i k>
•
•
e formulatio n o f communicatio n system s
—> | M
*0(»)
^
s^n)
•
•
U
W B >-
| M -+
F
M M) f
z
^HM_X(Z)\-+\JM
M-l( )
transmitting filters
|M
W*>
receiving filters
Figure 5 . 1 . Th e M-ban d filte r ban k transceive r system .
(a) s
sk(n-l) ^
♦f
^T
H H F k&
lowpass example
il
W i^W" ' sAn-\)
(d) , y\
M
sample s of lowpass^(n )
s (n)
\ Jf
(b)
Ht
k(n)
/
N
s
n
k(
) envelop
'■ f
\
(c)
e of bandpass/ (n)
/'
o ' -M v '
\.
Wt\
complex bandpass example
►
real bandpass \Fk\ exampl e
(e
)
Figure 5.2 . (a ) Puls e shaping filter; (b ) exampl e where Fk(z) i s lowpass; (c ) example s of lowpas s an d bandpas s Fk(z)\ (d ) exampl e wher e Fk(z) i s real-coefficien t bandpass ; (e) respons e o f a rea l bandpas s filter .
filter Fk(z) ca n als o b e a comple x bandpas s filter , a s show n i n par t (c) , o r a real bandpas s filter , a s show n i n par t (e) . Par t (d ) show s a n exampl e o f th e shaped signa l whe n fk(n) i s a real-coefficien t bandpas s filter . In Fig . 5.1 , th e outpu t x(n) o f th e transmittin g filte r ban k i s sen t ove r an LT I channe l wit h transfe r functio n C(z) an d additiv e nois e q(n). A t th e receiver end , th e filter s Hk(z) hav e th e tas k o f separatin g th e signal s an d reducing the m t o th e origina l rate s b y decimation . Th e transmitte d symbol s
5.1. Filte r ban k transceiver s
97
Sfe(n) ar e the n identifie d fro m th e signal s Sfe(n) , whic h i n genera l ar e distorte d versions o f the symbol s Sk(n) becaus e o f the combine d effect s o f channel, noise , and th e filters. T h e genera l goa l i s t o identif y th e symbol s Sk(n) accuratel y i n the presenc e o f these distortions . T h e choic e o f filters {Fk(z), H m(z)} depend s on th e detail s o f th e specifi c instance s o f application . W e wil l discus s detail s of thi s throughou t th e book .
5.1.1 Th
e multiplexin g operatio n
We ca n alway s regar d th e se t o f signal s Sk(n) a s th e polyphas e component s of a hypothetica l signa l s(n) , a s demonstrate d i n Fig . 5.3(a ) fo r M = 3 . T h a t is , w e ca n regar d Sfe(n ) a s component s o f a time-multiplexe d signa l s(n). O n th e othe r hand , wit h th e filters Fk(z) chose n a s a contiguou s se t of bandpas s filters (Fig . 5.3(b)) , w e ca n regar d th e transmitte d signa l x{n) as a frequency-multiplexe d versio n o f th e se t o f signal s Sk(n). T o se e thi s consider a n arbitrar y si(n) wit h a Fourie r transfor m a s demonstrate d i n Fig . 5.3(c). T h e expande r squeeze s thi s Fourie r transfor m b y a factor o f M, an d th e interpolation filter F\(z) retain s onl y on e cop y (Fig . 5.3(d)) . Thu s th e outpu t x(n) o f th e synthesi s ban k ha s a Fourie r transfor m a s show n i n Fig . 5.3(e) ; it i s jus t a concatenatio n o f squeeze d version s o f Sk(e^) fo r 0 < k < M — 1. Thus x(n) i s th e frequency-multiplexe d versio n o f {sk(n)}. T h e earlies t application s o f th e transmultiplexer s wer e indee d conversion s from time-divisio n multiplexin g ( T D M ) t o frequency-divisio n multiplexin g ( F D M ) o f telephon e signal s prio r t o thei r transmissio n [11 , 99]. T h e receive d signal i s the n converte d fro m a n F D M bac k t o a T D M format . S o transmulti plexers ar e ofte n referre d t o a s T D M t o F D M (an d vic e versa ) converters . T h e transmultiplexer i n Fig . 5. 1 i s sai d t o b e perfec t if , i n th e absenc e o f noise , Sfc(^) = Sk(n) fo r 0 < k < M. T h e theor y o f F I R perfec t transmultiplexer s was first studie d i n [168] , an d th e conditio n o f perfect transmultiplexe r wa s de rived fo r th e cas e o f C(z) = 1 . W h e n th e channe l C(z) i s frequency-selective , the desig n o f perfec t transmultiplexer s become s muc h mor e complicated . A s we shal l se e i n Chapte r 10 , n o F I R perfec t transmultiplexer s exis t whe n th e channel i s frequency-selective . T o ai d th e desig n o f transmultiplexe r systems , redundant sample s ar e ofte n introduce d t o th e transmittin g side . Thi s topi c will b e investigate d next .
5.1.2 Redundanc
y i n filte r ban k transceiver s
In wideban d communications , th e channe l C(z) i n genera l introduce s som e degree o f distortio n t o th e transmitte d signal . T o ai d th e tas k o f equalization , redundancy i s usuall y embedde d i n th e transmitte d signals . T o d o thi s w e ca n use a filter ban k transceiver , o f whic h th e numbe r o f subchannel s o r band s i s smaller t h a n th e interpolatio n ratio . Figur e 5. 4 show s suc h a system , wher e N > M. T o understan d ho w redundanc y i s embedde d i n th e t r a n s m i t t e d sequence x(n), suppos e t h a t th e samplin g perio d o f th e D / C converte r i s T.
5. Multirat
98
_u_ s0 s
(a)
x
2
<
(
e formulatio n o f communicatio n system s
<>
<>
>
t
0 1 23
4
5
_L__t_ 6
•
7
<> 4
JJ _U_
•• • ■►
i
Fo
FM-\
Fi
fW\ \y)
0
^^
1
2TI/M
^
2n
CO
2n c
o
2n c
o
output o f Fx
^
(d) 2n/M frequency multiplexe d signa l
X(^ro)
Ua (e)
Fo
•• •
U I U,
2 M ^ 2n/M
Figure 5.3 . Th e frequency-multiplexin g operatio n performe d b y the transmultiplexer . (a) Origina l T D M signal ; (b ) filte r ban k response ; (c ) Fourie r transfor m o f a signa l s i ( n ) ; (d ) outpu t o f th e correspondin g filte r F\(z)\ (e ) FD M signa l X(e juJ).
5.1. Filte r ban k transceiver s
s
oW ~+\i
N
s^n) -^ J f TV
W
11
n ^0 ^ i , u
11 H ^i
99
n(")
channel
/x
ivO
W
noise q(n
x(ri)
wZ
7 f_
-►| i/^z ) |-^ | | 7 V | — ► ^ r
W
•• •• ••
•• • W » ) - * | ft f
w £ 7 /- ^ w 1 A rw ? (V A —^| n 0 ^z; |-^- | + 7V | ^ • > QK"J r
\
*| ^Af-l W |
U|^l(z)|_^|^|_^^B
\
receiving filters
transmitting filters
Figure 5.4 . Th e M-ban d o r M-subchanne l filte r ban k transceive r syste m wit h redun dancy (J V > M).
Then, du e t o th e AT-fol d expander , th e modulatio n symbol s Sk(n) ar e space d apart b y NT seconds . Ther e ar e M informatio n carryin g symbol s i n on e inpu t block, bu t th e transmitte r send s ou t N sample s o f x(n) ove r th e channe l i n NT seconds. I n othe r words , fo r ever y bloc k o f M inpu t symbols , th e transmittin g filter ban k send s ou t N sample s o f x(n). T h e numbe r o f redundan t sample s inserted i s (N — M) pe r inpu t block . I f eac h modulatio n symbo l Sk(n) carrie s b bit s o f information , the n Mb bit s ar e sen t fo r ever y N sample s o f x(n) transmitted. T h e transmissio n rat e o f th e syste m i s give n b y
n
Mb ~N T
bits pe r secon d (bps) .
T h e transmissio n rat e ha s bee n reduce d b y a facto r o f N/M du e t o th e in troduction o f redundan t samples . T o compensat e fo r thi s reduction , w e ca n increase th e samplin g frequenc y o f th e D / C an d C / D converter s b y a facto r of N/M. Thi s mean s t h a t th e channe l bandwidt h ha s t o b e increase d b y a factor o f N/M. Fo r thi s reaso n th e rati o ( = N/M is als o know n a s th e bandwidt h expansio n ratio . I n practice , th e numbe r of redundan t sample s pe r bloc k (N — M) i s usuall y muc h smalle r t h a n M s o t h a t £ i s clos e t o one . We ca n als o understan d fro m a frequenc y domai n viewpoin t ho w th e in serted redundan t sample s coul d hel p i n th e equalizatio n process . Not e t h a t if th e filter s Fk(z) i n Fig . 5.3(b ) ar e no t perfectl y bandlimite d t o a widt h o f 2TT/M the n th e Fourie r transform s Uk(e^ UJ) wil l contai n leakag e fro m th e mul tiple image s create d b y th e expanders . Thu s th e spectr a Uk{e^) wil l overlap , and canno t i n genera l b e separate d perfectl y b y th e receivin g filter s Hk(z). Thus th e kth outpu t Sfc(n ) contain s contribution s fro m Si(n) fo r z ^ f c , whic h results i n interference . A simpl e wa y t o avoi d thi s i s t o hav e nonoverlappin g Uk(e^). Thi s ca n b e achieve d b y makin g th e interpolatio n rati o A " large r t h a n th e numbe r o f subchannel s M. T h e spectr a o f Sk(n) ar e no w squeeze d
100
5. Multirat
e formulatio n o f communicatio n system s
by a large r facto r N, an d th e M filter s Fk(e^) hav e a smalle r bandwidt h 2TT/N, whic h allow s a ga p betwee n th e filter s calle d guar d band s (Fig . 5.5) . As Uk(e^) ar e separate d b y th e guar d bands , w e ca n easil y remov e th e in terference fro m Si(n) foiiy^k b y designin g a receivin g filte r Hk(z) t h a t ha s the sam e passban d a s Fk(z). guard band s M) '
0
^
^
1I
,»
M
FM-\
r 2lZ
®
27l/i\T
Figure 5.5 . Illustratio
)
n o f guar d band s i n a redundan t transceiver .
Filter ban k transceiver s wit h N = M ar e calle d minima l transceivers , and thos e wit h N > M ar e calle d redundan t transceivers . I n practic e the filter s ar e no t perfectl y bandlimitin g filters ; i n fact , filter s o f fairl y smal l orders ar e used . I n suc h system s th e introductio n o f redundanc y i s stil l usefu l for th e purpos e o f channe l equalization . Thi s wil l b e explaine d i n mor e detai l in futur e chapters .
5.1.3 Type
s o f distortio n i n transceiver s
T h e receive d signal s Sfc(n ) i n genera l ar e differen t fro m Sk(n) fo r severa l rea sons. (1) First , ther e i s inte r subchanne l interference . Thi s mean s t h a t %(n) i s affected no t onl y b y Sfc(n) , bu t als o b y s m ( n ) , r a ^ k. (2) Second , fo r an y fixe d k, th e signa l Sfc(n ) depend s als o o n Sk(n — m) fo r m 7 ^ 0. Thi s i s du e t o th e effec t o f filterin g create d b y th e channe l an d the variou s filters . Thi s i s calle d intr a subchanne l interference . (3) Finall y ther e i s additiv e channe l noise.
1
T h e tas k a t th e receive r i s t o minimiz e th e effect s o f thes e distortion s s o t h a t the t r a n s m i t t e d symbol s Sk (n) ca n b e detecte d fro m s/ e (n) wit h a n acceptabl e low probabilit y o f error . W e wil l se e late r t h a t thi s tas k i s easie r whe n ther e is redundanc y (i.e . N > M). W e shal l presen t a quantitativ e stud y o f inte r subchannel an d intr a subchanne l interference s i n Sectio n 5.2 . 1 There ar e othe r source s o f errors , suc h a s th e nonlinea r distortio n i n analo g circuits , synchronization error , carrie r frequenc y offset , an d s o on , whic h ar e no t considere d here .
101
5.2. Analysi s o f filte r ban k transceiver s
5.2 Analysi
s o f filte r ban k transceiver s
In thi s sectio n w e sho w t h a t th e filte r ban k transceive r ca n b e represente d in a for m t h a t doe s no t requir e th e us e o f multirat e buildin g blocks . Suc h a representation give s th e channe l C(z) i n th e for m o f a specia l transfe r matri x called th e pseudocirculant . Thi s alternativ e mathematica l formulatio n i s very usefu l i n th e theoretica l stud y o f communicatio n systems . Belo w w e wil l carry ou t th e analysi s fo r th e redundan t filte r ban k transceiver s (N > M). T h e minima l transceive r cas e ca n b e obtaine d b y settin g N = M.
5.2.1 ISI-fre
e filte r ban k transceiver s
Consider th e p a t h fro m s m ( n ) t o s)fc(n ) i n Fig . 5.4 . I n th e absenc e o f channe l noise thi s p a t h i s simpl y a transfe r functio n sandwiche d betwee n a n expande r and a decimator , a s show n i n Fig . 5.6(a) . B y usin g th e polyphas e identit y (Fig. 4. 6 an d Theore m 4.1) , w e se e t h a t thi s p a t h ha s th e transfe r functio n Tkm(z)=
[H k{z)C{z)Fm(z)\iN. (5.1
)
Ignoring th e channe l nois e fo r a moment , th e transceive r ca n b e describe d s (n)
\N
Fm(z)
C(z)
Hk{z)
\N
h(n)
(a) (b)
s (n)
Tbniz)
sk(n)
s0(n) s An)
> s\(n)
W")
>W*>
(c)
Figure 5.6 . (a ) Th e pat h fro m th e m t h inpu t t o th e /ct h outpu t o f th e transceiver ; (b) th e equivalen t syste m wit h nois e ignored ; (c ) matri x representatio n o f the filte r ban k transceiver i n Fig . 5.4 , wit h nois e ignored .
as a simpl e transfe r matri x T(z). T h e off-diagona l element s o f th e matrix , namely Tkm(z),k ^ m , represen t th e p a t h fro m th e m t h inpu t t o th e /ct h output. I f th e matri x i s diagonal , t h a t is ,
Tkm(z)=0, k^m,
102
5. Multirat
e formulatio n o f communicatio n system s
then ther e i s no inter subchannel interference. W i t h inte r subchanne l interfer ence eliminated , eac h diagona l elemen t T kk(z) represent s th e transfe r functio n from Sk(n) t o s~ k(n). If , i n addition , Tkk(z) =
1,
then Sfc(n ) i s affecte d b y onl y a singl e sampl e o f s k(n) (e.g . s k(n — 1 ) doe s not affec t it) . Ther e i s no intra subchannel interference i n th e /ct h p a t h . Thu s the syste m i s fre e fro m b o t h interference s i f
T(z) =
I,
and w e se e t h a t ther e i s perfec t symbo l recover y o r th e ISI-fre e propert y %(n) =
s k(n) (5.2
)
in th e absenc e o f noise . Mor e generall y i n a n ISI-fre e transceiver , T(z) i s allowed t o b e a diagona l matri x wit h diagona l element s a kz~nk fo r som e integer n k an d som e a k ^ 0 , s o t h a t = Sfc(rc) a
ksk(n-nk).
W h e n th e transceive r i s ISI-free , th e channe l nois e i s th e onl y sourc e o f im perfection, whic h make s s~ k(n) differen t fro m s k(n). Nois e wa s ignore d i n th e preceding discussion , bu t w e shal l handl e i t wit h grea t car e i n late r chapter s where w e discus s differen t type s o f receiver s suc h a s th e zero-forcin g receiver , the MMS E receiver , an d th e minimu m B E R receiver . Interblock interferenc e (IBI ) Fro m th e abov e discussion , w e se e t h a t , although ther e ar e multirat e buildin g block s i n a filte r ban k transceiver , th e system fro m th e inpu t bloc k s(n ) t o th e outpu t bloc k s(n) i s i n fac t LT I wit h transfer functio n T ( z ) . Suppos e t h a t th e transceive r i s no t ISI-fre e an d i t ha s an F I R transfe r matri x T(z) = J2 k= o T kz~k. The n th e outpu t vecto r s(n) i s given b y s(n) = T 0 s ( n ) + T i s ( n - 1 ) + • • • + T j S ( n - J ) . T h e outpu t bloc k become s a mixtur e o f mor e t h a n on e inpu t block . Thi s phenomenon i s calle d interbloc k interferenc e (IBI) . W h e n T(z) = T 0 , th e output bloc k s(n) depend s onl y o n th e inpu t bloc k s ( n ) . I n thi s case , w e sa y t h a t th e transceive r i s IBI-free . Interchanging th e transmittin g an d receivin g filter s Suppos e w e inter change th e transmittin g filter s F k(z) an d th e receivin g filter s H k(z) an d le t T'f^z) b e th e transfe r matri x o f th e ne w transceive r system . Then , fro m (5.1) , we hav e TLW = [Fk(z)C(z)H m{z)} = T mk{z). iN In othe r words , T'(z) = T T(z). I n particular , thi s implie s t h a t interchanging the transmitting and receiving filters does not affect the IB I-free and ISI-free properties.
5.2. Analysi s o f filte r ban k transceiver s
5.2.2 Polyphas
103
e approac h
From th e precedin g discussions , w e kno w t h a t , i n th e absenc e o f channe l noise, th e transceive r i s an MIM O (multi-inpu t multi-output ) LT I syste m wit h transfer matri x T ( z ) , eve n thoug h i t ha s multirat e buildin g blocks . Belo w w e will redra w th e filter ban k transceive r i n a for m t h a t doe s no t requir e th e use o f an y multirat e buildin g blocks . T o deriv e suc h a representation , w e implement th e transmittin g an d receivin g filter bank s usin g th e polyphas e structures (Sectio n 4.4) . Ignorin g th e channe l noise , th e transceive r i n Fig . 5.4 ca n b e redraw n a s i n Fig . 5.7 . T h e N x M matri x G(z) i s know n a s th e transmitting (polyphase ) matrix , wherea s th e MxN matri x S(z) i s called th e receiving (polyphase ) matrix . Thes e matrice s ar e relate d t o th e transmittin g and receivin g filters b y
G(z)
G00(z) Gw(z)
G0i(z) Gu(z)
GO,MG\M- - i ( * )
G 7V-1,M-1 (*)
GN-I,O{Z) GN-I,I{Z)
and
Soo{z) S(z)
SQI(Z)
Sn(z)
Sio(z)
sM - l .0(2) S.
M-1,1
SlM-
S,M-l.N-l
(*)
(z)
where Gkm(z) i s th e kth Typ e 1 polyphas e componen t o f F Tn(z) an d Skm(z) is th e 772t h Typ e 2 polyphas e componen t o f Hk(z). T h e transmittin g an d receiving filters are , respectively , relate d t o th e transmittin g an d receivin g polyphase matrice s a s follows :
[ F 0(z) F!(z) and
^M-l
H0(z) Hi{z) H M-l
-N+l
[1
(z)
S(z»)
] G(z N)
1 z VN-1
(z)
Consider th e syste m show n i n th e gra y bo x i n Fig . 5.7 . W e wil l se e late r t h a t thi s i s a n LT I syste m wit h a n i V x i V transfe r matri x C ps(z) whic h de pends onl y o n th e scala r transfe r functio n C(z) an d th e intege r AT , bu t no t the transceive r used . Becaus e th e matri x C ps(z) operate s o n th e blocke d ver sion o f th e channe l inpu t x(n) t o produc e th e blocke d versio n o f th e channe l o u t p u t r ( n ) , w e sa y t h a t C ps(z) i s th e blocke d versio n o f th e channe l C{z). We wil l se e i n Sectio n 5. 3 t h a t th e transfe r matri x C ps(z) ha s a for m calle d the pseudocirculan t form . T h e subscrip t "ps v i s a reminde r o f "pseudocircu lant." Thu s th e entir e transceive r syste m ca n b e draw n a s a cascad e o f M I M O systems, a s show n i n Fig . 5.8 . Fro m th e figure w e se e t h a t th e overal l syste m from th e inpu t vecto r s(n ) t o th e outpu t vecto r s(n) ha s th e transfe r matri x
T(z) =
S(z)C
ps(z)G(z).
104
5. Multirat
xQ(n)
s0(n) - *
x An)
s^n) —> • • •
U
X{ri)
i
fN
e formulatio n o f communicatio n system s
—► A
k
• • •
■
1r
—► ^r
-I
* z
L
fN ►
\
C(z)
-i * ^Z Z
>
r{n)
IN
r0(n) ^
1"
r An)
c^oo
—►s1(«)
w
• • •
• • •
±N
blocked channe l
—► s (/f )
r
N-l W ^
X
Figure 5.7 . Th e M-ban d transceive r syste m i n polyphas e form .
s{(n)
r0(n) ^
x0(n)
s0(n)
* 0 (")
rx(ri)
G(z)
WB>
Cps(z)
• • •
^(AI)
i-iW W
transmitting polyphase matri x
S(z)
• • •
W
receiving polyphase matri x
Figure 5.8 . Th e M-ban d transceive r syste m i n matri x for m wit h transmittin g an d receiving polyphas e matrice s indicated .
In general , T(z ) i s a matri x dependen t o n z. Fro m thi s w e ca n com e t o a number o f conclusions : (1) inte r subchanne l interference i s eliminated i f and only if S(z)G is diagonal ;
ps(z)G(z)
(2) interbloc k interferenc e i s eliminate d i f an d onl y i f S(z)G ps(z)G(z) i constant matrix ;
sa
(3) th e transceiver has the ISI-free propert y if and only ifS(z)G ps(z)G(z) = I. Thi s conditio n i s equivalen t t o s(n) = s(n ) i n th e absenc e o f noise . More generally , sinc e Sfc(n ) = akSk(n — rik) i s acceptabl e fo r nonzer o a^, th e produc t S(z)G ps(z)G(z) ca n b e a diagonal matri x wit h nonzer o diagonal dela y elements .
5.2. Analysi s o f filte r ban k transceiver s
105
It turn s ou t t h a t th e ISI-fre e conditio n i s easie r t o satisf y wit h a redundan t filter ban k transceive r (N > M ) , a s w e shal l elaborat e later . Block transceiver s W h e n th e polyphas e matrice s G(z) an d S(z) ar e b o t h constant matrice s independen t o f z, th e filte r ban k transceive r i s called a bloc k transceiver. I n thi s case , th e processin g a t b o t h th e transmitte r an d receive r is don e i n a bloc k b y bloc k manner : x(n) =
G s ( n ) an d s(n) =
Sr(n) .
As th e polyphas e component s ar e constant , th e length s o f th e transmittin g and receivin g filter s d o no t excee d th e interpolatio n rati o N. T h e system s t o be discusse d i n Chapter s 6 , 7 , an d 8 ar e suc h bloc k transceivers . I n practice , the polyphas e matrice s ar e ofte n chose n t o b e matrice s t h a t allo w a low-cos t implementation, e.g . D F T matrices . Bloc k transceiver s wit h D F T polyphas e matrices ar e referre d t o a s DFT-base d transceivers . Man y moder n communi cation system s fal l int o thi s categor y (se e Chapte r 6) .
5.2.3 Channel-independen
t ISI-fre e filte r ban k transceiver s
Generally speaking , a n ISI-fre e filte r ban k transceive r i s channel-dependent . T h a t is , th e transmittin g filter s Fk(z) an d th e receivin g filter s Hk(z) depen d on th e channe l coefficient s c(n). I n man y applications , especiall y thos e in volving wireles s communications , th e channel s ar e ofte n time-varying . W h e n the channe l c(n) changes , w e hav e t o redesig n Fk(z) an d Hk(z) s o t h a t th e transceiver remain s ISI-free . Fo r thes e applications , i t i s desirabl e t o hav e ISI-free transceive r system s t h a t ar e channel-independent . I n particular , w e would lik e t o hav e a transceive r t h a t ca n b e ISI-fre e fo r an y z/th-orde r chan nel. Fro m earlie r discussions , w e kno w t h a t th e transfe r functio n fro m th e rath inpu t s m ( n ) t o th e kth outpu t i s give n b y Tkm(z) i n (5.1) . Fo r a n F I R channel o f orde r z/ , w e ca n writ e Tkm(z) a s V
Tkm(z)=
J2< n) [z~ n=0
n
Hk{z)Fm{z)]iN .
T h e channel-independen t ISI-fre e propert y implie s t h a t fo r an y c(n) , th e transfer functio n Tkm(z) = akZ~ Dk5(m — k). Thi s i s tru e i f an d onl y i f [z-nHk(z)Fm= (z)]iN
p knz-D"S(m -
k),
for n = 0 , 1 , . . . , v. Substitutin g thi s resul t int o th e expressio n fo r Tkm{z)-> find t h a t th e M h subchanne l gai n i s give n b y
we
n=0
In Chapte r 6 , w e wil l stud y a numbe r o f transceive r system s wit h suc h a channel-independent ISI-fre e property . W e wil l se e t h a t thes e system s belon g to th e clas s o f DFT-base d bloc k transceivers . T h e firs t non-DFT-base d bloc k transceivers wit h suc h a propert y wer e derive d b y Scaglion e an d Giannakis ,
106 5
. Multirat
e formulatio n o f communicatio n system s
and th e syste m i s called a Lagrange—Vandermonde (LV ) transceive r [132] . I n a L V transceiver , th e transmi t filter s ar e Lagrang e interpolatio n polynomi als, wherea s th e receiv e filter s ar e Vandermond e filters . A dua l syste m calle d a Vandermonde—Lagrang e (VL ) transceiver , wher e th e transmi t filter s ar e Vandermonde filter s an d th e receiv e filter s ar e Lagrang e filters , wa s derive d in [133] . Bot h th e L V and V L system s includ e th e widel y know n DFT-base d transceivers a s a special case . I n [46] , the L V and V L transceivers wer e gener alized t o cas e o f multiuse r transmission . I t wa s late r show n [84 ] that th e L V and V L system s ar e the onl y solution s o f block transceiver s wit h th e channel independent ISI-fre e property . Fo r filte r ban k transceiver s wit h transmittin g and receivin g filter s longe r tha n TV , the solutio n o f channel-independen t ISI free transceiver s i s stil l unknown . I n [117] , a metho d ha s bee n propose d fo r the desig n of near-ISI-free channel-independen t filte r ban k transceivers . Filte r bank transceiver s wit h hig h signa l t o interferenc e rati o an d goo d frequenc y response hav e bee n successfull y designe d therein .
5.3 Pseudocirculan
t an d circulan t matrice s
In thi s section , w e shall stud y tw o specia l classe s o f matrices, know n a s pseu docirculant an d circulan t matrices , whic h appea r frequentl y i n th e stud y of transceive r systems . Thes e matrice s ar e ver y usefu l i n th e discussio n o f transceiver system s usin g a matri x representation .
5.3.1 Pseudocirculant tems
s an d blocke d version s o f scala r sys -
Definition 5. 1 Pseudocirculan t matrice s A n N x N matri x C ps(z) i s sai d to b e a pseudocirculan t i f an y colum n (excep t th e leftmos t colum n whic h i s arbitrary) i s obtained fro m th e precedin g colum n b y performing th e followin g operations: (a ) shif t dow n b y on e element , (b ) recirculat e th e spille d elemen t to th e top , an d (c ) multipl y th e recirculate d elemen t b y th e dela y operato r
z~\ ■
Let [ CQ(Z) C±(Z) ••• CN-I(Z) ] b e th e firs t colum n o f C ps(z). Then on e ca n obtai n C ps(z) b y followin g th e mechanis m describe d i n th e above definition . Followin g th e procedure , w e ca n find th e followin g genera l form o f th e pseudocirculan t matri x C ps(z): rC
0(z)
Z-'CN-^Z)
Ci(z) C C2(z) d(z)
CN-2(z) C
[CN-^Z) C
Z^CN^Z) (z) Z^CN-^Z) 0
C
N-3(z)
N-2(Z)
C
C
■■■
■■■ (z) •• 0 N-i(z)
N-3(Z)
■■■ ••
z- 1C2(z) z^C^z)
1
z^diz) z-^iz) • z^C^z) z^Caiz) C • d(z
0(z)
Z^CN-^Z) )C
0(z)
J
(5.3) Note that alon g any lin e parallel to the mai n diagonal , al l elements ar e identi cal. Thu s a pseudocirculant i s also a Toeplit z matrix . Othe r mino r variation s
107
5.3. Pseudocirculan t an d circulan t matrice s
are possibl e i n th e definitio n o f pseudocirculants . Fo r example , w e ca n shif t a colum n up an d the n circulat e th e spille d elemen t wit h z~ x. Definitio n 5. 1 will b e use d throughou t thi s book . Pseudocirculant matrice s wer e originall y introduce d i n th e filter ban k lit erature i n th e contex t o f aliasfre e filter bank s an d blocke d LT I system s (se e [157] an d [159]) . I n ou r context , thei r importanc e arise s fro m th e fac t t h a t the blocke d channe l matri x C ps(z) i s a pseudocirculan t matri x o f exactl y th e form define d above . Thi s i s state d i n th e followin g theorem . x0(n)
w
\N
w
fJV
x(n) i
xx(n)
• •
W")
\N
—► A
C(z)
r(n)
\N
> r W
Q(n)
~^> \N
>W r
x(n)
w
zy
r
.z-1
z'
f
Li
• •
U|7^"
-+ 'N- ,(»)
Figure 5.9 . N x N blocke d versio n o f a channe l C(z
Theorem 5. 1 Blocke d version s an d pseudocirculant s Conside r th e A - i n p u t iV-output syste m show n i n Fig . 5.9 . Le t th e Typ e 1 polyphase representatio n of th e channe l b e C 0(zN) +
C(z) =
C 1{zN)z~1 +
N
• • •+ C
)z-N+1.
N-1(z
T h e n th e A x A syste m i s LT I an d it s transfe r matri x i s th e pseudocirculan t matrix give n b y C ps(z) i n (5.3) . ■
fN
Ciz)
Figure 5 . 1 0 . Pat
h fro m Xi{n) t o rj(n) o
x. in)
zJ
-+ r
|N
j(n)
f th e blocke d syste m i n Fig . 5.9 .
A sketc h o f th e proo f i s a s follows . Le t u s loo k a t th e syste m fro m Xi(n) to rj(n) i n Fig . 5.9 . I t i s simpl y z^~ lC{z) sandwiche d betwee n a n A-fol d expander an d a n AT-fol d decimato r (Fig . 5.10) . Accordin g t o th e polyphas e identity (Fig . 4.6) , thi s syste m i s LT I wit h transfe r functio n [z J ' - z C(z)] . Using th e polyphas e decompositio n o f C(z), i t i s no t difficul t t o sho w (Prob lem 5.2 ) t h a t 73~i
C(z)]IN
Cj-i(z),
if j-i>0;
CN+J-I(Z)I otherwise
,
(5.4)
108
5. Multirat
e formulatio n o f communicatio n system s
which proves that th e transfe r matri x o f the TV-inpu t TV-outpu t syste m i n Fig. 5.9 ha s th e pseudocirculan t for m C ps(z) i n (5.3) . Assume tha t th e channe l i s FI R wit h orde r is:
C(z)=J2c(^ i=0
For example , whe n v = 4 an d N = 3 w e hav e Co(z) = c(0 ) + c(3)z _ 1 , d(z)= c(l ) + c(4)z _ 1 , an d C 2{z) = c(2) . Substitutin g thes e int o (5.3) , th e pseudocirculant matri x i s c ^ + c^)*" c(2)
Cp S (z)
1
cfflz-1
c(l)z-1+c(4)z - 2 1 c(0) + c(3)z "- c ^ ) * " 1 c(l) + c(4> -- c(0 ) + c(3)z " 1 1
d i V = 6 , w e hav e
When
" c(0 ) 0 c(l) c(0 ) c(2) C (l) 0 c(2 ) 00 00
Cp S (z)
00 00 c(0) 0 c(l) c(0 ) c(2) c(l ) c(2)
^-^(2) ^-^(1 ) " 0 2-^(2 ) 00 00 c(0) 0 c(l) c(0 )
In man y practica l applications , th e bloc k siz e N i s usuall y muc h large r tha n the channe l order . Th e polyphas e component s o f C(z) wit h respec t t o N ar e all constan t (independen t o f z), an d onl y th e first (is + 1 ) polyphas e compo nents ar e nonzero . I n thi s case , C ps(z) ha s th e specia l for m c(0) 0
0
-^(i/)
-^(1)
c(l) c(0 ) z 1 c(z/) 0
c(i/)
(5.5)
Cp S (z)
0
c(0)
0
c(l)
c(0)
c(z/) c(z / — 1 )
0
c(0)
Observe tha t onl y th e is x is submatrix a t th e to p righ t corne r depend s o n z. W e wil l exploi t thi s propert y o f C ps(z) fo r IB I elimination . Som e deepe r properties o f pseudocirculan t matrice s wil l b e explore d i n Chapte r 10 .
5.3.2 Circulant
s an d circula r convolution s
If we substitute z = 1 into a pseudocirculant matri x C ps(z), w e get a constan t matrix C p s ( l ) . I t ha s a specia l for m know n a s circulant , whic h i s define d below.
5.3. Pseudocirculan t an d circulan t matrice s
109
Definition 5. 2 Circulan t matrice s A n M x M matri x C circ i s sai d t o b e a circulan t i f an y colum n (excep t th e leftmos t colum n whic h i s arbitrary ) i s obtained fro m th e precedin g colum n b y performin g th e followin g operations : (a) shif t dow n b y on e elemen t an d (b ) recirculat e th e spille d elemen t t o th e
top. ■
Similar t o th e cas e o f pseudocirculan t matrices , a circulan t matri x i s com pletely determine d b y an y o f it s column s o r an y o f it s rows . Lettin g it s firs t column b e [ c(0 ) c(l ) • • • c(M — 1) ] , th e M x M circulan t matri x i s c(0) c(M c(l) c(0
- 1 )• ) ••
c(M-2) c(M-3 c(M-l) c(M-2
• • c(2 ) c(l • c(3 ) c(2
) c(0 ) •• • c(l
) )
) c(M-l ) ) c(0 )
Observe t h a t , apar t fro m th e leftmos t column , al l th e othe r column s ar e ob tained b y shiftin g down th e previou s column . Thi s matri x i s therefor e calle d a circulant. 2 I t i s no t difficul t t o verif y (Proble m 5.5 ) t h a t a matri x i s circulan t if an d onl y i f it s (/c , ra)th entr y i s give n b y [Ccirc]krn=
c((k - ra))
M,
(5.6
)
where th e argumen t ((k — m))M i s interprete d modul o M , whic h i s a n inte ger betwee n 0 an d ( M — 1). Comparin g th e pseudocirculan t an d circulan t matrices, w e fin d t h a t pseudocirculan t matrice s hav e a for m simila r t o circu lant matrices , excep t t h a t th e recirculate d elemen t i s multiplie d b y th e dela y operator z~ x. Thi s explain s th e nam e "pseudocirculant. " An importan t propert y o f circulant s i s t h a t the y ca n b e diagonalize d b y the D F T matrix . T h e M x M D F T matri x W i s a matri x whos e (fc , ra)th entry i s give n by 3 Wkm [W]krn = - =, (5.7 ) where W = e _ j 2 7 r / M i s th e M t h roo t o f unity . Firs t conside r th e produc t C c i r c W^", wher e th e superscrip t f denote s transpos e conjugation . T h e (/c,£)t h element o f thi s produc t i s
wt
M-l
c fc m L = 47£ « - ))^~ M
-ml
in — \j
Making a chang e o f variabl e an d usin g th e fac t t h a t W M= the abov e expressio n a s [CoiroWt]^ = 2
- =Y:
c(m)W-^ =
-= ■ 2
1 , w e ca n rewrit e
cMW
"
Or , mor e specifically , a down circulan t (t o distinguis h fro m a n up circulan t whic h i s denned i n a n obviou s way ) I n thi s boo k th e ter m circulant alway s refer s t o a dow n circulant . 3 For convenience , w e have normalize d th e D F T matri x s o that i t i s unitary, W ^ W = 1M'-, the definitio n i s slightl y differen t fro m th e conventiona l one , fo r whic h th e (fc , ra)th elemen t is W km.
5. Multirat
110
e formulatio n o f communication system s
On t h e right-hand sid e o f t he above expression , t h e quantity insid e t h e bracke t is, i n fact, t h e £th D FT coefficien t o f c(n): M-l
Ci=Y^ c(n)W- n£
(5.8)
n=0
Note t h a t t h e D FT coefficient s ar e define d i n t he usua l manner , althoug h t he D F T matri x W i s normalized. Usin g t h e above expression , w e can writ e l^circ W
w
\k£
-k£
M
■Ce.
Writing t h e above expressio n fo r 0 < k < M — 1 , we get t he ^th column of the produc t C circW^:
l
Q = Q[Wt]
[{-'circW \{ w-(M-i)e
iT.
where [ W ^ = 1/y/M [ 1 W~ £ • • • W'^' 1^ f i s t he £th column of W"!". I n other words , t h e columns o f t he I D FT matri x W" ^ are t he eigenvector s of C circ a n d t he corresponding eigenvalue s ar e t he D FT coefficient s o f c(n). This i s true regardless of the values of c(n). Collectin g al l t he M column s for 0 < £ < M — 1, we have prove d
c arc wt = w+r, where T i s t he diagonal matri x
Co 0 0d 00
(5.9)
CM-l
•
Using t h e fac t t h a t W + W = I , we ge t
wc arc wt = r, or equivalentl y Ccirc =
W + r W ; (5.10
)
t h a t is , circulan t matrice s ar e diagonalized b y D FT matrices. Conversely , given an y matrix o f t he for m C = W ^ T W , we can write it s (/c, ra)th elemen t as M-l M-l k£ ktijrim 1 \k-m)£ c C [C]km M M vVM v ±VJ ±VJ
Y, < w~ w £=0
- = -£ 0
M£
^
Define t h e sequence c(n) a s t he I D FT coefficient s o f Ck- T h a t is, C(
1
M -- ll M £=0
111
5.4. Redundanc y fo r IB I eliminatio n
Then w e ca n writ e [C] fem = c(k — m). A s th e sequenc e c(n) i s periodi c wit h period M , w e hav e [C]km = c((k-m)) M, which prove s that th e matri x C i s circulant. A s a matrix o f the for m W ^ T W is circulant , it s diagona l entrie s ar e th e sam e an d the y ar e equa l t o c(0) , which i s th e averag e o f C^ , th e diagona l entrie s o f T. Thi s fac t wil l b e use d repeatedly later . Summarizing , w e have prove d th e followin g theorem . Theorem 5. 2 Diagonalizatio n o f circulant s I f C circ i s a circulan t matrix , it ca n b e diagonalize d a s i n (5.10) , wher e T i s th e diagona l matri x (5.9 ) o f the DF T coefficient s Ck define d i n (5.8) , an d W i s th e M x M normalize d DFT matri x define d i n (5.7) . Conversely, a produc t o f th e for m W^T W i s necessarily a circulant , an d it s leftmos t colum n i s [c(0 ) c(l ) • • • c(M — 1)] T , where c(n) = l / M ^ X " 1 C kW~kn ar e th e invers e DF T coefficient s o f th e diagonal element s Ck o f I \ ■ Circulant matrice s an d circula r convolution s Le t th e tw o M x l vector s • r( M - 1 ) ] T ,
r = [ r(0 ) r(l ) ••
s = [ s(0 ) s(l ) •• • s(M-l)
]
T
be suc h tha t r = C circs. Then , usin g (5.6) , w e ca n writ e th e nt h entr y o f r as r(n) =
M-l M-
£ [Carc]
= £=0 £ 0
nes(£)
=
l
£ c((n-£))
Ms(£).
That is , th e sequenc e r(n) i s th e circular convolution o f th e sequence s s(n) and c(n) , whic h correspond s t o th e entrie s i n th e first colum n o f C circ- Sub stituting (5.10 ) int o th e expressio n r — C c ^ rc s an d rearrangin g th e terms , w e get y/MWr = r (VMWs). (5.11 ) Note tha t th e entrie s o f th e vector s \ / M W r an d \ / M W s are , respectively , the M-poin t DF T coefficient s o f r{n) an d s{n). Equatio n (5.11 ) mean s tha t a circula r convolutio n i n th e tim e domai n become s a pointwis e multiplica tion operatio n i n th e DF T domain , whic h i s simpl y th e circula r convolutio n theorem [105] .
5.4 Redundanc
y fo r IB I eliminatio n
In a filter ban k transceive r th e channe l c(n) i n genera l introduce s interbloc k interference (IBI) . In this section, we will show that b y inserting enough redun dant sample s i n th e transmitte d sequence , IB I ca n b e avoide d o r eliminate d completely. I n th e followin g discussion , w e assume tha t th e numbe r o f redun dant sample s inserte d pe r inpu t bloc k i s v an d th e channe l i s causa l FI R o f order < i/ , i.e .
C(z) = J2c(.n)z~n- ( n=0
5 12
- )
112
5. Multirat
e formulatio n o f communicatio n system s
T h e las t fe w coefficient s ar e zer o i f th e orde r o f c(n) i s les s t h a n v. W e wil l also assum e t h a t ther e i s n o channe l noise , i.e . q(n) = 0 . Th e effec t o f nois e will b e studie d i n mor e detai l i n futur e chapters .
5.4.1 Zero-padde
d system s
One for m o f redundanc y i s th e insertio n o f zero s i n th e t r a n s m i t t e d sequence . This i s know n a s zer o padding . Figur e 5.11(a ) show s a syste m t h a t employ s the zero-paddin g schem e t o transmi t a sequenc e s(n). Figure s 5.11(b)-(d ) illustrate ho w th e zero-padde d syste m works . A t th e transmitter , th e inpu t s(n) i s firs t partitione d int o block s o f lengt h M (Fig . 5.11(b)) . The n v zero s are inserte d a t th e en d o f eac h bloc k t o obtai n th e zero-padde d resul t x(n) (Fig. 5.11(c)) . Thu s th e ne w bloc k siz e N i s give n b y TV = M + v. T h e zero-padde d sequenc e x(n) i s sen t ove r th e channel . Th e v zero s wil l be come nonzer o du e t o filterin g b y C(z). However , becaus e th e channe l ha s a t most orde r z/ , the N receive d sample s i n th e curren t bloc k o f r(n) depen d onl y on th e M t r a n s m i t t e d symbol s i n th e curren t bloc k o f s(n) (Fig . 5.11(d)) ; there i s n o IBI . I t shoul d b e emphasize d t h a t IB I i s completel y avoide d re gardless o f wha t th e impuls e respons e c(n) i s s o lon g a s it s orde r doe s no t exceed v.
Matrix formulatio n o f th e zero-padde d system s T h e zero-padde d syste m i n Fig . 5.11(a ) ca n b e represente d usin g multirat building blocks . Thi s give s ris e t o a usefu l representatio n t h a t ca n b e easil incorporated late r i n a matri x formulation . Th e insertio n o f zero s ca n b described usin g decimator s an d expander s a s i n Fig . 5.12 . Firs t th e inpu sequence s(n) i s partitione d int o block s o f siz e M usin g th e advanc e chai and decimators . Th e n t h bloc k s(n ) i s relate d t o th e sequenc e s(n) by 4 s(n)=[s(nM) s{nM
+ 1 ) ..
. s(n
M+ M - 1 ) ]
T
e y e t n
.
Each bloc k s(n ) i s padde d b y v zero s t o for m x ( n ) o f siz e N. The n x ( n ) i s interleaved usin g th e expande r an d th e dela y chai n t o for m th e sequenc e x(n), which i s t r a n s m i t t e d ove r th e channe l c(n). T o deriv e th e matri x represen tation, w e bloc k th e receive d sequenc e r(n) int o block s o f siz e N usin g th e advance chai n an d decimator s a s i n Fig . 5.12 . Th e n t h bloc k r ( n ) i s give n b y r(n)=[r(nN) r(nN
+
. r(nN
1 ) ..
+
N - 1 ) ]
T
.
Let u s loo k a t th e TV-inpu t TV-outpu t syste m fro m x ( n ) t o r ( n ) . Fro m Theore m 5.1, w e kno w t h a t i t i s simply th e N x N blocke d versio n C ps(z) o f the channe l c(n). Thu s w e hav e TZ(Z) =
C ps(z)xz(z) =
C
ps(z)
Sz(z) 0
4 By convention , whe n a sequenc e a(n) i s partitioned int o block s o f siz e P , th e /ct h bloc k begins a t a(kP) an d end s a t a(kP + P — 1) . Thi s conventio n wil l b e use d throughou t th e book.
113
5.4. Redundanc y fo r IB I eliminatio n
(a)
x(ri)
zero padding
s(n)
previous block
(b)
channel C(z)
-+ Hn)
current block
s(n) 2M
M
current block
previous block
(C)
nonzero samples due to channel filtering
(d)
r(n)
IM. Ml received samples receive due to du previous bloc k curren
Mill.
IN
d sample s e to t bloc k
Figure 5 . 1 1 . Zero-padde d system , (a ) Transmissio n schem e employin g zer o padding ; (b)-(d) illustratio n o f ho w IB I i s avoide d i n th e zero-padde d system .
where th e vector s r z(z), x^(z) , an d s z(z) ar e th e z-transform s o f th e vector s r(n), x(n) , an d s(n ) respectively . Recal l tha t th e matri x C ps(z) i s a n N x N pseudocirculant matrix . Becaus e th e bloc k siz e satisfie s N = M + v > v, th e matrix C ps(z) ha s the simpl e form i n (5.5) . Onl y the las t v column s o f C ps(z) are dependen t o n z. Thu s w e ca n writ e
rz(z) =
Ci
owsz(z),
114
5. Multirat
s(n) s(n)
|M
u
*,(«) -+
z|
x(n)
s0(n)
zv
5
| M
x(«)
fJV • •
|M
• •
e formulatio n o f communicatio n system s
^ A ")
i
kZ
i
^Z
/
r(n)
. ,i' cnannei C(z)
-1 -1
z
\N —►
1 r
|JV
^r
r(n)
0 (»)
_► > > )
A
\N
u
n block of v zeros
-1 A
•• •
-1
7V=M+v
TAH—*
UjT^"
- >r
AT-l(")
Figure 5 . 1 2 . Representatio n o f the zero-padded syste m usin g multirate buildin g blocks.
where Ci ow i s a n N x M matri x consistin g o f the firs t M column s o f C It i s the lowe r triangula r Toeplit z matri x give n b y c(0) c(l)
0 c(0)
0•
c(i/) C/ 0
•
0 0
c(0)
(5.13)
0 0
0 ••
ps(z).
0 c(y)
• c(0 ) c(l)
c(z/
)
•0
As C/ oli; i s independen t o f z , i n th e tim e domai n w e ca n writ e r(n) = C/
ow s(n);
the nt h receive d bloc k v(n) depend s onl y on the nt h inpu t bloc k s(n) , bu t no t other inpu t blocks . There is no IBL Thi s hold s fo r any causa l FI R channe l C(z) wit h orde r les s tha n o r equa l t o v. A s a result , symbo l recover y ca n be don e i n a bloc k b y bloc k manner . Wheneve r th e detail s o f blockin g an d unblocking are not relevan t t o our discussion , w e will adopt th e schemati c plo t shown i n Fig . 5.13 , whic h i s a mor e compac t descriptio n o f Fig . 5.12 . Th e legend unde r th e zero-paddin g bo x i s mean t t o indicat e tha t th e bandwidt h expansion facto r i s N/M (Sectio n 5.1.2) ; tha t is , N — M zero s ar e inserte d per bloc k o f siz e M. As the zero-padded system is free from IBI , symbol recovery can be achieve d by choosin g an y lef t invers e o f th e N x M matri x Ci ow. Th e ran k o f Ci ow
5.4. Redundanc y fo r IB I eliminatio n
115
.An)
, s(» )
r0(n)
*,(»>
^ •
p/s(iio
zero padding NIM
S
M-iW
C(z)
r(n)
rx(n) S/P(7V)
•
channel
ViW
Figure 5 . 1 3 . Schemati c plo t o f th e zero-padde d system .
is M a s lon g a s th e coefficient s c(n) ar e no t al l zer o (Proble m 3.1) . W e ca n always recove r th e inpu t vecto r s(n ) fro m th e receive d vecto r r ( n ) unles s th e channel C(z) i s identicall y zero . On e possibl e zero-forcin g solutio n i s [plow^lowj ^low->
which i n genera l ha s a n implementatio n complexit y o f O(MN). Othe r solu tions t h a t hav e a muc h lowe r implementatio n cos t wil l b e explore d i n Chapte r 6.
5.4.2 Cyclic-prefixe
d system s
Another wa y o f dealin g wit h IB I i s t o allo w a certai n amoun t o f IB I durin g transmission an d ensur e th e receive d sample s corrupte d b y IB I ar e remove d a t the receiver . Fo r a n F I R channe l o f orde r z/ , onl y th e firs t v sample s o f eac h received bloc k ar e corrupte d b y th e previou s transmitte d block . Therefor e IBI ca n b e remove d b y discardin g th e firs t v sample s o f eac h receive d block . This strateg y give s ris e t o anothe r for m o f redundanc y fo r IB I elimination , namely cycli c prefixing . I t i s th e mos t popula r for m o f redundanc y employe d in moder n communicatio n systems . W e shal l explai n thi s below . Figure 5.14(a ) show s a syste m t h a t employ s th e cyclic-prefixin g schem e to transmi t a sequenc e s(n). Figure s 5.14(b)-(e ) illustrat e ho w th e cyclic prefixed syste m work s fo r th e cas e o f v = 4 . A t th e transmitter , th e inpu t s(n) i s partitione d int o block s o f siz e M , a s i n Fig . 5.14(b) . T h e n w e cop y the las t v sample s a t th e en d o f eac h bloc k an d plac e t h e m a t th e beginnin g (Fig. 5.14(c)) . Thi s constructio n implicitl y mean s M > v. T h e ne w sequenc e x{n) ca n b e considere d t o hav e block s o f lengt h N = M + v, with th e firs t v sample s identica l t o th e las t v samples . T h e se t o f v sam ples a t th e beginnin g i s referre d t o a s a cycli c prefi x (CP) . T h e intege r v i s called th e C P length . T h e cyclic-prefixe d sequenc e x{n) i s the n sen t ove r th e channel c(n). A s th e channe l ha s orde r z/ , th e first v sample s o f eac h receive d block ar e corrupte d b y th e previou s transmitte d bloc k (Fig . 5.14(d)) . Thes e
116
5. Multirat
(a)
cyclic prefix
s(n)
e formulatio n o f communicatio n system s x(n)
previous block
(b)
discard prefix
r(n)
current block
2M
copy
x(n)
*** 0V
copy
hln i 11 ▼
11
in biillll Ihli l , l . , l l " N
*** 1
1
e bloc
\
\1
(d)
■
corrupted (discarded)
corrupted (discarded)
r(n)
2N
N+V
pr evioiJS curr t)lock
I0n . 1V
IlllllM.lll l l l l l . N N+V
M sample s M due only t o du previous bloc k curren
(e)
r(n)
ulikili
s(n)
I (c)
channel C(z)
m
IN
sample s e only t o t bloc k
llil previous curren block bloc
k
t
F i g u r e 5 . 1 4 . T h e Cyclic-prefixe d system , (a ) Transmissio n schem e e m p l o y i n g cycli c prefixing; ( b ) - ( e ) illustratio n o f ho w t h e cycli c prefi x i s adde d an d ho w IB I i s eliminate d in t h e cyclic-prefixe d system .
contaminated sample s ar e discarded , an d thi s operatio n i s denote d b y a bo x labeled "discar d prefix " (Fig. 5.14(a)) . Afte r discardin g th e prefix , w e obtai n the ne w sequenc e F(n) , whic h ca n b e viewe d a s a sequenc e consistin g o f block s of lengt h M (Fig . 5.14(e)) . Not e t h a t eac h receive d bloc k r(n) depend s onl y on th e correspondin g t r a n s m i t t e d bloc k o f th e inpu t signa l s ( n ) . IB I i s com -
117
5.4. Redundanc y fo r IB I eliminatio n
pletely eliminate d regardles s o f wha t th e impuls e respons e c(n) i s s o lon g a s its orde r doe s no t excee d v. discard prefix
^
v prefi x
samples
sin) z'
s0(n) ^ T
r
*,(»)
•••
1 F x
• • •
cp
x(n)
\N
•••
i1 —► fN
—►
77 77 •• •
—►
s(») x(« Figure 5.15 . Th notation.
\
-1
C(z)
channel
4
r{n)
IN
■ * r o<">
z
jiV
r
v-lW
A
±N
-+" r „(") \
A.
±N A.
t N=M+v
)
UH7
1v '
^ v+
\r(n)
■*rAL,W/
r(n)
e cyclic-prefixe d syste m show n i n term s o f multirat e buildin g bloc k
Matrix formulatio n o f th e cyclic-prefixe d system s T h e cyclic-prefixe d syste m i n Fig . 5.14(a ) ca n als o b e represente d usin g mul tirate buildin g blocks . Agai n thi s representatio n give s ris e t o a usefu l for mulation t h a t ca n b e convenientl y include d i n a matri x framework . Usin g multirate buildin g blocks , th e cyclic-prefixe d syste m i n Fig . 5.14(a ) ca n b e redrawn a s Fig . 5.15 . T h e inpu t sequenc e s(n) i s firs t blocke d int o block s o f size M usin g th e advanc e chai n an d decimators . T h e inpu t bloc k s(n ) i s the n multiplied b y a n N x M matri x F cp define d a s 0I \M-V 0 0L
v
(5.14)
,
T h e resul t i s a n N x 1 vecto r x(n) = F
cps(n).
It i s no t difficul t t o verif y t h a t multiplyin g s(n ) b y F cp i s equivalen t t o addin g a cycli c prefi x o f lengt h v. T h e vecto r x ( n ) i s th e cyclic-prefixe d versio n o f s(n) an d i t i s interleave d t o obtai n th e cyclic-prefixe d sequenc e x(n) a s i n Fig . 5.15. A t th e receiver , th e receive d sequenc e r{n) i s blocke d int o A ^ x l vector s r ( n ) . Afte r removin g th e firs t v sample s i n eac h vecto r r(n) (discardin g th e
118
5. Multirat
e formulatio n o f communicatio n system s
prefix), w e obtain M x l outpu t vector s ?(n) a s indicate d i n Fig . 5.15 . Not e that th e system from x(n ) t o r(n) i s the blocked version C ps(z) o f the channel . Their z-transform s ar e therefor e relate d b y TZ(Z) =
C
ps(z)xz(z).
After discardin g th e v prefi x sample s o f eac h receive d block , w e hav e TZ(Z) =
[ 0 I
M
] C
ps(z)xz(z),
where 0 i s a n M x v zer o matrix . A s the bloc k siz e satisfie s N = M + v > z/, the pseudocirculan t matri x C ps(z) ha s th e simpl e for m give n i n (5.5) . Onl y the first v row s of C ps(z) ar e dependent o n z. Th e produc t [ 0 \M ] C ps(z) is a constan t matrix . Therefore , i n th e tim e domai n w e ca n writ e r(n) = C
w p x(n),
(5.15)
where C up i s a n M x N matri x consistin g o f th e botto m M row s o f C From (5.5) , w e se e tha t C up ha s th e form : c(u) c ( z / - l ) . . 0 c(v)
'•
. c(0
)0
-.
• '•
.0
-.
•:
ps(z).
.0
: c(0) 0
0 0
c(iz)
c(0)
0
c(iz) ••
•
c(l) c(0 )
(5.16) The subscript "up" serve s as a reminder tha t th e matrix is an upper triangula r Toeplitz matrix . Substitutin g th e cyclic-prefi x relatio n x(n ) = F c p s(n) int o (5.15), w e ge t r(n) ^up*- cpSyTl)' The nt h outpu t bloc k ?(n) depend s onl y o n th e nt h inpu t bloc k s(n) . IBI is eliminated b y th e operatio n o f discardin g th e prefix . Th e outpu t an d inpu t vectors ar e related b y the M x M produc t matri x (C upFcp). I t turn s ou t tha t this produc t i s a circulan t matrix . T o se e this , observ e tha t multiplyin g C up with F cp fro m th e righ t i s equivalent t o addin g th e first v column s o f C up t o the las t v columns . Thi s operation wil l transform th e M x N uppe r triangula r Toeplitz matri x C up int o a n M x M circulan t matrix . T o understand this , le t us conside r th e exampl e o f v = 3 and M = 4 . I n thi s case , the M x N matri x V-Joinr) I S
c(3) c(2 0 c(3 00 00
) c(l ) ) c(2 ) c(3 ) 0
I c(0 I c(l | c(2 | c(3
)0 ) c(0 ) c(l ) c(2
0 )0 ) c(0 ) c(l
0 0 )0 ) c(0 )
Adding th e first thre e column s t o th e las t thre e columns , w e ge t
^up*- cp
c(0) c(3 c(l) c(0 c(2) c(l c(3) c(2
) c(2 ) c(3 ) c(0 ) c(l
) c(l ) c(2 ) c(3 ) c(0
) ) ) )
(5.17)
119
5.4. Redundanc y fo r IB I eliminatio n
which is an M x M circulan t matrix . Fo r general M an d v that satisf y M > v, the relatio n r(n) = C circs(n) (5.18 ) continues t o hold . I n thi s case , th e firs t colum n o f the circulan t matri x Q circ is th e sam e a s th e z/t h colum n o f C up. I n practice , M i s usuall y muc h large r than z/ , and i n thi s cas e man y o f th e entrie s o f Q circ ar e zero : c(0)
0
c(l)
c(0)
c{v)
. . . c(l
)
C(iz)
c(^)
0
0 0
(5.19)
c(0)
0 c(^
)
0 c(l) c(0 )
When M = z/ , th e matri x C i s stil l circulan t bu t i t i s slightl y differen t fro m the abov e expressio n (Proble m 5.16) . From Sectio n 5.4.2 , w e know tha t th e multiplicatio n o f a circulan t matri x and a vecto r ca n b e interprete d a s a circula r convolution . Equatio n (5.18 ) means that th e entrie s o f r(n) represen t th e circula r convolutio n o f the entrie s of s(n ) an d th e channe l impuls e response . Th e remova l o f the cycli c prefi x a t the receive r eliminate s IB I an d the insertion of the cyclic prefix at the transmitter converts a linear convolution to a circular one. Fo r symbo l recovery , we ca n simpl y inver t th e matri x Q circ (provide d tha t i t i s invertible ) t o ge t s(«) = C c -> c ?(n). From Theore m 5.3 , w e kno w tha t Q circ i s invertibl e i f an d onl y i f th e diag onal matri x V i s invertible . Th e diagona l entrie s o f V ar e th e M-poin t DF T coefficients o f th e channe l impuls e respons e c(n). Thu s zero-forcing equalization exists for a CP system if and only if the channel C(z) does not have zeros at the DFT frequencies 2kir/M. Wheneve r th e detail s o f blockin g an d unblocking ar e no t relevan t t o ou r discussion , w e wil l adop t th e schemati c plot o f th e cyclic-prefixe d syste m show n i n Fig . 5.16 . Agai n th e rati o N/M is th e bandwidt h expansio n ratio ; tha t is , (N — M) C P sample s ar e inserte d per bloc k o f siz e M. I n Chapte r 6 , w e will introduc e a numbe r o f transceive r systems tha t emplo y th e cyclic-prefixin g scheme .
5.4.3 Summar
y an d compariso n
The zero-padde d (ZP ) syste m an d th e cyclic-prefixe d (CP ) syste m ar e ver y similar, bu t ther e ar e als o som e differences . Her e i s a summar y an d compari son o f the tw o systems . (1) Bandwidth efficiency. Bot h system s hav e th e sam e bandwidt h expan sion ratio : M + z/ N M M
120
5. Multirat
s(w)
/
cyclic
• •
S
M-M
e formulatio n o f communicatio n system s
—►
ryri)
P/S(AQ —► prefix — ► C(z) NIM
channel
discard — ► S/P(M) prefix
• •
Figure 5.16 . Schemati c plo t o f th e cyclic-prefixe d system .
In practice , du e t o bandwidt h efficienc y th e bloc k siz e M i s usually muc h larger t h a n v. Als o not e t h a t , i n orde r t o eliminat e IBI , th e numbe r v has t o b e a t leas t a s larg e a s th e channe l order . Fo r som e applications , such a s DSLs , th e channel s ca n hav e ver y lon g impuls e responses . I n thi s case, a channel-shortening filter i s usuall y employe d a t th e receive r t o shorten th e channel s s o t h at th e equivalen t channe l i s well approximate d by a muc h shorte r filter . Th e desig n o f channel-shortenin g filter s wa s described i n Sectio n 3.5 . (2) Transmission power. I n a Z P syste m th e redundan t sample s ar e zeros , whereas i n a C P syste m th e redundan t sample s ar e th e las t v sample s i n each block , whic h ar e i n genera l nonzero . A s th e zer o sample s hav e zer o power, th e Z P syste m need s a smalle r transmissio n powe r t h a n th e C P system fo r th e sam e inpu t block . Th e differenc e i s smal l a s th e bloc k size M i s usuall y muc h large r t h a n v. (3) Symbol recovery. I n a Z P system , ther e i s n o IBI . Al l th e N sample s i n the receive d bloc k r ( n ) ca n b e utilize d fo r symbo l recovery . O n th e othe r hand, i n a C P syste m onl y th e M sample s i n th e vecto r r(n) ca n b e use d for symbo l recovery . S o whe n ther e i s n o constrain t o n th e complexit y of th e receiver , th e Z P syste m wil l outperfor m th e C P system . However , when th e receiver s o f b o t h system s ar e implemente d wit h roughl y th e same complexity , thei r performance s ar e almos t th e sam e (Chapte r 6) . (4) Channel equalization. I n b o t h systems , th e equalizatio n o f F I R channel s can b e achieve d b y invertin g a constan t matri x provide d t h a t th e trans mitter insert s enoug h redundan t samples . No IIR filtering is needed. Fo r ZP systems , zero-forcin g equalizer s alway s exis t a s lon g a s th e channe l is no t identicall y zero . O n th e othe r hand , fo r C P system s th e existenc e of zero-forcin g equalizer s depend s o n th e location s o f th e channe l zeros . For th e purpos e o f channe l equalization , th e Z P syste m i n genera l ha s a bette r performance t h a n th e C P system . However , th e inserte d C P sample s ca n b e exploited a t th e receive r fo r solvin g othe r issues , suc h a s timin g synchroniza tion [100 ] an d channe l shortenin g [87] . A mor e detaile d compariso n o f th e Z P and C P system s ca n b e foun d i n [96] .
121
5.4. Redundanc y fo r IB I eliminatio n
5.4.4 IBI-fre
e system s wit h reduce d redundanc y
Both th e Z P an d C P system s ar e IBI-fre e fo r an y causa l F I R channel s pro vided t h a t th e numbe r o f inserte d redundan t sample s pe r block , z/ , i s large r t h a n o r equa l t o th e channe l order . A Z P syste m avoids IB I b y insertin g enough zero s betwee n successiv e blocks . O n th e othe r hand , i n a C P syste m the channe l doe s incu r IB I o n successiv e blocks , bu t th e receive r eliminates IB I by discardin g th e C P samples . Instea d o f insertin g v zeros , on e ca n als o ap pend fewe r zero s a t th e transmitter , an d th e IB I du e t o insufficien t redundanc y can b e eliminate d b y removin g th e contaminate d sample s a t th e receiver . I t was show n i n [76 ] t h a t , b y doin g so , w e ca n hav e IBI-fre e bloc k transceiver s with fewe r redundan t sample s pe r block . Belo w w e wil l demonstrat e ho w t o achieve this . Let v b e orde r o f th e channe l C(z). Suppos e t h a t w e appen d p (wit h p < v) zero s t o eac h transmissio n bloc k s(n ) o f M samples . Le t r ( n ) denot e the receive d bloc k o f N = M + p samples . T h e n th e z-transform s o f r ( n ) an d s(n) ar e relate d b y rz(z)=
C
ps(z)
*z(z) 0
where C ps(z) i s th e N x N pseudocirculan t matri x i n (5.5 ) (assumin g t h a t N = M + p > v). Usin g th e fact s t h a t 0 i s a n p x 1 zero vecto r an d C ps(z) ha s the specia l for m i n (5.5) , onl y th e firs t (y — p) entrie s o f r z(z) ar e dependen t on z. I n othe r words , th e las t (N — v + p) receive d sample s i n r ( n ) depen d only o n th e curren t transmitte d bloc k s ( n ) . Therefor e b y discardin g th e firs t {y — p) o f eac h receive d block , w e wil l hav e a n IBI-fre e syste m a s show n i n Fig. 5.17 . Le t r ( n ) b e th e vecto r consistin g o f th e las t (N — v + p) entrie s o f r ( n ) . T h e n w e hav e r(n) = Cps(n) , where C p i s th e (N — v-\-p) x M Toeplit z matri x wit h firs t ro w [c{y — p) c{y — p-l) . . . c(0 ) 0 . . . 0 ] an d firs t colum n [c{y-p) c{y-p+l) . . . c{y) 0 . . . 0 ] T . To ensur e th e existenc e o f a zero-forcin g receiver , th e ran k o f C p shoul d b e at leas t M . Thu s on e necessar y conditio n i s N - v + p > M. Using N = M + p, th e abov e conditio n ca n b e rewritte n a s 2p — v > 0 . T h u s we ca n conclud e t h a t th e minimu m redundanc y neede d fo r th e existenc e o f ISI-free bloc k transceiver s i s Prnin=
J z//2 , fo r eve n is; \ (i / + l ) / 2 , fo r od d v.
T h e numbe r o f redundan t sample s neede d i s abou t half of that in the ZP and CP systems. Lik e th e C P system , th e existenc e o f a zero-forcin g equalize r de pends o n th e channe l c(n). T h e bloc k transceiver s wit h minimu m redundanc y were introduce d i n [76] . On e majo r drawbac k o f suc h transceiver s wa s thei r high complexity . Unlik e th e C P an d th e Z P systems , whic h hav e lo w com plexity DFT-base d implementation s (a s w e wil l sho w i n Chapte r 6) , th e cos t of implementin g th e lef t invers e o f th e Toeplit z matri x C p i s high . A reduced complexity implementatio n o f thes e minimu m redundanc y bloc k transceiver s
122
5. Multirat
/
discard
s(«) x(n)
fN
sQ(n)
fN fN ••
block of p zeros
u
0
C(z)
r(n)
j
IN
channel
fJV
0
e formulatio n o f communicatio n system s
-1 A A
zI
A.
zT
Li
—►
±N
-+
|tf
z|
N=M+p
>rv-P+l(M)^
—> |AT
^ v-p+2
v/
••
U{7^
\
m
■>V-iW
Tin)
Figure 5 . 1 7 . IBI-fre e syste m wit h reduce d redundancy .
was give n i n [26] . Th e minimu m redundanc y give n i n thi s subsectio n i s fo r block transceiver s wher e th e transmittin g an d receivin g polyphas e matrice s G(z) an d S(z) ar e constan t matrice s independen t o f z. I f w e allo w G(z) an d S(z) t o hav e memor y (^-dependent) , the n th e redundanc y ca n b e eve n lowe r (Chapter 10) .
5-5 Fractionall
y space d equalize r system s
In moder n communicatio n systems , th e receive r sometime s take s sample s o f the receive d continuous-tim e signa l a t a highe r rat e t o enhanc e th e syste m performance. Figur e 5.1 8 show s suc h a system , wher e th e transmitte r send s the sequenc e x(n) a t a rat e o f 1/ T (Hz ) bu t th e receive r take s th e sample s r(n) at a rate o f N/T (Hz) . Th e samplin g rat e o f the receive r i s N time s highe r t h a n t h a t o f the transmitter . W e wil l assum e N i s an intege r i n ou r discussion . I f w e look a t th e symbo l spacing , th e inpu t signa l x(n) i s spaced apar t b y T seconds , whereas th e receive d signa l r(n) i s space d apar t b y T/N seconds . Henc e th e system i s know n a s th e fractionall y space d equalize r (FSE ) system . W h e n the transmitte r an d receive r hav e th e sam e samplin g rate , t h a t i s N = 1 , th e F S E syste m reduce s t o th e symbo l space d equalize r (SSE ) system , whic h has bee n discusse d i n detai l i n Sectio n 2.1 . W e kno w t h a t th e SS E syste m ca n be describe d b y a n equivalen t discrete-tim e LT I system . W e shal l deriv e t h a t the F S E syste m als o ha s a discrete-tim e equivalen t system . A s th e samplin g rates ar e differen t a t th e transmitte r an d receiver , th e equivalen t syste m i s n o longer LTI , an d multirat e buildin g block s ar e needed . Let u s compar e Fig . 5.1 8 wit h Fig . 2.3 . W e find t h a t th e signal s w a{t) i n these tw o figures ar e identica l an d the y ar e give n b y (2.1) . Fo r convenience ,
123
5.5. Fractionall y space d equalize r system s
qa(t) x(n)
D/C
T
Xa(t)
Px{t)
^r
Ca(t)
a(t)
J P2(t)
VaVU C/D
T
r(n)
T/N
Figure 5 . 1 8 . Transmissio n syste m wit h oversamplin g a t th e receiver .
we reproduc e th e expressio n fo r w a(t) below :
va{t)= Y, w„.
oo
x{k)c
e{t-kT)
+
qe{t),
/c= —o o
where c e(t) = (pi * ca *P2)(f) an d q e(t) = (q a *P2)(t) ar e th e effectiv e channe l and noise , respectively . W h e n w a(t) i s uniforml y sample d ever y T/N second s as i n Fig . 5.18 , w e obtai n th e discrete-tim e signa l oo
r(n) = Y, x(k)c
e(nT/N-kT)
+
q
e(nT/N).
/ c = — oo
Define th e discrete-tim e equivalen t channe l an d noise , respectively , a s follows : c(ra) = ( p i * c a * p 2 ) ( t ) | , q(n) =
(q a*p2)(t)\ .
(5.20
\t=nT/N \t=nT/N
(5.21
) )
T h e n w e ca n rewrit e r(n) a s oo
r(n) =
N , x(k)c(n
—
kN) + q(n).
A;= —oo
T h e firs t t e r m simpl y represent s th e interpolatio n filte r operatin g o n th e inpu t x(n). I n term s o f discrete-tim e signa l processin g notations , th e precedin g equation ha s th e beautifu l interpretatio n show n i n Fig . 5.19 , wher e
C(*) = 5>(n)z"n. n
Let u s compar e th e discrete-tim e equivalen t channe l an d nois e o f th e F S E system wit h thos e o f the SS E syste m derive d i n Sectio n 2.1 . Fo r a n SS E syste m where th e receive r take s sample s a t th e samplin g rat e 1/T , th e discrete-tim e equivalent channe l an d nois e are , respectively , give n b y csse(n)= ( p qSsein) = {q
i * c a *P2)(t) a *P2)(* )
t=nT
t=nT
124
5. Multirat
^A
x(n)
>
AT
w
e formulatio n o f communicatio n system s
q(n)
y(n)
r*( -\ G
\UlA H
WI
i
w W
—•
r(ri)
Figure 5 . 1 9 . Discrete-tim e equivalen t mode l o f th e communicatio n syste m (i n Fig. 5.18 ) wit h oversamplin g a t th e receiver .
From (5.20 ) an d (5.21) , w e find t h a t th e tw o discrete-tim e model s ar e relate d by cSSe(n) =
c(Nn) =
[c(n)]
iN
an
dq
sse{n)=
q(Nn) =
[q(n)] iN ;
the channe l impuls e respons e an d nois e o f th e SS E syste m are , respectively , the TV-fol d decimate d version s o f th e channe l an d nois e o f th e associate d F S E system.
5.5.1 Zero-forcin
g FS E system s
In a n F S E system , th e receive d sample s r(n) ar e space d apar t b y T/N second s whereas th e inpu t sample s x(n) ar e space d apar t b y T seconds . Th e receive d d a t a rat e i s N time s th e inpu t d a t a rate . T o recove r x(n) fro m r ( n ) , w e ca n us e an TV-fol d decimatio n filter. Thi s i s show n i n Fig . 5.20 . Th e simples t wa y t o equalize th e effec t o f th e channe l i s t o choos e th e equalize r a s H(z) = 1/C(z). T h e n th e syste m i s zero-forcin g an d th e outpu t erro r i s simpl y q(n) passe d through 1/C(z) an d decimate d b y N. A ver y importan t observatio n her e i s t h a t i f w e wan t t o cance l ou t th e effec t o f C(z) completely , i t i s no t necessar y to us e 1/C(z) a s th e equalize r becaus e th e outpu t i s decimate d anyway . Fro m the polyphas e identit y (Fig . 4.6) , w e kno w t h a t i n th e absenc e o f nois e th e system fro m x(n) t o x(n) i s i n fac t LT I wit h transfe r functio n
T(z)= \C{z)H(z) Recall fro m Chapte r 4 t h a t [C(z)H(z)], component o f th e produc t C(z)H(z). Thu when
C(z)H(z)
N
i
IN
s simpl y th e zerot h polyphas e s th e syste m i n Fig . 5.2 0 i s ISI-fre e
IN
1.
Or, equivalentl y i n th e tim e domain , t(n) = ( c * h)(Nn)= 5{n), whic h means t h a t th e convolutio n ( c * / i ) ( n ) satisfie s th e Nyquist(TV ) propert y (Sec tion 4.3.2) . I n othe r words , th e syste m i s ISI-fre e i f an d onl y i f h{n) i s suc h t h a t th e impuls e respons e ( c * h){n) ha s th e zero-crossin g property. 5 Mor e 5 When tw o transfe r function s C(z) an d H(z) ar e suc h tha t [C(Z)H(Z)]±N = 1 , the y are calle d biorthogona l partners . Furthe r detail s abou t th e theor y an d application s o f biorthogonal partner s an d thei r extension s ca n b e foun d i n [160 ] an d [171] .
125
5.5. Fractionall y space d equalize r system s
generally, i f we allow a dela y i n th e syste m output , th e zero-forcin g conditio n becomes T(z)=\C(z)H(z)
HN
for som e non-negativ e intege r no. I n th e absenc e o f noise , th e outpu t i s x(n) = x(n — no). q(n)
y(n) x(n) ^
fN
C(z)
i
equalizer
r(n) I
►
H(z) \-^\±N\ ►
x(n)
Figure 5 . 2 0 . Decimatio n filte r fo r FS E channe l equalization .
Note tha t fo r th e FS E system , w e d o no t nee d C(z)H(z) = 1 to achiev e zero-forcing equalization . Onl y th e zerot h polyphas e componen t o f the prod uct C(z)H(z) need s t o b e unity ; al l th e othe r polyphas e component s ca n b e arbitrary. Thu s channe l inversio n ca n b e avoide d b y applyin g oversamplin g at th e receiver . I n fact , fo r mos t FI R channels , perfec t channe l equalizatio n can b e achieve d wit h a n FI R filte r H(z), a s w e will se e i n th e following . 5.5.2 Polyphas
e approac h
Let u s decompos e th e channe l C(z) an d th e filte r H(z) usin g Typ e 1 and 2 polyphase representation s respectively : N-l
C(z) =
k
Gk(zN),
] T z~ k=0 N-l
H{z) =
J2 z
k
Sk(zN).
k=0
Then th e zerot h polyphas e componen t o f th e produc t C(z)H(z) i s give n b y N-l
T(z) =
Y, G k(z)Sk(z). (5.22 /c=0
)
Given the channel C(z), th e equalizer is zero-forcing if and only if the polyphas e components Sk(z) satisf y N-l
Y,Gk{z)Sk{z)
/c=0
(5.23)
126
5. Multirat
e formulatio n o f communicatio n system s
A pictoria l vie w o f (5.22 ) W e ca n als o deriv e th e transfe r functio n T(z) in (5.22 ) pictorially . Usin g th e polyphas e implementatio n o f th e decimatio n and interpolatio n filter s i n Sectio n 4.4 , w e ca n redra w th e syste m i n Fig . 5.2 0 as i n Fig . 5.21(a) . Recal l fro m Fig . 4.10(b ) t h a t th e multi-inpu t multi-outpu t (MIMO) syste m i n th e gra y bo x i s a n identit y system . Thu s Fig . 5.21(a ) ca n be redraw n a s Fig . 5.21(b) , wher e Qkip) =
q(Nn + k)
is th e kth polyphas e componen t o f th e channe l nois e q(n). I t follow s fro m Fig. 5.21(b ) t h a t th e transfe r functio n fro m x(n) t o x(n) i s give n b y (5.22) . Figure 5.21(b ) als o give s a n interestin g interpretatio n o f th e F S E system . Looking a t it s to p branch , w e hav e a channe l Go(z) wit h nois e qo(n). Not e that G0(z) = [C(z)]i N an d q 0(n) = [q(n)] iN; these quantitie s wil l b e th e channe l an d nois e o f a n SS E system , respectively , if th e receive r take s sample s r(n) a t th e symbo l spacing . O n th e othe r hand , in a n F S E syste m w e hav e N branches . The FSE receiver has N copies of the transmitted signal, each going through a different "channel" Gk{z). A s a result, th e receive r o f a n F S E syste m ca n recove r x(n) usin g a n F I R equalize r H(z) fo r mos t F I R channel s C(z), a s state d b y Theore m 5.3 .
Theorem 5. 3 Existenc e o f a n FI R zero-forcin g equalizer . Give n a n F I R channel C{z) = Yln= o c (n)z~n, ther e exist s a n F I R H(z) suc h t h a t T(z)=\C(z)H(z)
=1
if an d onl y i f th e polyphas e component s Gi(z) o f C(z) d o no t hav e an y com mon facto r o f th e for m ( 1 — a z " 1 ) , a ^ O . ■ Proof Fro m (5.22 ) w e kno w t h a t T(z) = Y^JQ 1 G l(z)Sl(z). W h e n al l Gi(z) hav e a nontrivia l commo n facto r ( 1 — az -1), a ^ 0 , i t i s clea r t h a t T(z) wil l als o hav e suc h a commo n facto r i f Si(z) ar e FIR . Becaus e the polyphas e component s Si(z) o f F I R H(z) ar e als o FIR , w e conclud e t h a t ther e doe s no t exis t an y F I R H(z) suc h t h a t T(z) = 1 whe n al l Gi(z) hav e a nontrivia l commo n facto r ( 1 — a z " 1 ) , a ^ 0 . Conversely , suppose t h a t Gi(z) d o no t hav e suc h a factor . The n Euclid' s theore m states t h a t ther e exis t causa l F I R filter s S[(z) suc h t h a t N-l
J2 GiWSfr) = z~ K, i=0
where z~ K i s th e greates t commo n facto r o f al l th e polyphas e compo nents Gi(z). Lettin g S t(z) = z KS'i{z)1 the n H(z) = J ^ 1 z iSl(zN) i s F I R an d i t satisfie s [C(z)H(z)]= 1 .■ iN In practice , i t rarel y happen s t h a t al l th e polyphas e component s Gi(z) o f a channe l C(z) hav e a nontrivia l commo n facto r ( 1 — az~x). I f i t doe s happen ,
127
5.5. Fractionall y space d equalize r system s
(«)
(a)
~~**| O r
0 (z)
f
| ►
—►] G x{z) 1 * | \N \-*>
= * L
1
i
H
-l ^z z kZ
1r 1 ►w
, —^1 ▼ JV
H ^o( z) 1
1 r
—*\ ±N
-1 z
A
H ^(z ) | —
• ••
• • •
i
v
kn)
T A
*| S N_Y(z) | - J i d e n t i t y syste m
i
?o(") |
x(n)
G 0 (2 )
G t (z)
(b)
u
G
iV-l( Z )
U(») |
-»• M
M
W
-> x(n)
^(z)
K>> I 1
t
Figure 5 . 2 1 . (a ) Oversample d channe l C(z ) an d fractionall y space d equalize r H(z ) (b) polyphas e representation ; (c ) simplifie d equivalent .
then a n F I R zero-forcin g solutio n doe s no t exis t an d on e need s a n II R filter . Let Q(z) b e th e greates t commo n facto r o f al l Gi(z). T h e n Euclid' s theore m says t h a t w e ca n find F I R S[(z) suc h t h a t N-l
Y,Gi(z)S[(z)=Q{z). i=0
T h e n th e zero-forcin g conditio n ca n b e satisfie d i f w e choos e
Si(z) =
S'(z)/Q(z).
In thi s case , th e equalize r H(z) = X)z= o zl Si(zN) i s a n II R filter. I f Q(z) does no t hav e an y zero s o n th e uni t circle , the n H(z) i s stable . W h e n al l th e zeros o f Q(z) ar e insid e th e uni t circle , H(z) i s b o t h causa l an d stable . Causal equalize r T h e transfe r functio n H(z) = X)z= o z% Si{zN) ca n b e noncausal eve n whe n al l Si(z) ar e causal . T o obtai n a causa l equalizer , on e can multipl y a noncausa l H(z) b y z~ kN. Not e t h a t multiplyin g a zero-forcin g equalizer H(z) b y z~ m ca n destro y it s zero-forcin g propert y whe n m i s no t a multiple o f N. Thi s i s becaus e [z~ rnC(z)H(z)] , N i s no t a delaye d versio n o f [C(z)H(z)],N whe n m i s no t a multipl e o f N.
128
5. Multirat
e formulatio n o f communicatio n system s
Example 5. 1 Suppos e tha t th e discrete-tim e equivalen t channe l wit h over sampling facto r N = 2 is give n b y C(z) =
l + z~ 1 + z~ 2 -2z~
■z
-5
Note tha t (7(1 ) = 0 . S o ( 1 — z * ) i s a facto r o f C(z) an d th e invers e 1/C(z) is not stable . Th e polyphas e component s ar e l + z~ 1 an d G^z) =
G0{z) =
1 - 2z~ x - z~
2
.
One ca n verif y b y explici t calculatio n that , i f w e choos e S0(z) =
3 — z~ x 1 an d S^z)= -,
then w e will hav e Go(z)So(z) + H(z) i s give n b y H{z) =
Gi(z)S\(z) =
S 0(z2) + zS^z 2)=
1 . Th e zero-forcin g FI R filte r (z + 3 - z~
2
)/A.
The noncausalit y o f H(z) ca n b e remove d b y multiplyin g z~ 2. I n thi s case , the transfe r functio n become s T(z) = z~ x. I n th e cas e tha t th e receive r doe s not oversample , th e channe l o f th e SS E syste m woul d b e Csse(z) =
G 0(z) =
l + z-\
whose invers e 1/C sse{z) i s not stabl e becaus e C sse{—1) = 0 . However , w e ar e able t o find FI R zero-forcin g H(z) fo r th e FS E system . ■ Nonuniqueness o f th e zero-forcin g FI R equalize r Wheneve r i t exists , the zero-forcin g FI R equalize r i n a n FS E syste m i s no t unique . T o se e this , consider th e cas e o f N = 2 . Suppos e H(z) = So(z 2) + zS\(z 2) i s zero-forcin g so tha t G0(z)S0{z) + G 1(z)S1{z) = l. Note no w that , give n an y transfe r functio n A(z), th e followin g identit y hold s trivially: 0. Go(z)\G1(z)A(z)\ + G i ( ^ ) -G 0(z)A(z) This mean s tha t i f w e defin e a ne w transfe r functio n H(z) wit h polyphas e components S0(z) =
S 0(z) + G 1(z)A(z) an
d S^z) =
S^z) - G
0(z)A(z),
then T(z) = G 0(z)S0(z) + G 1(z)S1(z) i s als o equa l t o unity . Th e filter H(z) is a vali d zero-forcin g FI R filter a s well . Sinc e A(z) i s arbitrary , ther e exist s infinitely man y zero-forcin g FI R filters. I n th e absenc e o f channe l noise , al l zero-forcing equalizer s ar e equivalent. Thi s is not th e cas e when there i s noise. In practice , on e ca n optimiz e A(z) t o minimiz e th e powe r o f the outpu t noise . Further detail s o f this ide a ca n b e foun d i n [171] . In th e abov e discussions , th e oversamplin g facto r N i s assume d t o b e a n integer. Th e ide a o f representin g a n FS E syste m usin g multirat e buildin g blocks ca n als o b e extende d t o th e cas e o f rationa l oversamplin g factors . A n FSE syste m wit h a rationa l oversamplin g facto r ca n als o b e describe d usin g multirate buildin g blocks . Mor e detail s o n th e equalize r desig n fo r suc h a n FSE syste m ca n b e foun d i n [172] .
5.6. Concludin g remark s
5.6 Concludin
129
g remark s
T h e connectio n betwee n a perfec t reconstructio n filte r ban k an d a perfec t transmultiplexer wa s firs t recognize d b y Vetterl i [168] . Multirat e signa l pro cessing fo r channe l equalizatio n wa s reporte d i n [121] . T h e applicatio n o f nonmaximally decimate d (redundant ) filte r bank s t o precodin g wa s firs t pro posed b y Xi a [182 , 185] . Optimizatio n o f filte r ban k precoder s i s studie d i n [135]. Som e tutorial s o n thi s topi c ca n b e foun d i n [3 , 161 , 162] . Mor e topic s on filte r ban k transceiver s wil l b e explore d late r i n th e book . T h e matri x formulations o f th e Z P an d C P system s ar e usefu l fo r th e design s o f a numbe r of bloc k transceive r system s (Chapter s 6 , 7 , an d 8) . T h e filte r ban k structur e will greatl y facilitat e th e desig n o f transceive r system s wit h bette r frequenc y responses (Chapte r 9) . T h e multirat e theor y wil l b e applie d late r t o th e stud y of th e existenc e o f zero-forcin g F I R filte r ban k transceiver s (Chapte r 10) .
5.7 Problem
s
5.1 Conside r th e filte r ban k transceive r syste m show n i n Fig . 5.1 . Suppos e t h a t th e transmittin g filte r Fi(z) i s a rea l bandpas s filte r wit h frequenc y response give n b y F (e
^'
jnf
\
1 ,u
e [TT/M , 2 T T / M ] U [ - 2 T T / M , - T T / M ] ;
0 , otherwise
,
and th e Fourie r transfor m o f th e signa l s\{n) i s a s give n i n Fig . 5.3(c) . Draw th e Fourie r transfor m o f th e outpu t u\(n) o f F\{z). 5.2 Le t C ps(z) b e th e N x N blocke d versio n o f th e channe l C(z). Sho t h a t it s (z , j ) th entr y [Cpg^z)]^ i s give n b y (5.4) .
w
5.3 Suppos e t h a t w e bloc k a scala r channe l C(z) a s i n Fig . P5.3 . W e kno w t h a t whe n p = 1 , th e N x N syste m fro m s(n ) t o s(n) i s LT I wit h transfer matri x C ps(z). I n thi s problem , w e explor e th e cas e o f p > 1 . Let th e polyphas e decompositio n o f th e channe l b e C(z) = CQ (zN) +
z-^z") +
• •• +
Z-WCN-^Z*).
(a) Le t N = 3 an d p = 2 . Fin d th e 3 x 3 transfe r matri x T(z). Expres your answe r i n term s o f Ci(z). (b) Repea t (a ) fo r N = T(z)7 (c) Repea t (a ) fo r N =
p =
s
3 . W h a t ca n w e sa y abou t th e ran k o f
6 an d p = 2 . W h a t i s th e ran k o f T(z)7
(d) Mor e generally , le t th e greates t commo n diviso r (gcd ) o f N an d p be gcd(iV,p ) = K an d le t N/K = Q. Prov e t h a t th e ran k o f T(z) is a t mos t Q. 5.4 Suppos e t h a t w e bloc k a scala r channe l C(z), a s w e di d i n Fig . P 5.3 . Le t T(z) b e it s N x N transfe r matrix . Assum e t h a t th e integer s p an d N are coprime . Le t th e polyphas e decompositio n o f th e channe l b e C(z) =
130
5. Multirat
J
e formulatio n o f communicatio n system s
s(w)
s(«)
s0(n)
u
s^n)
fN ••
w*>
i — i\
Q(n)
'
T"~P
—► j i V
—► i
>s
jiV
C(z)
zp'
'z-*
z p
~ *
'+
r
•• i — i-
, ■
Figure P 5 . 3 . Blockin g o f a scala r channe l usin g mor e genera l dela y an d advanc e chains.
C0(zN) + z-1C1(zN) + define
... + z-N+1CN-1(zN).Foik =
O y 1, . . . , 7 V - 1 ,
kp = q kN + r k,
where q k an d r ^ are , respectively , th e quotien t an d remainde r o f kp divided b y N. Thes e integer s satisf y q k > 0 and 0 < r k < N — 1. (a) Prov e tha t r k + r^-k = ^ an dg % + ^TV—/ c = P — 1 (b) Sho w tha t ther e exist s a n i V x i V permutatio n P suc h tha t 0 1
ro 1 =P
_N-l_
ri
jnv-ij
(c) Fin d th e firs t colum n an d th e firs t ro w of T(z). Writ e you r answe r in term s o f Ci(z\ r^ , an d qi. (d) Sho w tha t T(z) i s a Toeplit z matrix . Tha t is , [ T ( Z)\kl
[T(z)]o,j-fc, i f Z > fc; [T(0)]fc_,,o, i f fc > Z.
Using th e resul t i n (c) , writ e T(z) i n term s o f Ci(z\ r^ , an d qi. (e) Sho w tha t
T(z) = B(z)PC
T
ps(z)P
B(z-1),
where C ps(z) i s th e pseudocirculan t matri x give n i n (5.3) , P i s a permutation matrix , an d D(z) i s a diagonal matrix with [D(z)]fcf c = znk fo r som e intege r n k. This problem shows that b y blocking a scalar channel using a delay chai n and a n advanc e chai n wit h z~ p1 th e resultin g transfe r matri x T(z) i s closely relate d t o C ps(z). Fo r example , al l th e column s o f T(z) ca n b e obtained fro m it s firs t colum n b y followin g a mechanis m tha t i s ver y similar t o tha t i n Definitio n 5.1 , except tha t th e recirculate d elemen t i s multiplied b y z~ p rathe r tha n z~ x.
5.7. Problem s 13
1
5.5 Prov e t h a t a matri x Q circ satisfie s Definitio n 5. 2 i f an d onl y i f it s (fc,ra)th entr y satisfie s (5.6) . 5.6 Sho w t h a t th e invers e (i f i t exists ) o f a circulan t matri x i s als o circulant . 5.7 I s th e produc t o f tw o N x N circulan t matrice s als o circulant ? Justif y your answer . 5.8 Variations of circulant matrices. A n M x M matri x C Up,drc i s s a i d t o be up circulant i f an y colum n o f G Up,circ i s obtaine d fro m th e precedin g column b y performin g a n up shif t followe d b y recirculatio n o f th e ele ment t h a t spill s over . Not e t h a t thes e matrice s ca n als o b e obtaine d b y applying a lef t shif t t o th e row s followe d b y recirculation . Henc e the y are als o calle d left circulant. I n vie w o f this , circulan t matrice s define d in Definitio n 5. 2 ar e als o know n a s th e down o r right circulan t matrices . (a) Giv e a n exampl e o f a 5 x 5 u p circulan t matrix . (b) Le t C o = C i J , wher e J i s the MxM reversa a 3 x 3 reversa l matri x i s
l matrix . Fo r example ,
" 0 0 1 " 01 0 10 0 Show t h a t C o i s a dow n circulan t matri x i f an d onl y i f C i i s a n u p circulant matrix . 5.9 Usin g th e result s o f Proble m 5.8(b) , modif y Theore m 5. 2 s o t h a t i t i s true fo r u p circulan t matrices . Prov e you r result . 5.10 Sho w t h a t th e matri x A = circulant.
W D W ^ , wher e D i s a diagona l matrix , i s
5.11 I n Subsectio n 5.3.2 , w e focu s onl y o n circulan t matrice s t h a t ar e in dependent o f z. On e ca n generaliz e Definitio n 5. 2 t o includ e circulan t matrices C circ(z) i n whic h al l th e entrie s o f C circ(z) ca n b e a functio n of z. A 3 x 3 exampl e i s C0(z) C 2(z) d(z) d(z) C 0(z) C C2(z) d(z) C Can w e expres s C circ(z) a
2(z)
0(z)
s
Ccirc(z) =
P~ 1r{z)P,
for som e diagona l matri x T(z)7 I f w e can , wha t i s P an d ho w i s T(z) related t o th e entrie s Ci(z) i n th e first colum n o f C circ(z)7 5.12 I n Fig . 5.9 , w e us e a dela y chai n t o bloc k th e inpu t an d a n advanc e chain t o bloc k th e outpu t o f th e channel . Suppos e instea d t h a t a n advance chai n an d a dela y chai n ar e use d t o bloc k th e inpu t an d outpu t of a channe l C(z), respectively . Fin d th e transfe r matri x C ps(z) o f th e blocked system .
132
5. Multirat
e formulatio n o f communicatio n system s
5.13 Le t tw o N x N matrice s A(z) an d B ( z ) b e relate d b y
B(z)=A(z)A(zN)A(z-1), where A(z) i s th e diagona l matri x diag[ l z~ x • • • z~ N+1\. Sho w tha t A(z) i s pseudocirculan t i f an d onl y i f B ( z ) i s circulant . Furthermore , show t h a t i f A(z) i s causal , the n B ( z ) i s als o causal . 5.14 Usin g th e result s i n Problem s 5.1 1 an d 5.13 , sho w t h a t th e produc t o f two N x N pseudocirculan t matrice s i s als o pseudocirculant . 5.15 Prov e t h a t an y N x N pseudocirculan t matri x C ps(z) satisfie = Cps(z) J
s
NCpS(z)JN,
where J AT i s th e N x N reversa l matri x (se e Proble m 5.8(b)) . Sup pose t h a t th e transceive r i n Fig . 5. 7 i s ISI-free . Sho w t h a t th e ne w transceiver wit h th e transmittin g matri x J T V S T ( 2 ; ) an d th e receivin g matrix G T(Z)JN i s als o ISI-free . 5.16 Sho w t h a t i n th e C P schem e whe n M = v th e outpu t vecto r r(n) i s stil l related t o s(n ) b y r(n) = C circs(n), for som e M x M circulan t matri x C c ^ r c . Expres s C c ^ r c i n term s o f th e channel impuls e respons e c(0) , c ( l ) , . . . , c(y). 5.17 Conside r tw o sequence s o f lengt h M , c(n) , an d s(n) , fo r 0 < n < M — 1. Define th e M-poin t sequenc e x(n) a s x n =
M-l
( ) ^2
s k c n
k=0
( ) (( - k))M>
Let s = [s(0 ) s ( l ) • • • s(M - 1)] T an d x = [x(0) x(l) • • • x(M - 1 ) ] T . Show t h a t ther e exist s a circulan t matri x Q circ suc h t h a t x = C c i r c s . Express th e entrie s o f C circ i n term s o f c(n). 5.18 CP system with v > M. Suppos e t h a t i n a C P syste m th e inpu t bloc k size i s M = 3 an d th e C P lengt h i s v = 5 . Le t s = [s(0 ) s ( l ) s(2) ] b e the inpu t block . The n w e ca n vie w th e followin g sequenc e o f lengt h 8 as th e resul t o f addin g v cyclic-prefi x sample s t o th e inpu t block : [ S (l) 8(2) S(0) S(l)
8(2) 8(0) 5(1 ) 8(2)].
Suppose t h a t th e channe l c(n) ha s orde r 5 . Sho w t h a t th e outpu t vecto r r an d th e inpu t bloc k s(n ) ar e relate d b y r(n) = C circs(n) fo r som e 3 x 3 circulan t matri x C circ. Writ e C circ i n term s o f c(n). 5.19 Cyclic-suffixed systems and cyclic-prefixed/suffixed systems. Le t s(n ) b e input block s o f siz e M. Le t u s cop y th e first v sample s a t th e beginnin g of eac h bloc k int o th e v location s a t th e end . Th e resul t i s th e vecto r [ so(n) V
v
si(n
) ••
•s
M-i(n)
' original bloc k suffi
:
V
s 0(n) •• v
•s
' x
1 / _i(n)
]
T
.
5.7. Problem s 13
3
This operatio n i s called cycli c suffixing . Mor e generall y w e ca n combin e cyclic prefixing wit h cycli c suffixing. T h e resul t i s a hybri d syste m whic h will b e calle d th e cyclic-prefixed/suffixe d system . Fo r example , w e ca n copy th e las t Z/ Q sample s a t th e en d o f eac h bloc k int o th e Z/ Q location s at th e beginnin g an d cop y th e first v\ sample s a t th e beginnin g o f eac h block int o th e v\ location s a t th e end , wher e z/ 0 + z/ i = ^ . T h e resul t i s the vecto r x ( n ) give n b y [ SM-VQ
'
"S
M-1 - S
OS
i ••
prefix origina
• SM-1
-S
O S
I ••
l bloc k suffi
•S
Ul-i
]
x
where w e hav e droppe d th e dependenc y o n n fo r simplicity . Suppos e t h a t th e receive r o f th e cyclic-prefixed/suffixe d syste m i s th e sam e a s t h a t o f th e cyclic-prefixe d syste m i n Fig . 5.15 . Sho w t h a t th e retaine d block r ( n ) i s fre e fro m IBI . W h a t i s th e relatio n betwee n r ( n ) an d s(n) ? 5.20 Conside r a minimum-redundanc y bloc k transceive r wit h M = 2 . Sup pose t h a t th e channe l i s C(z) = 1 + z~ x + z~ 2. W h a t ar e th e minimu m redundancy pmin an d th e Toeplit z matri x C p ? Doe s ther e exis t a zero forcing equalizer ? 5.21 Repea t th e abov e proble m fo r th e channe l C(z) =
1 + z~ x + z~ 2 + z~
5.22 Conside r a n F S E syste m wit h oversamphn g facto r N = T = 1 an d th e wavefor m (jpi * c a * P2)(f) i s give n b y / \,.\ f (pi*ca*p2)(t) =
{
0
3
.
2 . Suppos e t h a t
-0.5|t-2 |+ l , for0<£<4 ' ~ ~ otherwise.
;
(a) Fin d th e discrete-tim e equivalen t channe l C(z). (b) Fin d th e Typ e 1 polyphas e component s Go(z) an d Gi(z) o f th e channel C(z). (c) Fin d a zero-forcin g F I R filter
H(z).
(d) Giv e anothe r zero-forcin g F I R filter H'\z) t h a t i s no t a delaye d o r scaled versio n o f th e filter H(z) i n (c) . 5.23 Suppos e t h a t w e obtai n a n F S E syste m b y increasin g th e samplin g rat e of th e receive r o f a n SS E syste m an d keepin g th e sam e receivin g pulse . Show t h a t i f th e F S E syste m doe s no t hav e a causa l stabl e zero-forcin g equalizer, the n neithe r doe s th e SS E system . 5.24 Maximum ratio combining (MRC). Suppos e t h a t w e transmi t a symbo l s wit h signa l powe r £ s ove r a se t o f paralle l subchannels , a s show n i n Fig. P5.24 , wher e Q ar e th e subchanne l gain s (possibl y complex ) an d qi ar e th e subchanne l noise . Suc h a syste m o f paralle l subchannel s ma y be th e resul t o f oversamphn g o r employin g multipl e antenna s a t th e receiver. Assum e t h a t qi ar e zero-mea n an d uncorrelate d wit h variance s A/o, an d the y ar e uncorrelate d wit h th e signa l s . A t th e receiver , th e received sample s ar e Ti = Q S + ^ . Suppos e w e linearl y combin e Ti t o obtain s " as follows : ?=a
0r0
+
ai nH h
AM-I^M-I .
134
5. Multirat
e formulatio n o f communicatio n system s
We ca n writ e s~ = as + r , wher e th e quantit y r depend s onl y o n th e noise component s qi. a0 -0—
CO
i M-\
^
->
q
^ "M-l
Figure P 5 . 2 4 . Maxima l rati o combinin g transmissio n scheme .
(a) Defin e th e vector s c = [c 0 ci ... c a = [a 0 ai ... a q=[<7o <7 i •• •
M-i]
T
M-i]
9M-I]
,
T
T
,
.
Express a , r , an d th e signa l t o nois e rati o
E[\A2\ in term s o f th e vector s c , a , an d q . (b) Sho w that th e SNR is maximized when ai = c * for i = 0 , 1 , . . . , M1, an d th e maximize d SN R i s give n b y (icoP + ICi p + .- . + l c M - l l 2 ) ^ A/-0
Such a n optima l receive r i s know n a s th e maxima l rati o combinin g (MRC) receiver . Th e phas e o f th e MR C receive r ai = c * cancel s th e phase o f Q S O that th e signa l componen t a t th e outpu t o f th e receive r adds constructively . Moreover , th e MR C receive r put s mor e weightin g on th e subchannel s wit h large r receive d signa l powers . Th e expressio n of th e maximize d SN R show s tha t eac h on e o f th e M subchannel s i s useful, n o matte r ho w smal l th e subchanne l SN R is .
6 DFT-based transceiver s T h e mos t commonl y use d typ e o f bloc k transceive r i s th e so-calle d DFT-base d transceiver, i n whic h th e polyphas e matrice s o f th e transmitte r an d receive r are relate d t o low-cos t D F T matrice s i n a simpl e way . T h e DFT-base d transceiver ha s foun d application s i n a wid e rang e o f transmissio n channels , wired [12 , 154 ] o r wireles s [27] . I t i s typicall y calle d a D M T (discret e multi tone) syste m fo r wire d DS L (digita l subscribe r line ) application s [7 , 8 ] an d a n O F D M (orthogona l frequenc y divisio n multiplexing ) syste m fo r wireles s loca l area network s [54 ] an d broadcastin g applications , e.g . digita l audi o broad casting [39 ] an d digita l vide o broadcastin g [40] . I n a n OFD M o r DM T transceiver, th e transmitte r an d receive r perform , respectively , I D F T an d D F T computations . Anothe r typ e o f DFT-base d transceiver , calle d a single carrier syste m wit h cycli c prefi x (SC-CP) , ha s als o bee n o f grea t importanc e in wireles s transmissio n [55 , 128 , 130] . T h e SC-C P syste m transmit s a bloc k of symbol s directl y afte r insertin g a cycli c prefi x whil e th e receive r perform s b o t h D F T an d I D F T computations . For wireles s transmission , th e channe l stat e informatio n i s usuall y no t available t o th e transmitter . T h e transmitte r i s typically channel-independen t and ther e i s n o bi t o r powe r allocation . Havin g a channel-independen t trans mitter i s als o a ver y usefu l featur e fo r broadcastin g applications , wher e ther e are man y receiver s wit h differen t transmissio n paths . I n O F D M o r SC-C P systems fo r wireles s applications , ther e i s usuall y n o bi t an d powe r alloca tion. T h e transmitter s hav e th e desirabl e channel-independenc e property . T h e channel-dependen t par t o f th e transceive r i s a se t o f M scalar s a t th e receiver, wher e M i s th e numbe r o f subchannels . I n D M T system s fo r wire d DSL applications , signal s ar e transmitte d ove r coppe r lines . T h e channe l doe s not var y rapidly . Thi s allow s tim e fo r th e receive r t o sen d bac k t o th e trans mitter th e channe l stat e information , base d o n whic h bi t an d powe r allocatio n can b e optimized . Usin g bi t allocation , th e disparit y amon g th e subchanne l noise variance s i s exploited i n th e D M T syste m fo r bi t rat e maximization . T h e D M T syste m ha s bee n show n t o b e ver y efficien t fo r high-spee d transmission . In thi s chapte r w e wil l giv e a detaile d analysi s o f DFT-base d transceivers . 135
136
6. DFT-base d transceiver s
s
o>
s
lt
• • • $M-\
• • •
x
o so
►
q{n)
+ \
w
y
^0
\
P/S(M)|
cyclic prefix
1/C0
x(n) discard prefix
C(z)
A
>►
r
M-\
transmitter
• • • p.
^M-\
• •A • S
M-\
i/cM_,
receiver Figure 6 . 1 . Th e OFD M system .
6-1 O F D
M system s
Chapter 5 explain s t h a t wit h bloc k transmissio n an d prope r insertio n o f re dundant samples , ther e wil l b e n o interbloc k interferenc e (IBI) . Tw o commo n approaches o f adding redundan t sample s ar e zer o paddin g an d cycli c prefixing . T h e O F D M transceive r i s a n exampl e o f cyclic-prefixin g systems . W h e n th e channel orde r i s no t large r t h a n th e prefi x lengt h z/ , IB I ca n b e easil y remove d by discardin g th e prefi x a t th e receiver . Becaus e ther e i s n o IBI , w e wil l con sider th e transmissio n o f onl y on e block , i.e . one-shot transmission, i n th e discussion o f syste m performance . Althoug h th e block s ar e sen t ou t consecu tively i n actua l transmission , ther e i s n o los s o f generalit y i n considerin g th e simple one-sho t scenario . Figur e 6. 1 give s th e bloc k diagra m o f th e O F D M system, wher e a singl e inpu t bloc k an d a singl e outpu t bloc k ar e shown . T h e input o f th e transmitte r s i s a n M x 1 vecto r o f modulatio n symbols . A s there i s usuall y n o bi t an d powe r allocatio n i n th e O F D M system , th e in put symbol s Sk hav e th e sam e constellatio n an d th e sam e varianc e £ s. T h e symbols ar e assume d t o b e zero-mea n an d uncorrelated , whic h i s usuall y a reasonable assumptio n afte r prope r interleavin g o f th e inpu t bi t stream . T h e autocorrelation matri x o f th e inpu t vecto r s i s thu s give n b y R s — £s±MSI T h e channe l c(ri) i s a n F I R filter wit h orde r L < z/ , V
C(z) =
J2c(n)z-
n
,
where w e hav e use d inde x u p t o v fo r convenience . (Fo r L < z/ , c(L + 1 ) = c(L + 2 ) = • • • = c(y) = 0. ) Fo r wireles s applications , th e channe l nois e q(n) i s usually modele d a s a circularl y symmetri c comple x Gaussia n rando m proces s with zer o mea n an d varianc e A/o - Th e channe l nois e q(n) i s assume d t o b e uncorrelated wit h th e symbol s s^ .
6.1. OFD M system s
137
At th e transmitte r IDF T i s applie d t o th e inpu t symbols , x = W fs, where W i s th e normalize d M x M DF T matri x give n b y [W] m n = _ L e - ^ ™ W M ? o
< m,n < M - 1 .
Then th e vecto r x i s unblocke d an d a cycli c prefi x o f lengt h v i s inserte d before th e sequenc e i s transmitted t o th e channel . A t th e receiver , th e prefi x is discarded an d th e DF T matri x i s applied . Whe n ther e i s n o channe l noise , recognize that th e syste m fro m x t o r is the prefixe d syste m discusse d i n grea t detail i n Section 5.4 . Th e transfer matri x i s the M x M circulan t matri x C circ given i n Definitio n 5.2 . I n th e presenc e o f channe l noise , th e receive d vecto r r is r = C c i r c x + q , (6.1 ) where q i s th e blocke d channe l nois e vecto r o f siz e M. Th e DF T matri x a t the receive r ha s outpu t vecto r y give n b y y = W C a r c x + r , wher
er
= Wq .
We als o kno w fro m Theore m 5. 2 tha t circulan t matrice s ca n b e diagonalize d using DF T an d IDF T matrices , Ccirc =
W t r W. (6.2
)
The matri x T i s a diagona l matri x wit h diagona l element s correspondin g t o the M-poin t DF T o f c(n) , i.e . r = dia g [C 0 d where C
k
= C(z)
-"
\z=ej2irk/M —
C
M-i]
,
Yl c ( n ) e
-j27rkn/M
Q
The DF T decompositio n o f Q circ lead s t o th e followin g expressio n fo r y : y = W C a r c Wfs + r = T s + r , x
or equivalentl y
yk = C kskJrrk. (6.3
)
Each y k i s simpl y s k scale d b y C k plu s nois e r k. Th e channe l i s thu s diago nalized usin g a n IDF T matri x an d a DF T matrix . Equivalent paralle l subchannels Wit h the channel diagonalized, the scala r FIR channe l C(z) i s converte d t o M paralle l subchannel s (Fig . 6.2(a)) . Th e scalar C k i s called th e M h subchanne l gain . Ther e i s no inter subchanne l ISI , and equalizatio n ca n b e easil y don e b y multiplyin g y k wit h th e scala r 1/Cfc , which i s usuall y referre d t o a s a frequenc y domai n equalize r (FEQ) . I t i s so name d becaus e th e equalizatio n i s carrie d ou t afte r DF T computations . The overal l M x M transfe r matri x T fro m th e transmitte r inpu t vecto r s t o the receive r outpu t vecto r s i s the M x M identit y matri x an d th e receive r i s zero-forcing. Th e overal l transceiver ca n b e viewe d a s a system o f M paralle l subchannels (Fig . 6.2(b)) , wher e th e onl y distortion s ar e th e additiv e nois e sources.
138
6. DFT-base d transceiver s
Jo
J£
C„
yn
FE
i/c
^1
(a)
l/C,
-t>—►*-
ri
Q
i/c„
-► J o -►*,
"► * ,
equivalent additiv e nois e
(b)
e0=x0/C0
J^ I e M-r \
"► ^
0
-► s
A
/C\
ic
Figure 6.2 . Equivalen t parallel-subchanne l mode l fo r th e OFD M system .
Complexity I n vie w o f th e bloc k diagra m o f th e O F D M system , w e ca n see t h a t th e mai n computatio n o f th e transceive r come s fro m th e I D F T an d D F T matrices . Fo r b o t h matrice s th e Fas t Fourie r Transfor m ( F F T ) ca n be use d i n th e computatio n whe n M i s a powe r o f 2 , whic h i s usuall y th e choice o f D F T siz e i n practice . Th e complexit y i s i n th e orde r o f M l o g 2 M , i.e. i n th e orde r o f log 2 M pe r inpu t modulatio n symbol . Th e complexit y of th e transmitte r i s mostl y t h a t o f a n I D F T matrix . Fo r th e receiver , th e complexity i s t h a t o f a D F T matri x plu s M multiplication s fo r th e F E Q part. I n th e contex t o f syste m implementation , i t i s usuall y desirabl e t o hav e as fe w channel-dependen t component s a s possible . I n particular , a channel independent transmitte r mean s ther e i s n o nee d t o wai t fo r th e receive r t o send bac k th e channe l information . Thi s i s especiall y importan t i n wireles s applications, wher e th e channe l ca n chang e rapidl y an d leave s littl e tim e fo r updating th e transmitter . I t i s als o o f vita l importanc e i n broadcastin g ap plications wher e ther e ar e on e transmitte r an d man y receivin g ends . Fro m the transmitte r t o eac h receiver , ther e i s a correspondin g channel . I t woul d be ver y difficul t t o desig n th e transmitte r t o accommodat e differen t chan nels a t th e sam e time . Th e transmitte r o f th e O F D M syste m ha s th e muc h desired channel-dependenc e property . Th e whol e O F D M syste m i s channel -
6.1. OFD M system s
139
independent excep t fo r th e FE Q coefficients . Onl y th e FE Q coefficient s nee d to b e recompute d a s th e channe l varies . Cyclic prefix Th e cyclic prefix act s as a buffer betwee n consecutive blocks. When a signa l i s passe d throug h th e channel , wha t actuall y take s plac e i s linear convolution . Th e insertio n o f a cycli c prefi x allow s u s t o fabricat e cir cular convolution , whic h lend s itsel f t o channe l diagonalizatio n usin g simple , channel-independent DF T an d IDF T matrices . However , prefixin g als o lead s to bandwidt h expansio n b y a facto r o f ( = ( M + v)jM. Th e expansio n wil l be mino r i f M i s chose n t o b e significantl y large r tha n z/ , i n whic h cas e £ i s close t o unity . Fo r example , th e OFD M syste m i n th e applicatio n o f wireles s local are a network s [54 ] has M = 6 4 an d v = 16 , so ( = 1.25 , a 25 % increase in bandwidth . Peak t o averag e powe r rati o Th e pea k t o averag e powe r rati o (PAPR ) of a signa l x(n) i s define d as 1 ^ . ^^ max n
PAPR-
v n n | |x(n)|
2
E[\x{n)\*] '
2
where max n |x(n)| i s use d her e t o denot e th e maximu m absolut e o f al l pos sible value s o f th e rando m variabl e x(n). A transmitte d signa l wit h a larg e PAPR wil l translat e t o lo w efficienc y fo r th e powe r amplifier o f th e trans mitter [149] . PAP R i s a critica l issu e i n th e desig n o f transmitters . Fo r th e OFDM system , th e transmitte r outpu t signa l x(n) i s obtaine d b y unblockin g the IDF T outpu t vecto r an d addin g a cyclic prefix. Therefor e th e transmitte r output ha s th e sam e pea k powe r an d th e sam e averag e powe r a s th e IDF T outputs, whic h ar e denote d b y Xk i n Fig . 6.1 . W e ca n conside r th e rati o maxfc \x k\2 instead. Th e outpu t vecto r o f the IDF T matri x x = W^~ s has autocorrelatio n matrix R x = ^[xx^ ] give n b y RX =
W^R SW=
S SIM-
Therefore x n ar e uncorrelate d an d thei r variance s ar e th e same , equa l t o £ s. That i s E[\xn\2]=Ss, n = 0 , l , . . . , M - l . (6.4 ) To comput e th e pea k power , not e tha t M-i1 k=0 1
A mor e relevan t quantit y i n practic e i s th e continuous-tim e PAPR , whic h i s denne d a s PAPRt
~ E[\x
a{t)\*\
'
where x a(t) i s th e continuous-tim e transmitte d signal . I n thi s definition , PAPRt depend s on th e transmittin g puls e pi(t) a t th e transmitter . I f a PAP R reductio n metho d i s no t applied, PAPRt i s usuall y onl y slightl y large r tha n th e PAP R denne d ove r th e discrete time transmitte d signa l [149] .
6. DFT-base d transceiver s
140 and
kn| =
M - 11 lM-1
I
So we hav e
°~ '
1
M - M - 11
vMmax|sfc|
max |x n = v M m a x |sfc| . n
.
(6.6)
T h e maximu m valu e \/Mmaxf c |sfc | i s attaine d when , fo r example , al l inpu t symbols hav e th e sam e valu e an d | s n | = max/ e \sk\- Combinin g (6.4 ) an d (6.6) , we arriv e a t th e followin g expressio n o f P A P R :
PAPR = M^y
.
T h e rati o o n th e right-han d sid e i s th e P A P R o f th e inpu t symbols . T h e P A P R fo r th e O F D M syste m i s M time s th e P A P R o f th e inpu t modulatio n symbols. Fo r som e applications , e.g . fixed broadban d wireles s acces s system s [55], M ca n b e a s larg e a s 2048 , whic h lead s t o a 3 3 d B increas e i n P A P R . Many method s hav e bee n devise d t o reduc e th e P A P R o f th e O F D M system , e.g. codin g [57 ] an d ton e reservatio n [148] . A mor e complet e t r e a t m e n t o f thi s topic an d relate d reference s o n P A P R reductio n ca n b e foun d i n [48 , 149] . Continuous-time transmitte r outpu t Th e transmitte r outpu t x(n) indi cated i n Fig . 6. 1 i s a discrete-tim e signal . Th e continuous-tim e outpu t x a(t) is relate d t o x{n) b y (Sectio n 2.1 ) x
a(i) = ^2x(k)p1(t - fcT),
where pi(t) i s th e transmittin g pulse . Th e signa l x a(t) ca n b e furthe r writte n in term s o f th e inpu t symbol s Sk usin g th e relatio n i n (6.5) . I n th e literature , the outpu t o f th e O F D M transmitte r i s commonl y writte n a s
xa(t) = Y,s
2
ke-
^t
for som e frequenc y / o o r
xa(t) = J2 sk9(t)e-2*kfot if a puls e shapin g filter g(t) i s present . Th e expressio n is , i n fact , incorrect Such a n expressio n i s tru e onl y fo r a n all-analo g implementatio n o f O F D M system. I t ca n b e show n t h a t th e outpu t o f a n O F D M transmitte r t h a t i s implemented usin g a n I D F T matri x followe d b y a D / C converte r (wit h o r without cycli c prefixing ) doe s no t hav e th e abov e expressio n [79] .
6.1.1 Nois
e analysi s
As th e O F D M syste m i n Fig . 6. 1 i s zero-forcing , th e outpu t erro r e ^ = Sk — Sk comes solel y fro m th e channe l noise . T o analyz e th e effec t o f th e channe l noise, w e dra w i n Fig . 6. 3 th e nois e p a t h a t th e receiver , wher e w e conside r only th e channe l nois e an d n o signal . W e assum e th e nois e q(n) i s a circularl y
6.1. OFD M system s
141
1/Q q(n)
discard prefix
S/P(AQ
W
-► e o
1/C,
QM-I
-► ^M- l 1/^M-l
Figure 6.3 . Nois e pat h a t th e receive r o f th e OFD M system .
symmetric comple x Gaussia n rando m proces s wit h zero-mea n an d varianc e A/o. Th e element s o f th e blocke d nois e vecto r q ar e uncorrelate d an d henc e also independen t becaus e uncorrelate d Gaussia n rando m variable s ar e inde pendent. Th e autocorrelatio n matri x o f q i s H q = A/OIM - Th e nois e vector r at th e outpu t o f th e DF T matri x W ha s autocorrelatio n matri x Rr = JV
0WW
t
= JVoIM.
Therefore r ^ ar e als o uncorrelate d Gaussian , wit h varianc e A/o ; th e vecto r r ha s th e sam e statistic s a s q . Th e kth subchanne l erro r i s e ^ = r^jC^. The error s continu e t o b e uncorrelate d Gaussia n rando m variables , bu t wit h different variances . Th e varianc e o f e ^ i s ^ 0
(6.7)
\Ck 12' The averag e mea n square d erro r £ rr = £
A/o M
1 / M J ] fc=0 az i s henc e M-l
E \Ci\
(6.8)
As the subchanne l error s are independent, symbo l detection ca n be performe d for eac h subchanne l separatel y withou t los s i n performance . The SN R o f the fcth subchanne l /3(fc ) = £ a/
e7
= £ S/Af0.
(6.9)
We observ e tha t th e subchanne l SN R P 0fdm(k) ls dependen t o n th e channe l response. Fo r highl y frequency-selectiv e channels , PofdmiX) c a n var y signifi cantly wit h respec t t o k. I f the channe l profil e i s availabl e t o th e transmitter , the disparit y amon g subchannel s ca n b e exploite d usin g bi t an d powe r al location, whic h ca n improv e th e syste m performanc e significantly . Thi s i s routinely don e i n DM T system s fo r DS L applications , a s w e wil l explai n i n Section 6.6 .
142
6. DFT-base d transceiver s
A not e o n MMS E receptio n Th e us e o f a n MMS E receive r ca n minimiz e the outpu t erro r an d improv e th e syste m performanc e i n general . Unfortu nately, thi s i s no t th e cas e fo r th e O F D M system . W e ca n se e thi s b y exam ining it s MMS E receiver . Conside r Fig . 6.1 . Give n th e receive d observatio n vector r , th e MMS E receive r i s th e optima l estimato r o f th e vecto r s . I t ca n be show n t h a t (Proble m 3.9) , withou t los s o f generality , w e ca n conside r th e problem o f estimatin g s fro m th e vecto r y i n Fig . 6.1 . Fro m (6.3 ) w e kno w t h a t yk = CkSk + Tk- Bu t th e symbol s Sk ar e uncorrelate d an d s o ar e ?> . T h e optimal estimato r become s a diagona l matrix . T h a t is , th e vecto r MMS E receiver reduce s t o M scala r MMS E receiver s (Proble m 3.8) . W e kno w fro m Example 3. 6 t h a t scala r MMS E receiver s d o no t improv e th e erro r rat e per formance. Therefor e fo r th e O F D M syste m MMS E receptio n wil l no t b e used ; we wil l onl y conside r th e zero-forcin g receiver .
6.1.2 Bi
t erro r rat e
For wireles s applications , usuall y ther e i s n o bi t allocation ; al l th e symbol s carry th e sam e numbe r o f bits . I n thi s case , th e averag e bi t erro r rat e ca n b e obtained b y averagin g th e subchanne l erro r rates . Th e calculatio n o f bi t erro r rate depend s o n th e modulatio n schem e used . Fo r Q P S K symbol s wit h powe r Ss, th e symbol s hav e th e for m zb \j£ s j2 z b j^£s/2. I n thi s cas e th e B E R ca n be compute d exactl y (Sectio n 2.3) . Th e B E R o f th e i t h subchanne l i s
n )=0
' (v^) = 0 ( v / / 3 o / d m ( i ) )
T h e functio n Q(-) , a s define d i n Sectio n 2. 3 representin g th e are a unde r th e Gaussian tail , i s reproduce d belo w fo r convenienc e
i r°° Q(x) =
- =/
e"^
2
dr.
Using (6.9) , th e averag e B E R o f th e O F D M syste m i s give n b y 1
M-
l
Vofdm =MY.Q {VW) •
(6-10
For mor e genera l 26-bi t QA M symbols , w e ca n comput e V{i) usin g th e ap proximation i n (2.18 ) an d averag e t h e m t o obtai n th e averag e B E R o f th e system. Example 6. 1 W e assum e t h a t th e nois e i s AWG N wit h varianc e A/o . T h e modulation symbol s ar e Q P S K wit h value s equa l t o ±^/£ s/2 z b j^/Ss/2 an d SNR 7 = £ s/Afo. Th e numbe r o f subchannel s M i s 6 4 an d th e lengt h o f cycli c prefix v i s three . Tw o channels , a s i n Tabl e 6.1 , with fou r coefficient s ( L = 3 ) will b e used . Th e magnitud e response s (i n dB ) o f th e tw o channel s c\ (n) an d C2{n) ar e show n i n Fig . 6.4 . Th e channe l ci(n) ha s a n almos t flat magnitud e response whil e C2{n) ha s a zer o aroun d uo = 1.17 T an d it s magnitud e respons e shows mor e variations . Thes e tw o channel s hav e th e sam e energy . Th e B E R performance wil l b e obtaine d throug h Mont e Carl o simulatio n (Sectio n 2.3) .
)
6 . 1 . O F D M system s 14
3
n
ci{n)
C2(n)
0 1 2 3
0.3903 + J0.1049
0.3699+ J0.5782
0.6050 + J0.1422 0.4402+ J0.0368 0.0714+ J0.5002
0.4053+ J0.575
0.0834+ J0.0406
-0.1587 + J0.0156
Table 6 . 1 . Tw o channel s wit h fou r coefficients .
20
IC/ehl
10
"
0 _ -1 0 CO
-
-20
-50
S ' /
\1
\ / \ 1 1
-30 -40
ic2«>)f
-
\
-
0.5 1 1. 52 Frequency normalize d bv7i.
Figure 6.4 . Magnitud e response s o f th e tw o channel s c i ( n ) an d C2(n) .
For channe l ci(n) , th e subchanne l SNR s fo r 7 = 1 6 d B ar e show n i n Fig. 6.5(a) . Figur e 6.5(b ) show s th e correspondin g erro r rate s fo r individua l subchannels. Th e subchannel s wit h erro r rate s smalle r tha n 1 0 - 6 ar e no t shown in the figure . W e can see that som e subchannels hav e larger erro r rate s and these correspond to where the channel has dips in the magnitude response . For 7 = 1 6 dB , th e goo d subchannel s hav e erro r rate s les s tha n 1 0 - 6 . Th e error rate s o f the ba d subchannel s ar e as large as 1 0 - 2 , an d thes e subchannel s will b e dominatin g whe n th e averag e BE R i s computed . Th e averag e bi t error rat e a s a functio n o f 7 i s show n i n Fig . 6.5(c) . Th e averag e erro r rat e for 7 = 1 6 d B i s aroun d 1 0 - 3 . W e perfor m th e sam e se t o f experiment s o n channel C2(n) an d sho w the results in Fig. 6.6. Fo r 7 = 1 6 dB, the subchannel s near th e channe l nul l hav e a hig h erro r rat e o f aroun d 0.5 . A s a result , th e average error rat e i s dominated b y these subchannels . W e can se e the averag e error rat e stall s fo r 2 0 < 7 < 5 0 dB . Thi s i s becaus e i n thi s SN R rang e al l
144
6. DFT-base d transceiver s
the subchannel s hav e erro r rate s clos e t o zer o excep t fo r thos e clos e t o th e channel zero . Suppos e on e subchanne l ha s a n erro r rat e o f 0.5 , th e averag e error rat e wil l b e a t leas t 0 . 5 / M = 1/128 . T h e averag e erro r rat e wil l sta y aroun d thi s valu e unti l 7 i s larg e enoug h t o bring dow n th e erro r rat e o f th e wors t subchannels . Not e t h a t th e range s o f SNR i n Fig . 6.5(c ) an d Fig . 6.6(c ) ar e ver y different . Fo r example , t o obtai n a B E R o f 1 0 - 4 , w e nee d les s t h a n 2 0 d B fo r c\{n) bu t mor e t h a n 4 0 d B fo r C2(n), althoug h thes e tw o channel s hav e th e sam e energy . Th e differenc e i s mainly du e t o th e channe l null . ■ Many technique s hav e bee n develope d i n the literatur e t o desig n transceiver s more robus t t o channe l zero s o n th e uni t circl e (als o know n a s spectra l nulls) . For example , error-correctin g cod e i s generall y applie d t o th e bi t strea m be fore bit s ar e m a p p e d t o modulatio n symbols . W h e n mos t subchannel s hav e small erro r rates , i t i s easie r t o correc t th e bit s o f th e ba d subchannel s base d on th e bit s fro m th e goo d subchannels . W e ca n als o preven t th e performanc e being dominate d b y th e ba d subchannel s b y usin g a precode r (Chapte r 7) , o r using bi t allocatio n (Chapte r 8) . Anothe r wa y t o improv e robustnes s i s t o us e zero paddin g rathe r t h a n cycli c prefixing , a s w e wil l sho w i n Sectio n 6.2 .
6.1. OFD M system s
(a)
01
02
03 04 05 subchannel index
0
60
(b)
(c)
10 1 52 0 y=Es/NQ (dB )
Figure 6.5 . Exampl e 6.1 . Performanc e o f th e OFD M syste m ove r channe l c i ( n ) . (a ) Subchannel SNR s for 7 = 1 6 dB; (b ) bi t erro r rate s o f individua l subchannel s fo r 7 = 1 6 dB; (c ) averag e bi t erro r rate .
146
6. DFT-base d transceiver s
ou
20 m "O
rr
(a)
7 CO CD CD
" X /^
10
"
s. /
\/ \/ \/ \
U -10
r
O X)
-20
"
I
-30
01
0
20 3 04 0 subchannel index
50 6
0
01
0
20 3 04 0 subchannel index
50 6
0
(b)
(c)
y=E s /N 0 (dB )
Figure 6.6 . Exampl e 6.1 . Performanc e o f th e OFD M syste m ove r channe l C2(n). (a ) Subchannel SNR s for 7 = 1 6 dB; (b ) bi t erro r rate s o f individua l subchannel s fo r 7 = 1 6 dB; (c ) averag e bi t erro r rate .
147
6.2. Zero-padde d O F D M system s
6.2 Zero-padde
d O F D M system s
The insertio n o f guard interval s allow s u s to conside r th e simple r proble m of one-shot bloc k transmission . Cycli c prefixin g i s the mos t commo n typ e o f guard interval . Anothe r popula r for m o f guard interva l is zero padding. Whe n the numbe r o f zeros padde d t o each bloc k i s not smaller tha n th e order o f the channel, ther e i s no IBI and agai n w e only nee d t o conside r th e transmissio n of a singl e block . Th e transmitter o f a zero-padde d OFD M syste m (Fig . 6.7) is the same as that o f the prefixed OFD M system , excep t tha t cycli c prefixin g is replace d b y zero padding . Not e tha t i n Fig. 6.7 the syste m fro m x t o r as indicated b y the gray box is the zero-padded syste m discussed in Section 5.4.1 . The receive d vecto r N x 1 vector r , wher e N = M + z/ , is related t o x b y r = C iowx + q , where Ci ow i s the N x M lowe r triangula r Toeplit z matri x define d i n (5.16 ) and q is the N x 1 blocked channel noise vector. Whe n we substitute x = W^~s into th e abov e equation , w e have
„wt s -
Qo
(6.11)
q-
The tas k o f th e receive r i s t o recove r s fro m th e receive d vecto r r b y usin g an M x N receivin g matri x S , a s show n i n Fig . 6.7. Tw o types o f receivers , zero-forcing an d MMSE , wil l b e derived below . k
—>
I—> • •
• • •
P/S(AQ
zero
r
o
q{n)
x(n)
—> padding ->
C(z)
J,
/
A
^
so ►
A
si
••
S/P(A0
•• m
•A
S
M-\
rN-\
Kslow
transmitter
receiver
Figure 6.7 . Zero-padde d OFD M system .
6.2.1 Zero-forcin
g receiver s
To obtai n a zero-forcin g solution , w e observe fro m (6.11 ) tha t th e receivin g matrix S ca n be an y lef t invers e o f Qi owW^. Le t th e singula r valu e decom position o f Ciow be -'low
u
0
vt
148
6. DFT-base d transceiver s
where U an d V are , respectively , N x N an d M x M unitar y matrices . T h e columns o f U an d V correspond , respectively , t o th e eigenvector s o f Ci owCJow and Cj owCiow. Th e matri x A s i s diagona l an d th e diagona l element s ar e th e singular value s o f Ci ow. W e kno w Ci ow ha s ful l ran k s o lon g a s C(z) ^ 0 . Hence th e diagona l element s o f A s ar e nonzero . W e ca n choos e S t o b e an y matrix o f th e for m S=
WV[A;
1
j
A]U
, (6.12
)
where A i s a n arbitrar y M x L matrix . W e ca n freel y choos e A t o obtai n different solution s o f S . Althoug h thes e ar e al l zero-forcin g receivers , the y wil l behave differentl y whe n th e channe l nois e come s int o play . I n wha t follow s we conside r tw o zero-forcin g solutions .
/
q(n)
S/P(7V)
-► e
i
-► e
M-i
Figure 6.8 . Nois e pat h o f th e zero-padde d OFD M system .
The pseudo-invers e receive r On e possibl e solutio n i s t o choos e A = 0 . This particula r solutio n o f S i s th e pseudo-invers e o f C | o w W l I t ca n als o b e written a s
w(cLc, OT )
(6.13)
J
low'
For a zero-forcin g receiver , th e outpu t erro r e = i s — s come s onl y fro m th e channel noise . Th e overal l syste m ca n b e converte d t o a se t o f M paralle l additive-noise subchannels , simila r t o t h a t show n i n Fig . 6. 2 fo r th e prefixe d O F D M system . W h e n th e channe l nois e q(n) i s AWG N wit h zero-mea n an d variance A/o , th e autocorrelatio n matri x o f th e N x 1 blocke d nois e vecto r q in Fig . 6. 8 i s A/OIJV - Th e erro r vecto r e = i s — s ha s autocorrelatio n matri x Re == A/- 0 SS+ Af
1
0W(clwClowy
wK
T h e /ct h subchanne l erro r varianc e o\ k i s equa l t o th e /ct h diagona l elemen t of R e . I n fact , th e pseudo-invers e receive r i s als o th e zero-forcin g solutio n t h a t yield s th e smalles t tota l outpu t nois e whe n th e channe l nois e i s AWG N (Problem 6.8) . Moreover , a s th e matri x Q>\ ow has ful l rank , th e pseudo-invers e solution alway s exists , eve n i n th e presenc e o f channe l spectra l nulls .
6.2. Zero-padde d OFD M system s
149
A computationall y efficien t receive r T h e pseudo-invers e solutio n i s n o longer relate d t o th e D F T matri x a s i n th e prefixe d O F D M system . T h e receiver doe s no t hav e a s low-cos t a n implementation . Alternatively , w e ca n use a zero-forcin g receive r t h a t i s mor e computationall y efficient . T o deriv e such a receiver , w e observ e t h a t th e N xM matri x Qi ow ha s a lowe r triangula r Toeplitz structure . I t ca n b e turne d int o th e M x M circulan t matri x Q circ by addin g th e b o t t o m v row s o f Ci ow t o th e to p v rows . Thi s ca n b e don e b y using th e followin g matri x manipulation : TC/ For example , whe n M =
10 01 00 00
0 0 1 0
0 0 0 1
| 10 | 01 | 00 I0 0
C c i r c , wher
0
eT
4 an d v = 2 , th e produc t TCi c(0) 0 c(l) c(0 c(2) c(l 0 c(2
00 )0 ) c(0 ) c(l
(6.14)
IMow
i
s
' 0 )0 ) c(0
)
c(2) c(l 0 c(2
) )
c(0) 0 c(l) c(0 c(2) c(l 0 c(2
c(2) c(i 0 c(2 ) ) c(0 ) 0 ) c(l ) c(0
y ) )
which i s a 4 x 4 circulan t matrix . Usin g th e D F T decompositio n Q circ= W ^ T W i n (6.2) , w e ca n obtai n th e followin g computationall y efficien t re ceiver: 1 (6.15)
s = r wx.
Compared t o th e receive r o f th e cyclic-prefixe d O F D M system , fo r eac h re ceived bloc k ther e ar e onl y v extr a addition s du e t o th e matri x T . T h e complexity i s roughl y th e same . W i t h th e efficien t receiver , th e subchanne l noise variance s ca n b e verifie d t o b e (Proble m 6.9 ) 2
N
Afo
M\CZ
12'
T h e facto r N/M i s als o du e t o th e matri x T . T h e subchanne l SN R i s j3(i) ^ 7 | C i | 2 . T h e averag e mea n square d erro r S rr i s give n b y M-l
12 _L v" - — I — V— M i=0
i=0
W h e n w e mak e a compariso n wit h th e cyclic-prefixe d O F D M , w e ca n observ e the followin g properties . • T h e subchanne l SNR s o f th e zero-padde d O F D M syste m ar e reduce d b y a facto r o f N/M whe n compare d t o t h a t o f th e cyclic-prefixe d O F D M system. • W e ma y als o not e t h a t th e averag e transmissio n powe r fo r th e zero padding cas e i s slightl y differen t du e t o th e paddin g o f zeros . W h e n th e input symbol s Sk ar e uncorrelated , wit h zero-mea n an d varianc e £ S1 th e
6. DFT-base d transceiver s
150
o u t p u t s o f th e I D F T matri x als o hav e zero-mea n an d varianc e S s. Afte r zero padding , th e t r a n s m i t t e d outpu t x(n) ha s averag e powe r S SM/N", which i s slightl y les s t h a n th e cyclic-prefixe d O F D M syste m b y a facto r M/N. • Th e decreas e i n subchanne l SNR s wil l b e a mino r on e whe n M i s muc h larger t h a n th e prefi x lengt h v an d henc e th e facto r i s clos e t o one . W e can expec t th e performanc e o f th e zero-padde d O F D M syste m wit h th e efficient receive r t o b e ver y clos e t o t h a t o f th e prefixe d system . I f w e fix the averag e transmissio n powe r t o b e th e sam e a s t h a t i n th e prefixin g case, the n th e erro r rate s o f thes e tw o system s wil l b e jus t th e same .
6.2.2 Th
e MMS E receive r
T h e receive r outpu t erro r ca n b e minimize d b y usin g th e MMS E receiver . Un like th e prefixe d O F D M system , whos e MMS E receive r reduce s t o M scala r M M S E receivers , th e performanc e o f th e zero-padde d O F D M syste m ca n b e improved usin g MMS E reception , a s w e wil l see . W e assum e th e inpu t sym bols Sk ar e uncorrelated , wit h zero-mea n an d varianc e £ s. Als o assum e th e channel q(n) i s AWGN wit h zero-mea n an d varianc e A/o- B y th e orthogonalit y principle i n Sectio n 3.2 , th e mea n square d erro r i£[||s" — s|| 2 ] i s a t it s smalles t when th e erro r e = s " — s i s orthogona l t o th e observatio n vecto r r . T h a t is , E[er^} =
0.
T h e receivin g matri x t h a t satisfie s thi s conditio n i s S = i^sr^ ] (i£[rr"l"] ) As th e inpu t symbol s an d nois e ar e uncorrelated , i.e . ^[sq^ ] = 0 , w e hav e £ [ s r t ] = £ sWCJow an
d E[rr^] =
£ sClowC\ow +N^l
N.
Thus,
S = 7 W C L (jC
lowCJow
+ IN) .-
1
(6.16
)
T h e overal l transfe r matri x T fro m th e transmitte r inpu t s t o th e receive r output s " is T = SCj owWt. It i s no w n o longe r th e identit y matrix , an d th e diagona l element s o f T ar e not unity . Th e erro r e ^ = Si — Si doe s no t com e fro m channe l nois e alone . I t is a combinatio n o f s^ , channe l noise , an d othe r modulatio n symbols . Becaus e ei contain s th e t e r m s^ , th e receive r outpu t si i s a biase d estimat e o f si (Section 3.2.1 ) an d th e rati o fibiased{i)=
£s/c
ei
is a biase d SN R quantity . Th e unbiase d subchanne l SN R /3(i), a s explaine d in Sectio n 3.2.2 , i s give n b y P(i) =
Pbiased(i) ~ 1 .
We ca n the n us e f3{i) t o obtai n a n accurat e estimat e o f th e actua l erro r rate .
6.2. Zero-padde d OFD M system s
151
Remarks • T h e efficien t receive r i n (6.15 ) ha s ver y lo w channe l dependence . T h e channel-dependent par t i s a se t o f M scalars , lik e th e cyclic-prefixe d sys tem. T h e pseudo-invers e receive r an d MMS E receiver , however , depen d heavily o n th e channel . Also , matri x inversion s ar e neede d t o comput e the coefficient s i n th e receivin g matrices . Fo r th e MMS E receiver , th e SNR quantit y 7 i s als o require d i n computin g th e receiver . • T h e implementatio n cos t o f th e efficien t receive r i s comparabl e t o t h a t of th e prefixe d system . Fo r th e pseudo-invers e an d MMS E receivers , there i s n o correspondin g D F T structure . Althoug h th e implementatio n cannot b e carrie d ou t usin g a low-cos t D F T matrix , i t i s possibl e t o reduce th e complexit y fo r thes e tw o receiver s [26] . • W e kno w i n genera l th e MMS E receive r wil l becom e a zero-forcin g on e when th e SN R i s large . Fo r th e zero-padde d O F D M system , th e solutio n of zero-forcin g receiver s i s no t unique . Whic h zero-forcin g solutio n wil l the MMS E receive r reduc e t o a s th e SN R approache s infinity ? T o answe r this question , w e rewrit e th e MMS E receive r i n (6.16 ) a s (Proble m 6.7 ) S = W {c\
owClow
+
7 " 1 I M ) " 1 C L - (6-17
)
W h e n th e SN R 7 i s large , th e MMS E receive r reduce s t o th e pseudo inverse solutio n i n (6.13) ! A s a result , th e performanc e o f th e pseudo inverse receive r come s clos e t o t h a t o f th e MMS E receive r fo r a suffi ciently larg e SNR . Example 6. 2 Zero-padde d OFD M systems . T h e simulatio n environmen t and parameter s use d i n thi s exampl e ar e th e sam e a s thos e i n Exampl e 6.1 . For th e channe l c i ( n ) , Fig . 6.9(a ) show s th e B E R performance s o f th e zero padded O F D M syste m wit h th e thre e type s o f receiver s discusse d i n thi s section: th e efficien t receive r i n (6.12) , th e pseudo-invers e receive r i n (6.13) , and th e MMS E receive r i n (6.16) . T h e B E R s ar e denoted , respectively , b y Vofdm-zp (efficient) , V 0fdm-zP (pseudo) , an d V 0fdm-zP,mmse i n th e figure. For comparison , th e erro r rat e o f th e prefixe d O F D M syste m i s als o show n i n the figure. T h e B E R s ar e plotte d a s a functio n o f 7 = £ s/J\fo- Fo r th e sam e 7, th e averag e transmissio n powe r o f th e prefixe d syste m i s N/M time s large r t h a n th e zero-padde d case . T h e performance s fo r th e channe l C2(n) ar e show n in Fig . 6.9(b) . For b o t h channel s th e erro r rat e o f th e efficien t receive r i s slightl y large r t h a n t h a t o f th e prefixe d system . Thi s i s becaus e th e rati o N/M = 17/1 6 i s very clos e t o one . (W e ar e comparin g tw o system s wit h th e sam e 7 . I f th e comparison i s don e fo r th e sam e transmissio n power , th e performance s wil l b e exactly th e same. ) I n th e cas e o f channe l C2(n) , thes e tw o system s ar e badl y affected b y th e zer o clos e t o th e uni t circl e (tw o curve s indistinguishabl e i n the figure). Fo r th e zero-padde d O F D M systems , th e performance s o f th e pseudo-inverse an d th e MMS E receiver s ar e muc h les s affecte d b y th e channe l null. T h e curve s o f thes e tw o receiver s ar e ver y clos e fo r b o t h channels . Fo r high SNR , on e i s indistinguishabl e fro m th e othe r i n b o t h figures. W e als o note th e differenc e i n th e SN R range s betwee n th e to p an d b o t t o m plots . A
152
6. DFT-base d transceiver s
much highe r SN R i s neede d t o achiev e th e sam e erro r rat e fo r c
1(T
.■
10"
P , . (efficient ofdm-zp v ' ofdm-cp P
—
10"'
(a)
P
X
DC 3 LU 1 0 CO
): ■
ofdm-zp(PSeud°) i
M
ofdm-zp,mmse
^
V V
10
V V V V
10"'
,\ \
10"
10 1 52 02 y=E s /N 0 (dB )
53
0
W -■-Pofdm-zp(efficient):
P..
10"
P
(b)
e
\
DC _ ; LU 1 0 DO
V
\]
N
\\ \A
10" 10 ' 10"'
' ofdm-zp,mms
^^s>- -
10 '
ofdm-cp ofdm-zp(P S e U d °) !
10 2
\* \• \"1 \
03 04 05 y=E s /N 0 (dB )
*-
=
0
Figure 6.9 . Exampl e 6.2 . Zero-padde d OFD M system : BE R performance s fo r (a ) the channe l c i ( n ) an d (b ) th e channe l C2(n) .
6-3 Single-carrie CP)
r system s wit h cycli c prefi x (SC -
In th e O F D M system , modulatio n symbol s ar e sen t afte r I D F T operatio n an d adding a prefix . I f th e symbol s ar e sen t directl y afte r redundan t sample s ar e
153
6.3. Single-carrie r system s w i t h cycli c prefi x (SC-CP )
added, w e call it a block-based single-carrie r system . Redundan t sample s can take the form o f cyclic prefixing o r zero padding. Correspondingly , th e system is calle d th e single-carrie r syste m wit h cycli c prefi x (SC-CP ) o r th e single carrier syste m wit h zer o padding (SC-ZP) . I t i s called a single carrie r syste m because th e modulatio n symbol s ar e directly transmitte d afte r th e insertio n of redundan t samples , a s oppose d t o th e multicarrie r OFD M system , wher e the channe l i s divided int o multipl e subchannel s vi a the DFT matrix .
/»
/
\
v« -* «•
r
►
q(n)
—►
— ► sji/-ii
cyclic prefix
->
C(z)
1r rw w
discard prefix
-+
K^circ
transmitter
►
•• • rM-\
1/C0 P► 1/Cj
•• •
•• •
i/cM-.
receiver
Figure 6 . 1 0 . SC-C P syste m wit h a zero-forcing receiver .
The syste m bloc k diagra m o f th e SC-C P syste m i s show n i n Fig . 6.10. There i s no processing o f the modulation symbol s a t the transmitter. W e can see that th e system fro m th e vector s at the transmitter t o the vector r a t the receiver i s the familiar cyclic-prefixe d syste m show n in Fig. 5.1 6 (Section 5.4). The correspondin g transfe r matri x i s the M x M circulan t matri x C circ. W e can remov e IS I by inverting C circ. Th e resulting zero-forcin g receive r i s
c-L = w+r^w,
(6.18)
which i s a circulan t matri x a s i t i s a diagona l matri x sandwiche d betwee n an IDF T an d a DF T matri x (Theore m 5.2) . Th e block diagra m o f the zeroforcing receive r i s a s show n i n Fig . 6.10. A fe w observation s o n the SC-C P system ar e in order . • Channe l independence . Lik e the OFD M system , th e SC-C P syste m i s also a DFT-base d transceive r wit h a channel-independen t transmitter . A cycli c prefi x i s inserte d a s in the OFD M system . Th e onl y channel dependent par t i s a set of M scalar s 1/Cf c a t the receiver. Thes e scalar s are calle d FEQ s i n the OFDM receiver . Therefor e th e SC-CP syste m is also know n a s a single-carrie r syste m wit h frequenc y domai n equaliza tion (SC-FDE ) [129] . • PAPR . Th e SC-C P syste m ha s th e advantag e tha t th e PAP R o f th e transmitted signa l i s very low . Thi s i s because th e modulation symbol s
154
6. DFT-base d transceiver s
are t r a n s m i t t e d directly , withou t furthe r processing , an d th e P A P R o f the t r a n s m i t t e d signa l i s the sam e a s th e P A P R o f th e inpu t modulatio n symbols. Th e P A P R i s muc h lowe r t h a n t h a t i n th e O F D M system , especially fo r larg e M. • Implementation . I f w e compar e th e SC-C P syste m t o th e O F D M sys tem, i t i s lik e movin g th e computatio n o f th e I D F T matri x a t th e trans mitter t o th e receiver . I t ha s th e sam e overal l complexit y a s th e O F D M system. I n particular , th e transmitte r doe s no t requir e an y computa tion. Thi s i s ver y usefu l fo r application s wher e th e transmitte r doe s no t have a s muc h computin g resourc e a s th e receiver . Fo r example , i n a n uplink wireles s transmissio n (mobil e unit s t o th e bas e station) , th e re ceiver a t th e bas e statio n generall y ha s mor e computin g powe r availabl e t h a n th e transmitte r o f a mobil e unit . I n thi s cas e i t i s preferre d t o offload computation s fro m th e transmitte r t o th e receiver . O n th e othe r hand, i f th e transmitte r an d th e receive r hav e comparabl e computin g resources, the n th e O F D M syste m i s mor e suitable . • Performance . Th e B E R performanc e o f th e SC-C P syste m wit h a zero forcing receive r i s sensitiv e t o th e presenc e o f channe l spectra l nulls , as wil l b e explaine d i n Sectio n 6.3.1 . However , th e sensitivit y ca n b e dramatically reduce d i f a n MMS E receive r (Sectio n 6.3.3 ) i s used .
w
•• •
precoder W
•• •
w
OFDM transmission syste m
•• •
post-coder W+
Figure 6 . 1 1 . SC-C P syste m viewe d a s a precode d OFD M system .
We ca n als o loo k a t th e SC-C P fro m a n alternativ e viewpoint , whic h allow s us t o deriv e th e SC-C P syste m fro m th e O F D M system . Fo r a n O F D M system, i f w e cascad e a D F T matri x befor e th e transmitte r an d a n I D F T matrix a t th e en d o f the receive r (Fig . 6.11) , the n th e I D F T an d D F T matrice s at th e transmitte r cance l eac h othe r out . W h a t w e ge t i s a n SC-C P system ! T h u s th e SC-C P syste m ca n b e viewe d a s a n O F D M syste m wit h a D F T precoder an d I D F T post-coder . Havin g a precode r ca n significantl y alte r th e characteristics o f th e system , fo r exampl e P A P R . Anothe r exampl e i s th e behavior o f subchanne l noise , whic h wil l b e explaine d i n th e followin g nois e analysis. (Mor e detaile d discussio n o n precode d O F D M system s wil l b e give n in Chapte r 7. )
155
6.3. Single-carrie r system s w i t h cycli c prefi x (SC-CP )
6.3.1 Nois
e analysis : zero-forcin g cas e
T h e nois e p a t h a t th e receive r i s show n i n Fig . 6.12 . A compariso n wit h the nois e p a t h o f th e O F D M syste m i n Fig . 6. 3 show s t h a t th e nois e vecto r /j, ha s th e sam e statistic s a s th e outpu t nois e vecto r i n th e O F D M system . Thus ilk ar e uncorrelate d Gaussian , wit h variance s give n b y A/o/|C/c| 2 . T h e autocorrelation matri x R ^ i s a diagona l matrix , R /y
diag
JO, |2 | d | 2 \C
M-i\
T h e autocorrelatio n matri x o f th e outpu t nois e vecto r e i s give n b y Re =
WtR^W .
It i s a circulan t matri x a s R ^ i s diagona l (Theore m 5.2) . T h e diagona l ele ments o f R e ar e th e same , equa l t o th e averag e o f a 1. Therefor e th e sub channel nois e variances , correspondin g t o th e diagona l element s o f R e , ar e the sam e fo r al l subchannels . W e hav e 1
S
M ^ \Ct\ £=0 '
(6.19)
2
* '
'
T h e averag e erro r i s th e sam e a s t h a t i n th e O F D M system .
%
/x
q[n)
discard —> S/?(M) prefix
ii
• • •
T
o
w
^0
^1
1/C,
Wt
•• • MM-I
<1M-\
i/cM_,
►^M-l
Figure 6 . 1 2 . Nois e pat h a t th e receive r o f th e zero-forcin g SC-C P system .
Identical subchanne l nois e varianc e mean s identica l subchanne l SNRs . T h e subchannel SN R (3 sc_cp = S s/Srr i s give n b y Psc-cp=
1
Ml1 * 2 M Z^i= 0 7|Ci|
W h e n Q P S K modulatio n i s used , th e bi t erro r rat e o f th e SC-C P syste m i s
Vsc.cp = Q ( V / W) • (
6 2
- °)
In vie w o f th e subchanne l SNRs , th e SC-C P syste m i s ver y differen t fro m th e O F D M system . T h e subchanne l SNR s ar e th e sam e eve n whe n th e channe l i s not flat . W e wil l se e i n Chapte r 8 t h a t ther e i s littl e gai n whe n bi t an d powe r allocation ar e allowed .
156
6. DFT-base d transceiver s
Channels wit h spectra l null s W h e n w e examin e th e expressio n i n (6.19) , we observ e t h a t al l th e subchanne l error s wil l becom e ver y larg e whe n th e channel ha s a spectra l null . Suppos e on e D F T coefficient , sa y Cg 0 , i s equa l t o zero. Al l th e subchannel s ar e equall y affected ; th e subchanne l nois e variance s will g o t o infinit y an d th e bi t erro r rat e i s 0. 5 fo r al l th e subchannels . Suc h a disaster ca n b e avoide d b y usin g a n MMS E receiver , t o b e discusse d next .
6.3.2 Th
e MMS E receive r
For th e prefixe d O F D M system , th e MMS E receive r degenerate s t o M scala r M M S E receivers . Ther e i s n o performanc e improvement . I n th e cas e o f th e SC-CP system , ther e i s n o suc h degeneration . Th e us e o f MMS E receptio n does improv e th e performanc e i n term s o f BER . I t t u r n s ou t t h a t th e MMS E receiver ca n b e easil y obtaine d fro m th e zero-forcin g receive r b y replacin g th e channel-dependent scalar s wit h anothe r se t o f coefficients .
*f
1/
s
o
• •
q(n)
P/S(M) —> cyclic prefix
discard prefix
C(z)
S/P(M)
fM-l
SM-l
Figure 6 . 1 3 . Th e SC-C P system .
Suppose w e ar e give n th e transmitte r o f th e SC-C P syste m i n Fig . 6.13 , which doe s nothin g mor e t h a n inser t a cycli c prefi x t o eac h inpu t bloc k o f modulation symbols . W h a t woul d b e th e correspondin g MMS E receivin g matrix S t h a t minimize s th e subchanne l errors ? T o answe r thi s question , recall t h a t th e receive d vecto r ca n b e writte n a s r — Cc ^ r c s + q . Th e receive r output erro r i s e = s — s = S r — s. By th e orthogonalit y principl e i n Sectio n 3.2 , th e mea n square d erro r i£[||e|| 2 ] is minimize d i f S i s suc h t h a t th e erro r e i s orthogona l t o th e observatio n vector r . T h a t is , ^[er1] = 0 . We ca n verif y t h a t th e optima l S satisfyin g th e abov e conditio n ca n b e ex pressed i n th e for m (Proble m 6.10 ) S=
W
1
AW ,
(6.21)
157
6.3. Single-carrie r system s w i t h cycli c prefi x (SC-CP )
where A i s a n M x M diagona l matri x wit h th e ^t h diagona l entr y give n b y Xi
1 + 7|C *
r,
£
= 0,1,...,M -1 .
W h e n 7 approache s infinity , th e £t h diagona l entr y reduce s t o Ai = \jCt an d the MMS E receive r reduce s t o th e zero-forcin g case . T h e MMS E receivin g matrix ha s th e interestin g propert y t h a t i t i s als o a circulan t matrix . I t i s a channel-dependen t diagona l matri x A sandwiche d betwee n a n I D F T an d a D F T matri x (Fig . 6.14) . I t ha s th e sam e structur e a s th e zero-forcin g receiver . T h e implementatio n complexit y i s als o th e sam e a s th e zero-forcin g receiver : one M-poin t D F T , on e M-poin t I D F T , an d M channel-dependen t scalars . I n terms o f channe l dependence , th e SN R quantit y 7 = £ S/MQ i s als o neede d fo r the MMS E receiver , whil e th e zero-forcin g receive r require s onl y th e channe l response.
C discard prefix
>
S/P(AQ
>
►
►■
X0
W
-►
•••
•••
rn-i
XM-
w+
—►
r w
• •• ►
1
Figure 6.14 . Th e MMS E receive r fo r th e SC-C P system .
6.3.3 Erro
r analysis : MMS E cas e
W h e n th e MMS E receive r i s used , w e replac e th e receivin g matri x S i n Fig. 6.1 0 b y th e MMS E solutio n i n (6.21) . T h e overal l transfe r matri x T becomes T = S C a r c = W f A r W , (6.22 ) which i s n o longe r th e identit y matrix . T h e zt h outpu t erro r ei = si — si does no t com e fro m channe l nois e alone ; i t i s a mixtur e o f channe l nois e an d interference. T h e receive r outpu t vecto r i s s = T s + Sq . T h e outpu t erro r vector e = s — s ca n b e expresse d a s e = ( T - I ) s + S q = W f ( A r - I ) W s + Sq . V
v
/
D
W h e n th e signa l s an d nois e q ar e uncorrelated , th e autocorrelatio n matri x of th e outpu t erro r R e = ^[ee^ ] i s Re =
W ^ D Dt+
AA 0 AA f )W,
158
6. DFT-base d transceiver s
which i s a circulan t matri x a s th e matri x sandwiche d betwee n W ^ an d W i s diagonal. Thi s implie s tha t al l the subchanne l error s hav e th e sam e variance , just a s i n th e zero-forcin g case . Th e subchanne l erro r variance s ar e equa l t o their averag e £ rr. Usin g th e definition s o f D an d A , w e ca n verif y tha t th e average erro r £ rr i s give n b y M-l
We can use the subchannel error variance to compute the biased SNR, fiuased — £s/£rr. Th e unbiase d SN R /3 can b e obtaine d usin g /3 = ^biased — 1 Comparing wit h th e zero-forcin g receiver , w e ca n se e fro m (6.23 ) tha t the MMS E receive r alway s ha s a smalle r erro r du e t o th e extr a unit y i n th e denominator. W e ma y als o observ e tha t eac h o f th e term s i n th e summatio n in (6.23 ) i s les s tha n one . Th e averag e erro r wil l b e uppe r bounde d b y £ s. The subchanne l error s will b e bounde d eve n i f the channe l ha s spectra l nulls . This is a marked differenc e fro m th e zero-forcin g case , where the averag e erro r can g o t o infinit y i f th e channe l ha s on e DF T coefficien t equa l t o zero . Th e MMSE receive r wil l b e muc h mor e robus t agains t channe l spectra l null s a s demonstrated i n th e followin g example . Example 6. 3 Th e SC-C P system . Th e simulatio n environmen t an d param eters use d i n thi s exampl e ar e th e sam e a s thos e i n Exampl e 6.1 . Fo r th e SC-CP system , th e erro r rate s ar e the sam e acros s al l subchannels. Fo r chan nel ci(n), Fig . 6.15(a) show s the BE R performances o f the SC-C P syste m wit h the zero-forcin g receive r i n (6.18 ) an d wit h th e MMS E receive r i n (6.21) . Th e MMSE receive r give s a slightly lowe r BE R tha n th e zero-forcin g receiver . Fo r comparison, th e erro r rat e o f the prefixe d OFD M syste m i s also shown. Com paring th e OFD M an d th e SC-C P systems , bot h wit h zero-forcin g receivers , the tota l outpu t error s ar e th e sam e bu t th e SC-C P syste m outperform s th e OFDM syste m for a moderate value of BER. Th e differenc e i s significant whe n SNR i s large. Th e performance s fo r channe l C2(n) are show n i n Fig . 6.15(b) . Although ther e i s n o significan t differenc e i n term s o f performanc e betwee n zero-forcing an d MMS E receiver s fo r a n almost-fla t channe l lik e ci(n) , fo r channels wit h spectra l null s th e ga p betwee n th e tw o i s muc h larger . Th e MMSE receive r considerabl y improve s th e erro r rate , a s w e ca n se e fro m Fig. 6.15(b) . ■
159
6.3. Single-carrie r system s wit h cycli c prefi x (SC-CP )
10" p
ofdm-cp sc-cp,zf sc-cp,mmse
(a)
7=E s /N Q (dB )
20 2
5
30
(b)
10 2
03 04 05 y=E s /N 0 (dB )
06
0
Figure 6 . 1 5 . Exampl e 6.3 . Th e SC-CP system : BE R performances fo r (a ) channe l c i ( n ) an d (b) channel C2(n) .
160
6. DFT-base d transceiver s
We observ e fro m Fig . 6.1 5 that , fo r bot h channels , th e BE R curve s o f th e MMSE SC-C P syste m li e unde r thos e o f th e prefixe d OFD M syste m fo r al l SNR 7 . Thi s is , i n fact , tru e fo r al l channels . Th e MMS E SC-C P syste m always ha s a lowe r BE R tha n th e prefixe d OFD M system , a s w e wil l se e i n Chapter 7 .
6-4 Single-carrie ZP)
r syste m wit h zero-paddin g (SC -
The single-carrie r syste m wit h zer o paddin g show n i n Fig . 6.16 , muc h lik e the SC-C P system , als o send s ou t modulatio n symbo l directly , bu t th e guar d intervals ar e filled wit h zeros . Simila r t o th e zero-padde d OFD M system , w e know th e transfe r matri x fro m th e inpu t vecto r s a t th e transmitte r t o th e vector r a t th e receive r i s th e lowe r triangula r Toeplit z matri x Ci ow define d in (5.16) . Th e receive d vecto r r i n Fig . 6.1 6 ca n b e expresse d a s follows : r = Q ows + q .
(6.24)
We can obtain a zero-forcing receive r by choosing the receivin g matrix S to b e any lef t invers e o f Ci ow. Tw o receivers , on e wit h a n efficien t implementatio n and on e correspondin g t o th e pseudo-invers e solution , wil l b e discussed .
/i
^ r
o
q(n)
P/S(M)
sM-
M
zero padding - ► C(z)\
S
S/P(A0
Kslow
•••
•*•
rN-\
transmitter
Figure 6 . 1 6 . Th e SC-Z P system .
One zero-forcing choic e of S can b e obtained b y first turnin g Ci ow int o th e M x M circulan t C circ usin g (6.14) . W e ca n the n tak e th e invers e o f C circThe resultin g solutio n i s the followin g receiver :
s = wt r 1 w i ,
(6.25)
where T i s the MxN matri x defined i n (6.14) . Th e above receiver differs fro m the zero-forcin g SC-C P receive r onl y i n T , whic h require s onl y v addition s
6.4. Single-carrie r syste m wit h zero-paddin g (SC-ZP )
161
per block . Suc h a receive r ha s th e advantag e t h a t i t ca n b e implemente d efficiently. Not e t h a t th e matri x T ha s littl e effec t whe n M is much large r t h a n v. A s a result, th e performanc e o f th e SC-Z P syste m wit h th e efficien t receiver wil l b e simila r t o th e SC-C P syste m wit h th e zero-forcin g receiver . Both receiver s contai n th e t e r m T~ , i.e . th e inverse s o f th e channe l D F T coefficients. W h e n th e channe l ha s on e D F T coefficien t equa l t o zero, th e performance ca n b e seriousl y affected . Another possibl e zero-forcin g choic e o f S i s th e pseudo-invers e o f Ci ow, S = (C\owClow)-1Clow. (6.26
)
This doe s no t hav e th e simpl e D F T - I D F T structur e an d canno t b e imple mented a s efficientl y a s th e receive r i n (6.25) i n general. I n term s o f nois e analysis, thi s i s als o th e zero-forcin g receive r t h a t lead s t o th e smalles t tota l output nois e (simila r t o th e zero-padde d O F D M case) . Lik e th e zero-padde d O F D M system , th e pseudo-invers e solutio n alway s exist s irrespectiv e o f chan nel spectra l nulls . Thi s stand s i n contrast t o the efficien t receive r give n in (6.25), whic h canno t recove r th e transmitte d symbol s whe n th e channe l ha s one o r mor e D F T coefficient s equa l t o zero . MMSE receive r W i t h th e receive d vecto r r given i n (6.24) , w e ca n di rectly obtai n th e MMS E receive r usin g th e orthogonalit y principle . I t i s give n by
S = 7C L ( 7 Q 0 W C L + I N ) - 1 . (6.27
)
T h e overal l transfe r matri x fo r th e MMS E receive r i s
T = 7C L ( 7 Q ™ C L +
IJV^CW
T h e subchanne l SN R /3(i) ca n b e compute d usin g a n approac h simila r t o t h a t for th e zero-padde d O F D M system . Unlik e th e MMS E receive r o f th e SC-C P system i n (6.21) , th e MMS E receive r o f th e SC-Z P i s no t a circulant matrix . It canno t b e implemente d efficientl y usin g D F T an d I D F T matrices . Example 6. 4 Th e SC-Z P system . T h e simulatio n environmen t an d param eters use d i n thi s exampl e ar e th e sam e a s thos e i n Exampl e 6.1 . Figur e 6.1 7 shows th e B E R performance s o f th e SC-Z P syste m wit h th e efficien t receive r in (6.25) , wit h th e pseudo-invers e receive r i n (6.26) , an d wit h th e MMS E re ceiver i n (6.27) . T h e B E R s o f th e thre e receiver s fo r th e SC-Z P syste m ar e denoted, respectively , b y V sc-zp (efficient) , V sc-zp (pseudo) , an d V Sc-zP,mmse in th e figure . Fo r comparison , th e erro r rate s o f th e SC-C P syste m wit h th e zero-forcing an d th e MMS E receiver s ar e als o shown . T h e erro r rat e o f th e zero-padded syste m wit h th e efficien t receive r i s ver y clos e t o th e SC-C P sys t e m fo r b o t h channels , a s w e ha d expected . Fo r th e channe l c i ( n ) , al l th e curves ar e ver y close . T h e pseudo-invers e an d MMS E receiver s d o no t hav e much gai n ove r th e efficien t receiver . Fo r th e secon d channel , th e MMS E an d pseudo-inverse receiver s ar e les s affecte d b y th e zer o t h a t i s clos e t o th e uni t circle. T h e pseudo-invers e receive r ca n achiev e a gain o f aroun d 1 5 d B ove r the efficien t receiver . T h e MMS E receive r ha s a n eve n mor e significan t gain .
162
6. DFT-base d transceiver s
H— P (efficient 1
px '
sc-z
sc-cp,zf
P (pseudo p
sc-zp
x r
'
)
i
)
sc-cp,mmse
P
sc-zp,mmse
y = E s / N 0 (dB
)
25
30
. P (efficient p
sc-zp v ' sc-cp,zf
■ P (pseudo
sc-zp Vl ^ ' ■P sc-cp,mmse
P
sc-zp,mmse
y = E s / N 0 (dB
)
Figure 6 . 1 7 . Exampl e 6.4 . Th e SC-Z P system : BE R performance s fo r (a ) channe l c i ( n ) an d (b ) channe l C2(n) .
)
)
163
6.5. Filte r ban k representatio n o f O F D M system s
6.5 Filte
r ban k representatio n o f O F D M system s
In thi s section , w e wil l loo k a t th e O F D M syste m fro m th e viewpoin t o f filter banks. Thi s viewpoin t allow s u s t o se e ho w th e transmitte d powe r spectru m is relate d t o th e inpu t modulatio n symbols , whic h wil l giv e u s mor e insigh t into ho w th e symbol s ar e transmitte d i n th e frequenc y domain . |<7(»)
i
s,(n)
W">
X\Yl)
-+> fiV
*o(»)
-+> fiV
• • •
—► A
• • •
I
W C(z)
- 1
z
-1
^z
\
(a)
*,(») *,(»)
q(n)
x( n)
p(») —►
~^> \N
'U
AJL
-+> [0 W ]
• • • |JV -+-
r(t 0
• • •
transmitting filter ban k
• • •
/ W-M-X
)f
C(z) FcpW
W">- 11 4T NV \1 *to l
\ICX
v(/i)
I/C0
zi
4N
1/C0
r
z ^r
i
^
|JV -+> ,
zi
[0 W ]
*4
J* frW
h(n)
(b)
i/ci
J " ! _ ♦ _ _ i/c„_, receiving filter ban k
Figure 6 . 1 8 . Derivation s o f th e filte r ban k representatio n fo r th e OFD M system .
To deriv e th e equivalen t filter ban k structure , le t u s redra w th e O F D M system usin g th e matri x representatio n i n Fig . 6.18(a) , wher e th e detail s o f blocking an d unblockin g mechanisms , an d th e insertio n an d remova l o f prefix , are show n explicitly . T h e N x M matri x ~F cp i s th e prefi x insertio n matri x defined i n (5.14) . A s F cp an d W ^ ar e constan t matrices , w e ca n exchang e these tw o matrice s an d th e expanders . T h e resultin g transmitte r i s a s show n in Fig . 6.18(b) , wher e w e hav e lumpe d F cp an d W ^ together . Similarly , w e can exchang e th e decimator s an d th e matri x [ 0 W ] a t th e receive r t o yiel d the receive r show n i n Fig . 6.18(b) . W e not e t h a t th e M - i n p u t one-outpu t system fro m p ( n ) t o x(n) a t th e transmitte r i s a n LT I system , whic h w e cal l the transmittin g filter bank . Denotin g th e syste m b y f T(z) an d it s kth filter
164
6. DFT-base d transceiver s
by Fk(z), w e hav e fT(z)=[F0(z) F^z)
..
.F
M-i(z)\.
Similarly, a t th e receivin g end , th e one-inpu t M-outpu t syste m fro m r(n) t o v(n) i s als o LT L W e cal l th e M x l syste m h(z) an d denot e it s /ct h filte r a s Hk(z). Thu s T h(z)=[H0(z) H^z) ...H M-i(z)] . We thu s arriv e a t th e filte r ban k representatio n o f th e OFD M syste m show n in Fig . 6.19 .
s0(n) s^n)
t"
—►
fjv
—► F^z)
Fo®
•
Vi<">-
fiV
uQ(n) x(n) k"
\q(n) C(z)
A *9
ux(n)
-+
i
•
>r
k
H0(z)
-^> \N
Hx(z)
—►
r
•
IN
*. w
1/C0 ^w
•
i/q •
I*
1/CW
U
—►
Fu-l®
M-l^ |
(*)
^\~"M~I
transmitting filters
receiving filters
Figure 6 . 1 9 . Filte r ban k representatio n fo r th e OFD M system .
The transmittin g ban k show n i n Fig . 6.18(b ) i s the interconnectio n o f th e matrix F ^ W ^ an d a dela y chain , s o we hav e iT(z)=
-(N-
[1
- x) ] F c p W t .
The /ct h transmittin g filte r i s determine d b y F k{z) = Its impuls e respons e i s fk(n) =
-LwM*-")*, n VM
J2i= o
= 0,l,...,N-l,
z
iFcpW^fc .
(6.28)
where v i s the lengt h o f the cycli c prefix. Th e transmittin g filter s ar e all DF T filters. Th e filte r fo(n) i s a rectangula r windo w o f lengt h N. Al l th e othe r transmitting filter s ar e scale d an d frequency-shifte d version s of the prototyp e Fk(z) =
W" kFQ(zWk) o
rF
JUJ
k(e
)=
W ^F^e3^-2k^/M)y
^
2Q^
The transmittin g filter s for m a DFT bank. Figur e 6.20(a ) show s th e magni tude respons e oiFo(z). A s fo(n) i s a rectangular windo w of length N, F 0 (e ja; ) has zero s a t intege r multiple s o f 2TT/N (excep t uo = 0) . Th e magnitud e re sponses o f th e othe r filter s ar e shifte d version s o f |F 0 (e j a ; )|, th e shift s bein g
165
6.5. Filte r ban k representatio n o f O F D M system s
integer multiple s o f 2TT/M. Figur e 6.20(b ) give s a schemati c plo t o f th e mag nitude response s o f th e transmittin g filters . T h e kth filte r Fk(z) i s a bandpas s filter wit h th e passban d centere d aroun d 2kix jM. T h e stopban d attenuatio n of th e filter s i s aroun d 1 3 d B an d th e stopban d o f F 0 ( e j a ; ) decay s slowl y wit h frequency [105] .
(a)
l*b I l* i I (b)
2(M-l)n M
Figure 6 . 2 0 . (a ) Th e magnitud e respons e o f th e prototyp e FQ{Z)\ (b ) schemati c plo t of th e magnitud e response s o f th e transmittin g filter s i n th e OFD M system .
Similarly, th e receivin g ban k i n Fig . 6.18(b ) i s th e interconnectio n o f a n advance chai n an d [ 0 W ] , s o w e hav e
I
h(» = [ o w ]
z
VN-1
T h e impuls e respons e o f th e kth filte r i s give n b y hk(n)
1 M
yy-(n+v)k
-W + l , - i V + 2 , . . . , - i / .
(6.30)
T h e receivin g filter s ar e als o D F T filters . The y ar e noncausa l du e t o th e advance chai n bu t ca n b e mad e causa l easil y b y introducin g enoug h delay s i n the receiver . T h e filte r HQ(Z) i s als o a rectangula r window , bu t o f lengt h M . All th e othe r receivin g filter s ar e scale d an d frequency-shifte d version s o f th e
6. DFT-base d transceiver s
166 p r o t o t y p e HQ(Z),
Hk(z)=
W-" kH0(zWk) o
rH
JUJ
k(e
)=
W
-1/fc
iJo(ej(w~2WM)).
A schemati c plo t o f the magnitud e response s o f the receivin g filter s woul d loo k very simila r t o t h a t o f transmittin g filter s i n Fig . 6.20 . A sligh t differenc e i s t h a t th e firs t zer o o f H^{e^) i s a t 2TT/M (instea d o f 2TT/N) a s it s lengt h i s M. Th e kth filte r i s bandpas s wit h th e passban d centere d a t 2kir/M. A numerica l exampl e o f th e transmittin g an d receivin g filter s fo r M = 8 and v = 2 i s show n i n Fig . 6.21 . Th e magnitud e respons e o f th e tw o proto type filter s FQ(Z) an d HQ(Z) ar e draw n wit h a thicke r line . Th e magnitud e responses o f al l th e othe r filter s ar e shifte d version s o f thei r respectiv e proto types. W e ca n se e t h a t th e firs t sidelob e ha s a n attenuatio n o f aroun d 1 3 d B only. Th e attenuatio n i s no t adequat e i n man y applications . Mor e detail s o n the importanc e o f th e frequenc y selectivit y o f th e transmittin g an d receivin g filters ar e discusse d i n Chapte r 9 .
6.5.1 Transmitte
d powe r spectru m
T h e powe r spectru m o f th e t r a n s m i t t e d signa l x(n) i n Fig . 6.1 9 depend s o n the transmittin g filter s a s wel l a s th e transmitte r inputs . Assum e th e input s of th e transmitte r Sk(n) ar e white , zero-mea n WS S rando m processe s wit h power spectru m S Sk(e^) = £ s. Furthermore , assum e th e inpu t processe s ar e jointly WS S an d u n c o r r e c t e d , 2 E[sk(n)s^(i)] =
0 , Vn,i
, an d k ^ j .
T h e outpu t o f th e kth transmittin g filter , Uk(n), ha s a powe r spectru m give n by (Appendi x B ) SUk(eJ= n
^\Fk(e
J
n\2-
W h e n th e input s o f th e transmitte r ar e jointl y WS S an d uncorrelated , th e t r a n s m i t t e d signa l x(n) i s CWSS(A^ ) wit h powe r spectru m
Sx{en = I J2 \ F^n\2, (6-31 keA
)
where A i s the se t o f subchannels t h a t ar e actuall y use d fo r transmission. 3 T h e contribution fro m th e /c-subchanne l occupie s th e frequenc y bi n correspondin g to th e passban d o f F k(z), i.e . th e bi n centere d a t 2kir/M. Thu s th e d a t a fro m each subchanne l occup y a separat e frequenc y ban d an d th e subchannel s ar e separated vi a th e frequency-selectiv e propert y o f th e transmittin g filters . W h e n w e pas s th e discrete-tim e t r a n s m i t t e d signa l x(n) throug h th e D / C converter, w e obtai n th e continuous-tim e t r a n s m i t t e d signa l x a(t). W e ca n use S x{e^) t o obtai n th e powe r spectru m o f x a{t). W h e n x{n) i s CWSS(A^ ) 2 T h e transmitte d powe r spectru m ma y als o b e derive d wit h a mor e relaxe d conditio n using th e resul t i n [131] , whic h use s onl y th e assumptio n tha t th e input s for m a WS S vecto r process. 3 In som e application s o f th e OFD M system , fo r exampl e wireles s loca l are a network s [54], certai n subchannel s ar e reserve d an d onl y a subse t o f th e subchannel s ar e actuall y used fo r dat a transmission .
6.5. Filte r ban k representatio n o f OFD M system s
167
15
10
(a)
H
^imho
-5 ft -10
hih
0.5 1 1. 5 Frequency normalize d by%.
00 ■D
(b)
-10
0.5 1 1. 5 Frequency normalize d by7i.
F i g u r e 6 . 2 1 . M a g n i t u d e response s o f t h e t r a n s m i t t i n g an d receivin g filter s i n a n O F D M system fo r M =
8 an d v =
2 : (a ) t h e t r a n s m i t t i n g filters ; ( b ) t h e receivin g filters .
and th e samplin g perio d o f th e D / C converte r i s T , th e transmitte d signa l xa(t) i s a continuous-tim e CWS S proces s wit h perio d NT, i.e . CWSS(A/T ) (see Appendi x B. 4 fo r details) . Assum e th e transmittin g puls e pi(t) a t th e transmitter i s a n idea l lowpas s filte r wit h cutof f frequenc y n/T an d a constan t unity gai n i n th e passband , wher e T i s th e underlyin g sampl e spacing . T h e n the powe r spectru m o f x a(t) i s (Appendi x B.4 )
SXaUty
£WT),
o,
\n\ < TT/T, otherwise.
Recall t h a t th e kth transmittin g filte r i s a bandpas s filte r centere d a t UJ = 2nk/M. A s a result , th e kth subchanne l correspond s t o th e frequenc y bi n
6. DFT-base d transceiver s
168 centered a t 12k7T
Ik
A
I
T
7^^r~r r a d / s , o r—— — Hz TM ' ' MT with respec t t o th e frequenc y o f th e continuous-tim e Fourie r transform . Thu the d a t a o f eac h subchanne l ar e t r a n s m i t t e d i n a separat e frequenc y bin In practice , th e transmittin g puls e pi(t) i s no t a n idea l lowpas s filter . It stopband attenuatio n i s finit e an d th e passban d gai n i s no t constant . I n thi case, xa (t) i s CWSS(NT) an d it s powe r spectru m i s (Appendi x B ) SXa(JV)=
^ 5 x ( e ^ T ) | P i C / n ) | 2 . (6.32
s . s s
)
T h e shap e o f S Xa(jQ.) i s determine d jointl y b y Pi(jQ) an d S x{e^T). A s Sx{e^T) i s periodi c wit h perio d 27r/T , th e stopban d o f Pi(jQ) wil l dictat e the rollof f o f th e t r a n s m i t t e d spectru m beyon d th e Nyquis t frequenc y TT/T. T h e abov e derivatio n doe s no t assum e Fk(z) t o b e D F T filters . T h e power spectru m expressio n i n (6.31 ) i s vali d fo r genera l transmittin g filters . W h e n th e firs t filte r F 0(z) i s use d a s a prototyp e t o generat e th e othe r filter s by frequenc y shiftin g a s i n (6.29) , th e spectru m S x{e^) i s simpl y th e su m
^(e^) = | E l ^ (e ^ " 2 W M ) ) | 2 keA
It consist s o f shift s o f th e spectru m o f F
6.5.2 ZP-OFD
0(z).
M system s
We ca n obtai n th e filte r ban k structur e fo r th e Z P - O F D M syste m i n a ver y similar manner . Th e zero-padde d O F D M syste m ca n als o b e represente d a s in Fig . 6.19 . I t t u r n s ou t t h a t (Proble m 6.16 ) th e transmittin g filter s Fk(z) are als o frequency-shifte d version s o f th e firs t transmittin g filte r F 0(z) (proto type) excep t fo r som e scalars . Th e prototyp e i s no w a rectangula r windo w o f length M. W h e n th e receive r i s chose n t o b e th e on e t h a t ha s th e efficien t im plementation i n (6.15) , th e firs t receivin g filte r HQ(Z) i s a rectangula r windo w of a longe r lengt h N. Al l th e othe r receivin g filter s ar e frequency-shifte d ver sions o f HQ(Z), excep t possibl y fo r som e scalars . Fo r othe r type s o f receivers , b o t h pseudo-invers e an d MMS E receivers , th e frequency-shiftin g propert y n o longer hold s i n general . A s th e derivatio n o f th e t r a n s m i t t e d powe r spec t r u m i n (6.31 ) doe s no t requir e assumption s o n th e lengt h o r coefficient s o f the transmittin g filters , th e t r a n s m i t t e d powe r spectru m fo r th e Z P - O F D M system als o ha s th e for m i n (6.31) .
6-6 D M
T system s
T h e discret e multiton e (DMT ) system , ofte n considere d fo r wire d applica tions, i s ver y simila r t o th e O F D M system . Th e D M T syste m ha s bee n successfully applie d t o high-spee d d a t a transmissio n ove r digita l subscribe r lines (DSL ) suc h a s ADS L (asymmetri c digita l subscribe r lines ) [7 ] an d VDS L (very-high-Speed digita l subscribe r lines ) [8] . I n DS L application s th e chan nel ca n hav e a ver y lon g impuls e response . T o avoi d IBI , w e nee d t o hav e
169
6.6. D M T system s
q(n)
-+
^0
Wti\
Trn
cyclic prefix —► C(z)
j
T(z) —► discard prefix
equivalent channe l X
M-l
(a)
x(n)=(q*t)(n)
x0
x(n) P/S(M)
cyclic prefix
r> V "
r PISH
discard prefix
}
s An)
X,
••
X
(b)
M-l
Figure 6 . 2 2 . (a ) Th e D M T syste m wit h a time domai n equalizer ; (b ) th e D M T syste m with th e equivalen t shortene d channel .
a cycli c prefi x tha t i s a s lon g a s th e longes t impuls e response . A lon g pre fix ca n seriousl y reduc e th e transmissio n rate . I n thes e applications , a tim e domain equalize r (TEQ ) t{n) i s usuall y include d a t th e receive r t o shorte n the effectiv e channe l impuls e response , a s show n i n Fig . 6.22(a) . TE Q play s an importan t rol e in improvin g th e transmissio n bi t rat e o f the DM T system . There ha s bee n extensiv e researc h o n th e desig n o f TE Q [1 , 6, 10 , 90] . Se e Section 3. 5 fo r TE Q designs . Upon tim e domai n equalization , th e impuls e respons e o f th e shortene d channel i s c eq(n) = t(n) * c(n) an d th e equivalen t channe l nois e i s r(n) = q(n) *t(n ) (Fig . 6.22(b)) . Th e origina l channe l nois e q(n) i n DSL application s is ofte n colore d du e t o crosstal k (t o b e explaine d later) . Thi s i s differen t from th e OFD M syste m fo r wireles s applications , wher e th e nois e can usuall y be assume d t o b e white . Eve n i f th e origina l channe l nois e i s white , th e equivalent nois e r(n) i s i n genera l colore d du e t o tim e domai n equalization . With th e equivalen t syste m i n Fig . 6.22(b) , w e ca n obtai n th e filter ban k representation fo r th e DM T syste m jus t lik e i n th e OFD M case . Th e filter bank representatio n wil l b e th e sam e a s tha t i n Fig . 6.1 9 excep t tha t th e channel C(z) i s replace d b y th e shortene d channe l C eq(z). Th e subchannel s in DM T system s ar e ofte n referre d t o a s tones . For wire d transmission , th e channe l doe s no t chang e a s rapidly a s in wire-
6. DFT-base d transceiver s
170
less applications . Thi s allow s th e transmitte r t o sen d ou t symbol s firs t t h a t are als o know n t o th e receiver . Thes e ar e calle d pilot s o r trainin g symbols . W i t h prope r training , th e receive r ca n obtai n th e channe l stat e information . Based o n thi s information , bi t loadin g ca n b e compute d fo r a give n erro r rate t o maximiz e th e transmissio n rate . Th e receive r ca n the n sen d bac k bi t and powe r loadin g informatio n t o th e transmitte r throug h a revers e channel . More discussio n o n bi t an d powe r allocatio n o f th e D M T syste m fo r DS L applications wil l b e give n late r i n thi s section . Real-valued transmitte d signal s I n th e D M T system , th e outpu t o f th e transmitter i s sen t directl y t o th e channe l withou t furthe r frequenc y modu lation, i.e . baseban d transmission . Th e t r a n s m i t t e d signa l x(n) i s real . T h e inputs o f th e transmitte r s k ar e typicall y comple x QA M modulatio n symbols . To hav e a rea l t r a n s m i t t e d signal , th e o u t p u t s o f th e I D F T matri x shoul d be rea l too . Thi s mean s th e input s o f th e I D F T matri x nee d t o hav e th e conjugate symmetri c property , i.e . so i s rea l an d s k =
s* M_kl k
= 1 , 2 , . . ., M — 1 . (6.33
)
If M i s even , the n th e abov e equatio n implie s t h a t s M/2 i s als o real . T h e modulation symbol s assigne d t o th e secon d hal f o f th e M subchannel s ar e th e complex conjugate s o f thos e i n th e firs t hal f subchannels , wit h th e exception s of s o an d s M / 2 • W e onl y nee d t o conside r th e firs t hal f subchannel s an d th e second hal f ar e implicitl y determined . Bi t allocatio n i s performed fo r onl y hal f the subchannels . Usuall y th e subchannel s correspondin g t o th e D C frequenc y and Nyquis t frequency , i.e . zerot h an d ( M / 2 ) t h subchannels , ar e reserve d fo r other purposes . I n thi s case , onl y tone s 1 to ( M / 2 — 1) will b e considered . Du e to th e conjugat e symmetri c propert y o f th e inputs , i t i s possibl e t o comput e the transmitte r o u t p u t s usin g onl y rea l addition s an d rea l multiplication s instead o f comple x computations . I t ca n b e show n t h a t th e I D F T operatio n for conjugat e symmetri c input s ca n b e compute d usin g D C T (discret e cosin e transform) matrice s an d D S T (discret e sin e transform ) matrice s usin g onl y real computation s [174 ] (Proble m 6.19) . The transmitte d powe r spectru m I n th e formulatio n o f th e t r a n s m i t t e d power spectru m fo r th e O F D M system , w e hav e assume d t h a t th e input s ar e uncorrelated WS S rando m processes . Fo r th e D M T system , w e ca n n o longe r make suc h a n assumptio n a s th e input s ar e i n conjugat e pair s an d henc e strongly correlated . However , wit h a sligh t modificatio n o f th e assumption , the t r a n s m i t t e d powe r spectru m o f D M T syste m ha s a for m simila r t o t h a t o f the O F D M syste m i n (6.31) . T o b e mor e specific , le t E[\s k(n)\2}= S s?k. W e have adde d th e subscrip t k t o indicat e t h a t th e subchannel s ca n hav e differen t symbol powers . Fo r thos e tone s wit h comple x symbols , w e expres s s k(n) i n terms o f it s rea l par t a k (n) an d imaginar y par t b k (n) a s sk(n) = a
k(n)
+
jb
k(n).
We assume , reasonably , t h a t thes e rando m processe s a k(n) an uncorrelated, jointl y WSS , wit h powe r spectru m Sak(ejoJ)=
S bk(e?u)=
^£ s,k, k
d b k(n) ar e
= 1,2,.. . ,M/ 2 - 1 .
6.6. DM T system s
171
T h e scala r 1/ 2 i s include d s o t h a t ^[|s/c(n)| 2 ] = £ s,kConsider th e M h an d th e ( M — /c)t h subchannels . T h e input s ar e a con jugate pai r an d s o ar e th e filte r coefficients , fM-k(n)=
f
k(n).
T h e correspondin g output s Uk(n) an d UM-k(n) (indicate d i n Fig . 6.19 ) ar e thus als o a conjugat e pair . T h e su m o f thes e tw o subchannel s i s give n b y u'k(n) =
u k(n) +
u M-k(n) =
2Real{u
k(n)}.
This ca n b e writte n a s u'k(n) =
2 ] T (a k(e)fk,r(n -
Nl) - b k[i)fk^[n -
Nl)\ , (6.34
)
where fk,r(n) an d fk,i(n) are , respectively , th e rea l an d imaginar y part s o f /fe(n). A s a k(n) an d b k(n) ar e uncorrelated , w e hav e S
<^) =
S
-f(^\FkAen\2 +
^\FkAen\
2
)-
Using th e fac t t h a t /M-fe(^ ) a n d fk{n) for m a conjugat e pair , w e ca n verif y t h a t th e su m i n th e abov e parenthesi s i s equa l t o \Fk(e^ UJ)\2 + |i*M-fc(e J ' a; )| 2 . As a resul t w e hav e
s
^n=
s
-f(\Fk(en? + \FM-k(en\2)-
Summing u p th e contribution s fro m individua l subchannels , w e ca n obtai n the transmitte d powe r spectrum . W h e n frequenc y divisio n multiplexin g (t o be explaine d i n th e following ) i s used , no t al l th e subchannel s ar e used . I n this case , S*{e?») =
^ E
ZsAFk{en\\ keA
(6.35
)
where A i s th e se t o f subchannel s t h a t ar e actuall y use d fo r transmission . (W e can als o arriv e a t th e sam e resul t usin g propertie s fro m [131]. ) Lik e the O F D M case (give n i n (6.32)) , th e powe r spectru m o f th e transmitte d continuous-tim e signal depend s o n th e samplin g perio d an d th e transmittin g pulse .
Remarks • Not e t h a t th e transmitte d powe r spectru m o f th e D M T syste m i s o f th e same for m a s t h a t o f th e O F D M syste m i f a n equa l powe r allocatio n i s used, i.e . S s^k = £ s for al l k. • I n th e abov e derivatio n o f th e transmitte d powe r spectrum , th e trans mitting filter s onl y nee d t o b e i n conjugat e pair s {JM-kip) — fk( n))T h e transmittin g filter s ar e no t necessaril y D F T filter s fo r th e expressio n in (6.35 ) t o hold . The y ca n b e arbitrar y filter s t h a t hav e th e conjugate pair property .
172
6. DFT-base d transceiver s
Frequency divisio n multiplexin g I n a D M T system , signal s ar e usuall y sent i n b o t h direction s simultaneousl y ove r th e sam e transmissio n mediu m such a s th e twiste d pair s i n ADS L systems , fro m th e custome r t o th e networ k (upstream direction ) an d fro m th e networ k t o th e custome r (downstrea m direction). Th e downstrea m an d upstrea m signal s ca n b e separate d b y usin g frequency divisio n multiplexin g (FDM) , i n whic h differen t frequenc y band s are allocate d fo r eac h direction . Th e D M T syste m lend s itsel f nicel y t o F D M , since th e subchannel s ar e divide d i n a frequenc y base d manner . T o d o this , we ca n se t asid e som e tone s fo r upstrea m transmissio n an d som e fo r down stream transmission . Fo r example , i n th e ADS L system , th e tota l numbe r o f subchannels M i s 512 . Tone s 1-3 1 ar e use d fo r upstrea m transmissio n an d the res t o f th e highe r frequenc y tone s (32-255 ) fo r downstrea m transmissio n as illustrate d i n Fig . 6.23 . (Usuall y a fe w close-to-D C tone s ar e reserve d fo r voice an d guar d band. ) I n downstrea m transmission , onl y th e downstrea m tones ar e used ; th e input s o f th e upstrea m tone s ar e zero . Similarly , th e in puts o f th e downstrea m tone s ar e zer o i n upstrea m transmission . B y havin g a muc h wide r bandwidth , th e downstrea m directio n ca n hav e a muc h highe r transmission rat e t h a n th e upstrea m direction . T h a t i s why ADSL , wher e "A " stands fo r asymmetric, i s s o named . Th e narrowe r upstrea m ban d i s place d in a lowe r frequenc y regio n t h a n th e downstrea m band . Thi s wa y upstrea m transmission i s stil l possibl e eve n whe n th e channe l i s muc h a t t e n u a t e d i n high frequency , whic h i s usuall y th e cas e wit h lon g DS L lines .
upstream downstrea tones tone <►
^►
13
2 25
s
m
5 tone
s index
Figure 6 . 2 3 . Upstrea m an d downstrea m tone s i n th e ADS L system .
In th e standard s fo r ADS L an d VDS L system s [7 , 8 ] th e powe r spectru m for eac h transmittin g directio n i s constraine d throug h a spectru m mask . T h e spectrum mas k set s a n uppe r boun d fo r th e spectru m o f the t r a n s m i t t e d signa l as a functio n o f frequency . Figur e 6.2 4 show s a n exampl e o f spectru m mas k for th e downstrea m ADS L transmission . Th e mas k i s given i n d B m / H z , wher e 1 d B m = 101og(10~
3
watts) .
Also show n i s an exampl e o f the downstrea m signal' s spectrum , whic h i s belo w the spectru m mas k fo r al l frequency . Noise impairment s I n a DS L environment , a numbe r o f nois e source s contribute t o th e channe l nois e q(n). Usuall y DS L line s ar e bundle d togethe r
173
6.6. D M T system s
11
— - spectru m mask signal spectrum ■ ■ ■ ■ NEX T crosstalk ■ — ■ FEX T crosstalk AWGN I RF I
HI
It
•/ 1
if ii
1r
"1 ' \ 0 4
mYi
'~""" "
^~
25.875 60
~
* -■» . j ^ i
i
0 110 4 3093 1104
kHz
i
0
Figure 6.24 . Exampl e o f spectru m mas k fo r downstrea m ADS L transmissio n an d th e spectrum o f a downstrea m signal . Als o show n i n th e figur e ar e nois e impairment s i n a DSL environment , includin g NEX T crosstalk , FEX T crosstalk , AWGN , an d RFI .
in multiple s o f 2 5 i n cable s an d thi s make s th e channe l pron e t o crosstal k from othe r line s i n th e sam e cable . T h e crosstal k ma y b e near-en d crosstal k (NEXT) o r far-en d crosstal k ( F E X T ) . N E X T come s i n whe n th e disturbin g signal i s traveling i n th e directio n opposit e t o th e receive d signal . Fo r example , in Fig . 6.25 , th e upstrea m signa l o f lin e 1 couple s int o lin e 2 an d become s the sourc e o f N E X T i n downstrea m transmissio n o n lin e 2 . F E X T refer s to th e cas e whe n th e disturbin g signa l i s travelin g i n th e sam e directio n a s the receive d signal . Figur e 6.2 5 show s th e downstrea m signa l o f lin e 1 a s a F E X T sourc e fo r th e downstrea m directio n o n lin e 2 . F E X T i s usuall y the dominatin g nois e whe n F D M i s use d becaus e th e spectru m o f th e signa l traveling i n th e opposit e directio n occupie s a differen t frequenc y band . I n fact , it i s possibl e t o eliminat e interferenc e fro m th e othe r directio n an d N E X T by usin g a longe r cycli c prefi x an d carefull y synchronizin g downstrea m an d upstream transmissio n [139] . Bot h type s o f crosstal k ca n b e determine d fro m the disturbers ' powe r spectru m an d crosstal k transfe r function s [7 , 8] and the y can b e modele d a s colore d Gaussia n noise s [122 , 144] . In addition , th e channe l nois e wil l contai n AWG N du e t o therma l noise . Also, radi o frequenc y signal s suc h a s amateu r radi o (HAM ) an d A M radi o may interfer e an d resul t i n radi o frequenc y interferenc e (RFI) . Thes e R F I signals ar e o f a narrowban d n a t u r e bu t hav e a larg e amplitud e (i n th e fre quency domain) . Man y tone s i n th e vicinit y o f th e R F I carrie r frequenc y ma y be affecte d du e t o th e poo r frequenc y separatio n o f D F T filters o f th e D M T receiver. Figur e 6.2 4 show s th e variou s nois e impairment s i n a DS L environ ment, includin g F E X T an d N E X T crosstalk , AWGN , an d R F I . T h e AWG N noise i n th e figure i s —14 0 d B m / H z an d th e R F I i s aroun d 60 0 kH z wit h a n amplitude o f aroun d —5 5 dBm .
174 6
. DFT-base d transceiver s
DMT transceiver 1
line 1
DMT transceiver 2
line 2
cable bundle
ft u
_ ^ ) NEXT
/FEXT
—
n~~I
network
^ ^
4-■
Figure 6.25 . Illustratio n o f near-en d an d far-en d crosstalks .
FEQ coefficient s Afte r prope r tim e domai n equalization , th e equivalen t channel c eq(n) ha s mos t o f th e energ y concentrate d i n a windo w o f v taps . Say th e windo w span s th e rang e fro m n o t o n o + v. W e ca n obtai n th e F E Q coefficient sA & by computin g th e reciprocal s o f th e M-poin t D F T o f th e channel coefficients . T h a t is , no+v
Xk = 1/C e ,, fc , wher
eC
eq,k =
£ c
kn M
eq(n)e-^
^ . (6.36
)
A precis e calculatio n o f F E Q coefficient s involve s th e coefficient s o f th e chan nel outsid e th e windo w to o [81] . Bu t (6.36 ) give s a ver y goo d estimation , especially whe n th e T E Q doe s a goo d jo b o f shortenin g th e channel . Bit allocatio n Althoug h tim e domai n equalizatio n ca n shorte n th e chan nel, th e equivalen t channe l c eq(n) wil l stil l hav e nonzer o coefficient s outsid e the window . Thes e nonzer o coefficient s resul t i n residua l interferenc e fro m adjacent block s an d als o fro m othe r subchannel s o f th e sam e block . I n par ticular, i f w e comput e th e erro r efc(n ) = Sfc(n ) — Sfc(n), i t wil l contai n channe l noise a s wel l a s interferenc e fro m othe r symbols . W e ca n obtai n th e SN R o f the fcth ton e b y computin g
where th e symbo l powe r i£[|sfc(7i)| 2] i s constraine d b y th e spectru m mas k i n DSL application s (Proble m 6.20) . Althoug h e ^ i s a mixtur e o f interferenc e and noise , th e Gaussia n distributio n render s a nic e approximatio n a s w e hav e explained i n Chapte r 3 . W e ca n us e (6.37 ) t o comput e th e numbe r o f bit s t h a t ca n b e loade d t o th e fcth tone , h=
l 0g2 ( l +
^ ),
(6.38
)
where T i s th e SN R ga p fo r a give n erro r rat e a s describe d i n (2.21) . Fo r those tone s t h a t ar e no t use d fo r transmission , n o bit s ar e allocated , bi = 0 . T h e tota l numbe r o f bit s t r a n s m i t t e d i n eac h bloc k i s J2i= o &* • Eac h bloc k
6.6. DM T system s
175
takes u p a transmissio n tim e o f NT seconds , wher e T i s the underlyin g sampl e spacing o f th e D / C converter . T h e transmissio n rat e R^ i s give n b y
Rb = Y 0
NT
M-l
^h
^ i=0%
T h e bi t assignment s compute d i n (6.38 ) ar e no t integer s i n general . W e ca n use, fo r example , rounding , t o hav e a n intege r bi t allocation . I f roundin g i s used, th e QA M symbo l o f th e kth ton e wil l carr y round log 2 I 1 + r
1
bits .
(6.39)
T h e valu e bk ca n b e eve n o r odd . Afte r rounding , th e erro r rat e wil l b e slightl y different fro m th e targe t erro r rate . A mino r adjustmen t o f th e subchanne l powers ca n b e carrie d ou t t o compensat e fo r th e roundin g erro r (Proble m 21) . In ADS L an d VDS L application s [7 , 8] , th e receive r compute s bi t allocatio n and powe r adjustment , an d send s th e informatio n bac k t o th e transmitter . Example 6. 5 A D M T syste m fo r ADS L application . T h e D F T siz e M = 51 2 and th e cyclic-prefi x lengt h v = 40 . T h e numbe r o f block s t o b e t r a n s m i t t e d per secon d i s 400 0 [7] . T h e samplin g rat e i s (512 + 40 ) x 400 0 = 2.20 8 x 10 6 samples/s . We wil l conside r downstrea m transmissio n i n thi s example . T h e tone s use d are 33-255 . T h e transmissio n powe r use d i s — 4 d B . Figur e 6.26(a ) show s a typical ADS L channe l - loo p 6 [7] . T h e lengt h o f th e loo p i s 18,00 0 feet . T h e magnitud e respons e o f th e channe l i s show n i n Fig . 6.26(b) . T h e channe l has a typica l lowpas s characteristic , mor e attenuate d i n high-frequenc y band . T h e low-frequenc y tone s ca n carr y mor e bit s t h a n high-frequenc y tones . Fig ure 6.26(c ) show s th e powe r spectru m o f th e channe l noise , whic h comprise s N E X T an d F E X T a s wel l a s AWG N noise . T h e channe l nois e i s mor e promi nent i n th e frequenc y ban d use d fo r upstrea m transmissio n du e t o N E X T . After tim e domai n equalization , th e resultin g equivalen t channel , show n i n Fig. 6.26(a) , ha s energ y muc h mor e concentrate d an d henc e muc h "shorter " t h a n th e origina l channel . T h e T E Q ha s si x tap s an d th e MSSN R metho d in Sectio n 3. 5 [90 ] ha s bee n use d t o desig n th e T E Q . Equation s (6.37 ) an d (6.39) ar e use d t o comput e th e subchanne l SNR s an d th e bi t loadin g o f QA M symbols, shown , respectively , i n Fig . 6.26(d ) an d (e) . T h e targe t symbo l erro r rate i n thi s cas e i s 1 0 - 7 . T h e tota l numbe r o f bit s transmitte d i n eac h bloc k is 174 2 an d th e transmissio n rat e i s 7. 0 Mbits/sec . ■
176
6. DFT-base d transceiver s
original channel respons e shortened channel respons e
(a)
50 10
0
200
time index
(b)
0.2 0. 4 0. 6 0. Frequency normalize d by7i.
8
-65
(c)
-100
0.2 0. 4 0. 6 0. Frequency normalize d by7i.
8
177
6.6. D M T system s
100 15 0 tone index
250
100 15 0 tone index
250
Figure 6.26 . Exampl e 6.5 . A D M T system , (a ) Th e impuls e response s o f the channe l and th e equivalen t shortene d channel ; (b ) th e magnitud e respons e o f th e channel ; (c ) the channe l nois e spectrum ; (d ) th e subchanne l SNRs ; (e ) th e subchanne l bi t loading .
6. DFT-base d transceiver s
178
6-7 Channe l estimatio n an d carrie r frequenc y synchronization In al l the previou s discussions , i t wa s assume d t h a t th e channe l i s known a t th e receiver an d t h a t th e carrie r frequencie s a t th e transmittin g an d receivin g side s are perfectl y synchronized . I n mos t practica l applications , th e receive r need s to estimat e th e channe l impuls e respons e o r frequenc y respons e an d i t als o needs t o synchroniz e th e carrie r frequency . Thi s sectio n give s a brie f coverag e of th e estimatio n o f channe l an d th e synchronizatio n o f carrie r frequenc y fo r O F D M systems .
6.7.1 Pilo
t symbo l aide d modulatio n
Channel estimatio n i s ofte n don e wit h th e ai d o f know n symbols. 4 I t i s know n as pilo t symbo l aide d modulatio n (PSAM) , a t e r m coine d b y Caver s [17] . T h e pilo t symbol s ca n b e inserte d eithe r i n th e tim e domai n o r frequenc y domain. I n O F D M systems , th e pilo t symbol s ar e ofte n multiplexe d wit h information-bearing d a t a i n th e frequenc y domain . Th e existenc e o f pilo t symbols simplifie s th e challengin g tas k o f receive r design . A t th e receiver , these pilo t symbol s ca n b e exploite d fo r channe l estimation , synchronization , receiver adaptation , an d othe r purposes . Belo w w e wil l briefl y discus s channe l estimation usin g th e P S A M scheme . I n th e literature , ther e ar e man y article s on thi s topic . Intereste d reader s ar e referre d t o [17 , 22 , 35 , 101 , 103 , 125 , 153] , to nam e jus t a few . Consider th e O F D M syste m show n i n Fig . 6.1 . I n a P S A M scheme , som e of th e subchannel s d o no t carr y information-bearin g data . Instead , thes e subchannels (know n a s pilo t tones ) ar e use d fo r sendin g th e pilo t symbols . Suppose t h a t P entrie s o f th e M x l inpu t vecto r s contai n th e pilo t symbols . For simplicity , w e assum e t h a t P divide s M an d M/P = J. I t wa s foun d [35, 153 ] t h a t a goo d strateg y i s t o distribut e th e pilo t symbol s uniforml y among al l subchannels ; i.e . th e indexe s n o f th e pilo t tone s s n ar e give n b y n = kJ + no , fo
r0 < k < P,
for som e 0 < n o < J . Assumin g t h a t th e channe l orde r i s no t large r t h a n th e cyclic-prefix length , fro m earlie r discussion s w e kno w t h a t th e signal s y n afte r the D F T operatio n a t th e receive r (se e Fig . 6.1 ) ar e give n b y Vn ^n^n
~
r Qui
where C n i s th e gai n o f th e n t h subchanne l an d q n i s th e nois e term . A s the pilo t symbol s Skj+ no ar e known , th e receive r ca n estimat e th e subchanne l gains fo r thes e pilo t tone s a s p VkJ+np ^kJ+no —
• SkJ-\-no
To obtai n th e subchanne l gai n C n fo r n ^ kJ + no , on e ca n us e th e interpo lation schem e [36] . W h e n th e channe l ha s a smoot h frequenc y response , a n 4 It i s als o possibl e t o estimat e th e channe l blindly . However , blin d channe l estimatio n is seldo m employe d i n practic e du e t o it s hig h complexit y an d lowe r estimatio n accuracy . Readers intereste d i n thi s topi c ar e referre d t o [21 , 45, 147 , 183] .
6.7. Channe l estimatio n an d carrie r frequenc y synchronizatio
n 17
9
accurate estimat e o f C n ca n b e obtaine d b y interpolation . Anothe r wa y o f getting al l C n i s b y th e followin g tim e domai n approach . Usin g th e fac t t h a t Cn ar e th e D F T coefficient s o f c(£) , w e ca n writ e V
£=0
for 0 < k < P. Le t v b e th e P x 1 vecto r containin g Ckj+ no fo r 0 < / c < P and le t u b e th e {y + 1 ) x 1 vector containin g d(£). T h e n th e abov e syste m o f linear equation s ca n b e re-expresse d a s v = W u , wher e W i s a P x (i / + 1 ) matrix wit h (fc , Z)th entr y W^ kJ+n^1. W h e n P > v, w e ca n solv e th e matri x equation fo r d(£) an d obtai n u=
(WtW^WW .
Taking th e M-poin t D F T o f c(£) , w e ge t C n fo r al l 0 < n < M. T h e com plexity o f th e tim e domai n approac h i s usuall y highe r t h a n th e interpolatio n scheme, bu t i t give s a mor e accurat e estimat e o f C n , especiall y whe n P i s much large r t h a n v.
6.7.2 Synchronizatio
n o f carrie r frequenc y
For passban d communications , th e transmitte d signa l i s modulate d t o a car rier frequenc y a t th e transmitter . A t th e receiver , w e nee d t o perfor m carrie r demodulation t o ge t th e baseban d signal . T h e frequenc y o f th e oscillato r a t the receive r i s usuall y slightl y differen t fro m t h a t o f the oscillato r a t th e trans mitter. Thi s result s i n carrie r frequenc y offse t (CFO) . T h e O F D M syste m i s very sensitiv e t o C F O [118] , whic h destroy s th e orthogonalit y o f subchannel s and degrade s th e performanc e o f O F D M system s significantl y [118] . Thus , a t the receive r ther e i s a nee d t o perfor m a n accurat e synchronizatio n o f carrie r frequency. Man y approache s hav e bee n develope d i n th e past . Fo r example , see [9 , 53 , 71 , 85 , 91 , 93 , 95 , 136 , 166] . I n man y practica l applications , a periodic trainin g sequenc e know n t o th e receive r i s sen t a t th e beginnin g o f transmission. B y exploitin g th e periodi c n a t u r e o f th e trainin g sequence , w e can us e a simpl e metho d fo r correctin g th e C F O , a s w e shal l explai n below . Consider Fig . 6.1 . Suppos e t h a t tw o identica l vector s s ar e sen t consec utively a t th e transmitter . W h e n ther e i s n o C F O , th e correspondin g tw o received vectors , denote d b y r i an d r 2 , respectively , wil l hav e th e followin g form (se e (6.1)) : ri =
C
circx
+ qi , r
2
=
C
circx
+ q 2 , (6.40
)
where x = W^~s . The y ar e differen t onl y becaus e o f th e nois e vectors . How ever, whe n ther e i s C F O , th e tw o receive d vector s ar e n o longe r give n b y (6.40). Le t f c b e th e oscillato r frequenc y a t th e transmitte r an d le t f c — Sf be th e oscillato r frequenc y a t th e receiver . T h e n du e t o th e mismatc h o f th e oscillator frequencies , th e receive d baseban d signa l wil l b e modulate d b y a n extra exponentia l t e r m e j e n . T h e quantit y e is th e normalize d C F O paramete r and i t i s relate d t o Sf b y e = 2TrS fT,
180
6. DFT-base d transceiver s
where T i s the sampl e spacing . A s a result , th e tw o receive d vector s becom e [53, 71 , 85] ?! = D r l 7 ? 2 = e ^ e D r 2 , (6.41 ) where D i s a matri x diagonal , give n b y D = diag[e" e e(
v+1 £
> ... e ^ "
1
^],
N = M + z/ , and v i s the lengt h o f cycli c prefix . Fro m (6.41) , i t i s clea r fro m (6.40) an d (6.41 ) tha t w e ca n obtai n a n estimat e o f th e CF O e as
r-^(ifc), where Lz denote s th e angl e o f th e comple x numbe r z. Th e abov e method , though simple , give s a ver y accurat e estimat e o f th e CF O [95] . Fo r mor e advanced approache s t o carrie r frequenc y synchronization , se e [9 , 53 , 71 , 85, 91, 93 , 95, 136 , 166].
6.8 A
historica l not e an d furthe r readin g
The us e o f frequenc y divisio n multiplexin g (FDM ) t o divid e th e channe l int o narrower subchannel s an d sen d dat a i n paralle l date s bac k t o th e 1950s . Fo r each subchannel , th e bandwidt h i s narrower. Th e transmitte d signa l "sees " a flatter channel , and equalization becomes easier. Th e system is potentially les s sensitive t o channe l distortio n an d wideban d impulsiv e noise . Th e first stud y on designin g continuous-tim e transmittin g filters fo r maintainin g orthogonal ity amon g the subchannel s a t th e receive r wa s made i n [18] , and late r i n [127 ] and [19] . Wit h th e advance s o f digita l signa l processing , digita l implementa tion wa s mad e mor e practica l a t a lowe r cost . A n efficien t implementatio n o f FDM system s using Discrete Fourier Transfor m (DFT ) wa s developed in [178] . This eliminate d th e nee d fo r a n expensiv e arra y o f oscillator s an d coheren t demodulators use d i n earlier works. Bi t loadin g optimization fo r multichanne l (multitone) system s was derived i n [59 , 60]. I n earlier FD M system s there wa s no prefix . Th e orthogonalit y amon g th e subchannel s wa s usuall y destroye d after channe l filtering, an d th e receive r coul d no t remov e IS I a s easily . Th e task o f equalizin g th e FD M syste m wa s greatl y simplifie d whe n cycli c pre fixing wa s introduce d a t th e transmitte r i n [116] . Th e insertio n o f a cycli c prefix allowe d the channe l t o b e equalize d usin g low-cos t IDF T an d DF T ma trices. Bit s wer e assigne d accordin g t o subchanne l SNR s t o achiev e a highe r transmission rat e [116] . The first practica l applicatio n o f cyclic-prefixe d DFT-base d transceiver s was propose d i n [154 ] fo r DS L applications . Th e cyclic-prefixe d DFT-base d transceiver wa s terme d th e discret e multiton e (DMT ) syste m i n [24] . Perfor mance evaluation s o f the DM T syste m wer e give n i n [24 ] for HDS L (high-bit rate digita l subscribe r lines ) application , an d i n [25 ] for ADS L an d VDS L ap plications. Subsequentl y th e DM T syste m ha s bee n repeatedl y demonstrate d to outperfor m othe r competin g single-carrie r solution s significantly . I t wa s later adopte d i n the standard s fo r ADS L [7 ] and VDS L [8 ] transmission. Mul tichannel FD M transmissio n wa s proposed fo r wireles s environment b y Cimin i
6.9. Problem s
181
[27], an d th e syste m i s terme d O F D M (orthogona l frequenc y divisio n multi plexing). T h e O F D M syste m i s no w on e o f th e mos t popula r technique s fo r wireless transmission . Cyclic-prefixe d O F D M system s hav e bee n adopte d i n standards suc h a s digita l audi o broadcastin g [39] , digita l vide o broadcastin g [40], wireles s loca l are a network s [54] , an d broadban d wireles s acces s [55] . Comparisons o f cyclic-prefixe d an d zero-padde d O F D M ca n b e foun d i n [96] . Frequency domai n equalizatio n fo r single-carrie r system s wa s first pro posed i n [173] . Ther e wa s n o redundanc y i n th e transmitte d signals . T h e introduction o f cycli c prefi x t o single-carrie r system s first appeare d i n [129] . As equalizatio n wa s performe d a t th e receive r afte r D F T operation , i t wa s also calle d a single-carrie r syste m wit h frequenc y domai n equalizatio n (SC F D E ) . Extensiv e performanc e evaluatio n give n i n [41 , 175 ] showe d t h a t th e SC-CP syste m compare s favorabl y wit h th e O F D M system . T h e SC-C P sys t e m enjoy s a smalle r P A P R , especiall y fo r a larg e numbe r o f subchannels . I t is als o les s sensitiv e t o th e so-calle d carrie r frequenc y offset , whic h i s a n im p o r t a n t issu e i n applyin g O F D M t o wireles s channel s [118 , 175] . A n analyti c comparison o f th e SC-C P an d th e O F D M system s wa s give n i n [77 , 78 ] t o show t h a t th e SC-C P alway s ha s a smalle r uncode d BER . T h e SC-C P syste m is no w par t o f th e broadban d wireles s acces s s t a n d a r d [55] . Fo r zero-padde d single-carrier systems , a n extensiv e stud y wa s give n i n [104] , an d a n analyti c B E R compariso n wit h SC-C P syste m wa s als o give n therein .
6.9 Problem
s
6.1 Le t M = 4 an d le t th e cyclic-prefi x lengt h v = 2 . Suppos e th e channe l is C(z) = 1 + z - 1 , th e channe l nois e i s AWGN , th e SN R quantit y 7 = S s/Afo i s 1 0 dB , an d th e modulatio n symbol s ar e Q P S K . Comput e the subchanne l SNR s /3(z) , th e averag e mea n square d erro r £ r r , an d th e average bi t erro r rat e V o f (a ) th e O F D M system , (b ) th e SC-C P syste m with a zero-forcin g receiver , an d (c ) th e SC-C P syste m wit h a n M M S E receiver. 6.2 Channel estimation error. I n th e O F D M system , th e receive r need s t o estimate th e channe l t o comput e th e F E Q coefficients . W h e n ther e i s channel noise , ther e wil l b e estimatio n error . Suppos e th e estimate d channel i s c(n) = 0.9£(n) , whil e th e actua l channe l i s c(n) = S(n). We us e th e channe l estimat e c{n) t o comput e th e F E Q coefficient s an d implement th e receiver . Assum e th e res t o f th e settin g i s th e sam e a s in Proble m 6.1 . W h a t ar e th e bi t erro r rate s fo r th e thre e system s i n Problem 6.1 ? 6.3 Modified OFDM system with rotated cyclic prefix. I n thi s problem , w consider a n O F D M syste m wit h a differen t typ e o f prefi x - a rotate cyclic prefix . W h e n w e cop y th e las t v sample s o f eac h bloc k an d plac t h e m a t th e beginnin g o f th e block , w e appl y a n extr a rotatio n e j M ( 9 t the prefix . T h e rotate d prefi x sample s ar e e-^XM-i-
e d e o
182
6. DFT-base d transceiver s
Assume th e channe l C(z) i s a n F I R filte r o f orde r < v. Th e ne w prefixe d system a s show n i n Fig . P6. 3 i s th e sam e a s t h a t i n Fig . 5.1 6 excep t fo r the rotate d prefix .
/
/
->
x
o
-> r ,
q{n) x(n)
-►
•• •
P/S(itf)
rotated prefix j
discard prefix
C(z)
S/P(M)
"► r,
- >
Figure P 6 . 3 . A
r
i
prefixe d syste m wit h rotation .
(a) Fin d th e transfe r matri x C fro m x t o r . (b) Sho w t h a t A ( e ^ ) C A ( e " ^ ) i s a circulan t matrix , wher e A(z ) i s th e diagonal matri x A(z)
diag [ l z
-(M-i)i
(c) Desig n a transmittin g matri x G an d a receivin g matri x S fo r th e new prefixe d syste m s o t h a t th e overal l syste m i s ISI-fre e b y modi fying th e receive r o f th e O F D M system . I n suc h a modifie d O F D M system, th e ISI-fre e propert y continue s t o hol d fo r an y F I R channe l of orde r v. Th e transmitte r i s stil l channel-independen t an d th e only channel-dependen t elemen t i s a se t o f scalar s a t th e receiver . (d) Suppos e w e alread y kno w t h a t th e channe l ha s a spectra l nul l a t 2TT/M. Choos e 0 s o t h a t th e receive r ca n recove r al l th e d a t a i n the absenc e o f channe l noise . I n thi s way , th e spectra l nul l ca n b e avoided b y choosin g 0 carefully . 6.4 Differential OFDM [39] . I n th e O F D M syste m show n i n Fig . 6.1 , onl y one bloc k o f input s i s shown . I n consecutiv e transmission , w e denot e the kth subchanne l inpu t o f th e n t h bloc k b y Sfc(n) . Suppos e th e Q P S K symbols t o b e t r a n s m i t t e d ar e a k (n) = ±l/\/2±j/V2, fo r0 < k < M and n > 0 . W e se t Sfc(-l) = y/£ 3/2(l-\-j), s
k(n)=
s k(n - l)a k(n), n
> 0.
Assume th e channe l doe s no t chang e ove r tw o consecutiv e blocks . I n the absenc e o f channe l noise , sho w ho w th e receive r ca n retriev e ak(n) without knowin g th e channe l impuls e response .
6.9. Problem s
183
6.5 OFDM with cyclic prefix and suffix. I n th e transmitte r o f th e O F D M system i n Fig . 6.1 , a cycli c prefi x o f lengt h v i s adde d an d th e transmit ted signa l x(n) ha s N = M + v sample s i n eac h block . I n particular , i f we conside r on e block , th e sample s ar e V
XM-v •
• • XM-1
%0
/
v
V
prefix IDF
v
Xl
•
'
• • XM-1
T output s
Now suppose , instea d o f cycli c prefix , th e transmitte r add s a combina tion o f cycli c prefi x o f lengt h I/Q an d cycli c suffi x o f lengt h v — VQ as i n Problem 5.19 . T h e resultin g transmitte d signa l is , XM-VQ • •
^_
• XM-1
_ -/
vv
prefix IDF
XQ
v_
XI
v
..
.X
_ -/
M-1
XQ
T output s suffi
v_
XI
..
.X _ -*
v-v^-\
x
Modify th e receive r i n Fig . 6. 1 s o t h a t zero-forcin g i s achieved . 6.6 Conside r th e Z P - O F D M syste m i n Fig . 6.7 . (a) Sho w t h a t th e receive r i s zero-forcin g i f an d onl y i f th e receivin g matrix S i s give n b y (6.12) . (b) Sho w t h a t whe n A = 0 , th e receive r become s th e pseudo-invers e solution i n (6.13) . 6.7 Sho w t h a t th e MMS E receive r o f th e Z P - O F D M syste m i n (6.16 ) ca n b e rewritten a s th e matri x i n (6.17) . (Hint . T h e identit y B ( B ^ B + I n ) = ( B B ^ + I m ) B hold s fo r a n arbitrar y m x n matri x B. ) 6.8 Suppos e th e channe l i s a MIM O syste m wit h a constan t m x n (m > n) transfe r matri x C . T h e transmitte d vecto r s i s a vecto r o f siz e n. T h e outpu t o f th e channe l i s r = C s + q . T h e channe l nois e q ha s autocorrelation matri x A/oI m . T h e receive r S i s a zero-forcin g receive r with S C = I n . T h e outpu t nois e i s e = S r — s . (a) Fin d th e zero-forcin g receive r t h a t lead s t o th e smalles t outpu t noise E ^ e ] . (Hint . Us e th e singula r valu e decompositio n o f C. ) (b) Sho w t h a t th e receive r i n (a ) als o minimize s th e nois e i n eac h sub channel. 6.9 W h e n th e Z P - O F D M syste m ha s th e DFT-base d efficien t receive r a s given i n (6.15) , sho w t h a t th e zt h subchanne l nois e varianc e i s o\. = M \Ci\
2
'
6.10 Fo r th e SC-C P system , deriv e th e MMS E receive r i n (6.21 ) usin g th e orthogonality principle . 6.11 I n th e erro r analysi s o f MMS E SC-C P syste m (Sectio n 6.3.3) , w e de rived th e autocorrelatio n matri x o f the outpu t erro r an d use d i t t o obtai n the subchanne l erro r variance s an d subchanne l unbiase d SNRs . Alter natively w e ca n obtai n thes e value s usin g diagona l element s tu o f th e overall transfe r matri x T (6.22 ) usin g (3.20 ) an d (3.18) .
184
6. DFT-base d transceiver s (a) Expres s tu i n term s o f th e channe l D F T coefficient s Ct. (b) Fin d f3(i) an d f3
biased(i).
(c) Fin d o\. . 6.12 I t i s know n t h a t th e magnitud e respons e o f a n LT I filte r C(z) stay s th e same whe n w e replac e on e zer o a o f C(z) b y it s conjugat e reciproca l 1/a*. I n thi s cas e onl y th e phas e respons e i s altered . Sho w t h a t fo r th e cyclic-prefixed O F D M an d SC-C P systems , th e performanc e i s affecte d only b y th e magnitud e response , bu t no t th e phas e response . Thi s mean s t h a t th e syste m performanc e i s th e sam e i f th e channe l i s replace d b y another t h a t ha s th e sam e magnitud e response . Therefor e i t doe s no t m a t t e r i f th e channe l ha s zero s outsid e th e uni t circle . Th e performanc e of th e syste m remain s th e same . 6.13 Conside r th e SC-Z P system . Suppos e th e channe l nois e i s AWG N wit h variance A/ o an d th e receive r i s th e on e wit h a n efficien t implementatio n as i n (6.25) . Comput e th e subchanne l nois e variance s a^. a t th e o u t p u t of th e receiver . 6.14 Th e SC-Z P transmitte r pad s v zero s fo r ever y bloc k o f siz e M. Suppos e we hav e a receive r t h a t ha s a computatio n mechanis m t h a t ca n perfor m Appoint D F T an d I D F T , wher e N = M + v. Fo r suc h a receive r ca n we us e Appoin t D F T an d I D F T instea d o f M-poin t D F T an d I D F T to achiev e zero-forcing ? I f so , find a bloc k diagra m o f th e receivin g end t h a t ca n recove r th e t r a n s m i t t e d symbol s i n th e absenc e o f channe l noise. Fin d th e solution s o f al l zero-forcin g receiver s i n thi s case . 6.15 DFT-based MMSE SC-ZP system. Conside r th e SC-Z P syste m i n Fig. P6.15 , where th e receive r ha s a DFT-base d structure . Th e matri x T i s a s de fined i n (6.14) . (a) Le t r b e th e outpu t vecto r o f th e matri x T a s indicate d i n th e figure. W e ca n writ e r a s C circs + r , wher e r come s entirel y fro m the channe l noise . Determin e th e autocorrelatio n matri x R r . (b) Le t y =
W s . Sho w t h a t E[\\y-y\\2]=E[\\s-s\\%
where y i s a s indicate d i n th e figure. (c) Us e th e orthogonalit y principl e t o find th e scalar s Ao , A i , . . ., \M-I such t h a t E[\\s — s|| 2 ] (o r equivalentl y E[\\y — y|| 2 ]) i s minimized . T h e MMS E receive r fo r th e SC-Z P syste m i n Sectio n 6. 4 doe s no t hav e an efficien t implementatio n lik e th e SC-C P system . I n thi s proble m th e receiver i s optimized t o minimiz e MS E subjec t t o a n efficien t DFT-base d receiver. Th e reader s ca n verif y b y simulation s t h a t th e performanc e comes ver y clos e t o th e unconstraine d MMS E receiver . 6.16 Zero-padded OFDM systems. Conside r th e Z P - O F D M syste m wit h th e efficient zero-forcin g receive r i n (6.15) .
185
6.9. Problem s
s
o —
s
, —
•• •
► ►
?(«)
P/S(AQ
-+
zero padding -> C(z)
^r
*/
M-\ ^
Figure P 6 . 1 5 . DFT-base d MMS E SC-Z P syste m
(a) Deriv e th e transmittin g an d receivin g filters Fk(z) an d Hk(z) i n the correspondin g filter ban k representation . Sho w t h a t FQ(Z) and HQ(Z) ar e b o t h rectangula r window s o f lengt h M an d AT , re spectively. Expres s th e othe r transmittin g (receiving ) filters a s frequency-shifted version s o f th e first transmittin g (receiving ) filter excep t fo r a scalar . (b) Usin g th e filter ban k representation , th e M h nois e p a t h a t th e receiver i s a s show n i n Fig . P6.16 . Suppos e th e channe l nois e q(n) is AWG N wit h varianc e A/o - Us e th e receivin g filter Hk(z) i n (a ) to obtai n th e subchanne l nois e varianc e a 2Pi .
q(n).
Hk{z)
±N
va
■>e
k
Figure P 6 . 1 6 . Nois e pat h fo r th e /ct h subchannel .
6.17 Deriv e th e filter ban k representatio n fo r th e SC-C P syste m b y finding the transmittin g filters Fk(z) an d receivin g filters Hk(z). Expres s Fk(z)
6. DFT-base d transceiver s
186
in term s o f FQ(Z) an d sho w t h a t al l th e transmittin g filter s ar e shifte d versions o f th e prototyp e FQ(Z) excep t fo r som e scalars . Similarly , ex press Hk(z) i n term s o f HQ(Z) an d sho w t h a t al l th e receivin g filter s ar e shifted version s o f H 0(z) excep t fo r som e scalars . 6.18 W e kno w t h a t whe n th e input s o f th e D M T syste m i n Fig . 6.2 2 hav e th e conjugate symmetri c propert y i n (6.33) , th e transmitte r outpu t x(n) ha s real coefficients . I s th e conjugat e symmetri c propert y als o a necessar y condition fo r havin g a rea l transmitte r o u t p u t ? Justif y you r answer . 6.19 I n a D M T transmitter , th e I D F T o u t p u t s Xfi a r e relate d t o th e input s Sk b y M-l
Suppose M i s even an d th e input s satisf y th e conjugat e symmetri c prop erty i n (6.33) . W e expres s Sk i n term s o f rea l an d imaginar y part s b y Sk = &k + jbk(a) Us e th e conjugat e symmetri c propert y o f Sk t o writ e x n i n term s of ak an d bk, an d sho w t h a t x^ ca n b e compute d usin g onl y rea l additions an d rea l multiplications . (b) I t i s know n t h a t a typ e I D C T (discret e cosin e transform ) matri x [187] C / o f dimension s (K + 1 ) x (K + 1 ) i s give n b y [C/]nm =
« n a m W — CO S (j7nm) >
where an =
<
(lA/2, n
0 < k,17l< K,
= 0,K,
11, otherwise . A typ e I D S T (discret e sin e transform ) matri x S j o f dimension s (K — 1) x (K — 1) i s give n b y [Si]nm =
J — s i n ( - | : ( r a + l ) ( n + l ) ) , 0<m,n
Use (a ) t o sho w t h a t x^ ca n b e compute d usin g C j an d S j matrices , which ar e know n t o hav e fas t algorithms . If w e ar e t o comput e x n usin g th e I D F T formul a directly , comple x ad ditions an d comple x multiplication s ar e required . Exploitin g th e conju gate symmetri c propert y o f th e inputs , w e ca n us e D C T an d D S T wit h only rea l computations . 6.20 Fo r th e D M T system , th e powe r spectru m o f th e t r a n s m i t t e d discret e time signa l x(n) i s a s give n i n (6.35) . W h e n FQ(Z) i s th e rectangu lar windo w an d al l th e othe r transmittin g filter s ar e frequency-shifte d versions o f FQ(Z), w e ca n approximat e S x(e^2k7r^M) b y
Sx(e^lM)K£-f\F»{ei°)\\
6.9. Problem s
187
where k G A. Us e thi s expressio n t o obtai n a n approximatio n fo r SXa(jk-^pj;). Suppos e i t i s require d t h a t S Xa{jkj^) < A k. Fin d th e m a x i m u m allowabl e valu e £ s,k6.21 I n (6.39 ) w e hav e use d roundin g t o obtai n intege r bi t assignment s fo r a give n se t o f subchanne l SNR k. T h e resultin g subchanne l erro r rate s will b e slightl y smalle r o r highe r t h a n th e targe t erro r rate . (a) Sho w t h a t w e ca n maintai n th e sam e erro r rat e b y adjustin g th e kth symbo l powe r b y a facto r o f T(2Sfc - l)/SNR= k
(2
hk
- l)/(2
6fc
- 1) .
(b) Sho w t h a t th e maximu m amoun t o f powe r compensatio n i s aroun d ± 1. 5 d B . 6.22 I n th e D M T syste m show n i n Fig . 6.22 , a tim e domai n equalize r ( T E Q ) is include d t o shorte n th e channel . Thi s affect s th e subchanne l signa l to nois e ratios . Assum e th e T E Q ha s shortene d th e channe l s o t h a t IBI i s negligible . Denot e th e M-poin t D F T coefficient s o f T(z) b y T kl /c = 0 , l , . . . , M - l . (a) Fin d th e subchanne l gains . (b) Fin d a n expressio n fo r subchanne l nois e variances i n terms o f T(e^) and receivin g filter s H^e^). (c) Suppos e th e T E Q ha s on e D F T coefficien t equa l t o zero , sa y T ko = 0. Sho w t h a t th e ko subchanne l nois e variance i s not zer o i n general . This demonstrate s t h a t includin g a T E Q wil l chang e th e subchanne l SNRs. 6.23 Zipper-avoidance of NEXT crosstalk [139] . Conside r a D M T syste m with transmitte r outpu t signa l x(n). T h e channe l c(n) ha s orde r L. Due t o N E X T crosstalk , th e receive d signa l i s r(n) =
x(n - A ) * c(n) + x'{n) * c
next(n),
where A i s th e propagatio n delay . T h e signa l x'{n) i s th e t r a n s m i t t e d signal o f anothe r D M T syste m t h a t come s int o th e receive d signa l be cause o f coupling , an d c next(n) i s th e impuls e respons e o f th e crosstal k transfer functio n (orde r les s t h a n L). Suppos e th e cyclic-prefi x lengt h v satisfies v > L + A . T h e receive r retain s th e followin g M samples : r = [r(v) r(z
/ + l ) ••
• r(N
- l)]
T
and applie s D F T o n r t o obtai n y = W r . (a) Sho w t h a t r depend s o n sample s o f x(n) t h a t ar e fro m th e sam e transmitter inpu t bloc k an d o n sample s o f x'(n) t h a t ar e als o fro m the sam e transmitte r inpu t block . Thi s mean s t h a t w e ca n conside r one-shot transmissio n i n thi s case .
6. DFT-base d transceiver s
188
(b) Le t th e inpu t symbol s o f th e tw o D M T transmitter s be , respec tively, Sk an d s k. Ho w i s yk relate d t o Sk an d s k, wher e yk i s th e kth elemen t o f y ? (c) W h e n F D M i s used , th e downstrea m an d upstrea m signal s us e different tones . Th e tone s ar e divide d int o tw o sets , A an d A!. T h e tone s i n A ar e use d b y x(n) an d th e tone s i n A' ar e use d b y x'(n). Ho w i s yk relate d t o Sk an d s k, fo r k G A an d fo r k G A'! (d) Suppos e a combinatio n o f cycli c prefi x an d suffi x (Proble m 5.19 ) is use d a t th e transmitter , i.e . a suffi x o f lengt h A an d a prefi x o f length (y — A ) . Ho w i s yk relate d t o Sk an d s k now ? Suc h a D M T system i s know n a s a zipper. In (c ) an d (d ) th e N E X T crosstal k ca n b e completel y avoide d b y usin g extra redundan t sample s (extr a A samples ) an d b y carefull y synchro nizing th e downstrea m an d upstrea m signals . 6.24 Oversampled OFDM system [150] . I n Sectio n 5.5 , w e considere d over sampling receivers . Figur e P6.24(a ) show s a n O F D M syste m wit h a n oversampling receiver . Th e transmitte r send s ou t a sampl e ever y T sec onds, wherea s th e C / D o f th e receive r take s sample s ever y T/Q seconds , where Q ( a positiv e integer ) i s th e oversamplin g factor . A t th e receiver , the numbe r o f prefi x sample s discarde d no w i s Qu an d th e D F T siz e is MQ du e t o oversampling . Fro m Sectio n 5.5 , w e kno w th e syste m from x(n) t o r(n) ca n b e replace d b y a discrete-tim e equivalen t channe l model, a s show n i n Fig . P6.24(b) . Th e discrete-tim e channe l i n thi s cas e is c(n) = (p i * c a*p = 2)(t)\t nT/Q, where p\(t) i s th e transmittin g puls e an d P2(t) i s th e receivin g pulse . Assume c{n) ha s orde r < Qu. Th e equivalen t bloc k diagra m o f th e oversampled O F D M syste m i s show n i n Fig . P6.24(c) . (a) Sho w t h a t th e transmitte r i n Fig . P6.24(c ) ha s th e alternativ e representation give n i n Fig . P6.24(d) . Th e input s ak show n i n Fig. P6.24(c ) ar e relate d t o Sk b y ak =
~7D
S((/C)) M k
'
=
° ' *' * * *' M ® ~ 1'
where th e ( ( - ) ) M denote s modul o M operation . T h a t is , th e input s of th e ne w transmitte r ar e th e origina l input s repeate d Q times . These tw o transmitter s produc e th e sam e outpu t x e(n). (b) Sho w t h a t , i n th e absenc e o f channe l noise , th e receive r o u t p u t s yk are give n b y Vk = C kak =
-j= C kS((k))M i
A
: = 0 , 1 , . . . , MQ - 1 ,
where Ck ar e th e M Q - p o i n t D F T o f c(n). Therefor e th e o u t p u t s VkiVk+M, • • • >2/fc+(Q-i) M a r e an " from th e sam e transmitte r inpu t Sk, bu t scale d b y differen t D F T coefficient s o f th e channel .
6.9. Problem s
189
(c) Suppos e th e channe l nois e q(n) i s AWG N wit h varianc e A/o - Give n 2/fe, 2/fe+M, • • • , 2/fe+(Q-i)Mj n n d t n e optima l estimat e o f s fe, fo r 0 < k < M. W h a t i s th e kth subchanne l SN R usin g th e optima l es t i m a t e o f s/c ? Compar e wit h th e cas e withou t oversamphng , i.e . Q = 1 , an d determin e th e gai n o f oversamphn g i n term s o f sub channel SNRs .
190
6. DFT-base d transceiver s
yMQ-i
\q(n)
x(n)
te
c(n)
-*•-
r(n)
(b)
q(n)
*0 vX (vi^ n
cyclic
— ► s l
—> prefix -+ (v)
• •
tc
e\ ;
—►
— ► SM-l
c(n)
^0
discard prefix (0v)
a
•• • <*MQ-1
1
-6 o 3'
3
— t
l
•• •
Tt
o
•• •
yMQ-i
(c)
u
-+
P/S (M0
—►
cyclic prefix
X g («)
(d)
Figure P 6 . 2 4 . (a ) Oversample d OFD M system ; (b ) discrete-tim e equivalen t channel ; (c) equivalen t bloc k diagra m o f (a) ; (d ) alternativ e representatio n o f th e transmitte r i n (c).
191
6.9. Problem s
6.25 Th e OFD M syste m ca n als o be use d i n dat a transmissio n fo r mor e tha n one user. I n thi s case , it i s known a s OFDM-FDM A (frequenc y divisio n multiple access ) o r OFDMA . Eac h use r i s allocate d a subse t o f th e M subchannels. A s an example , suppos e there ar e K user s and eac h user i s assigned P = M/K subchannel s fo r transmission , assumin g K divide s M. Fo r th e fcth user , subchannel s fcP, kP + 1 , . . . , kP + P — 1 are use d and th e channe l fro m th e kth use r t o th e bas e statio n i s Cfc(z) , wit h order les s than th e lengt h o f cyclic prefix, a s shown i n Fig. P6.25. Wha t the receive r ha s i s th e su m o f signal s fro m al l th e user s plu s channe l noise. (a) Determin e th e output s o f the receive r fo r th e first P subchannels . (b) Ho w ca n w e recove r th e dat a o f th e kth use r i n th e absenc e o f channel noise ? zeroth user ftp' kP+1
• •
kP+P-\
M-pt IDFTW
—>> • • •
x An) * cyclic P/S(M) —> prefix -> cm 1(») J
\
u
r{n)
y* / /
(A-l)thuser/
transmitter o f th e Ath user
receiver o f th e bas e station
Figure P 6 . 2 5 . OFDM A system .
192 6
. DFT-base d transceiver s
7 Precoded OFD M system s In practica l communicatio n systems , th e transmitte r usuall y send s ou t train ing symbol s t o th e receiver , base d o n whic h th e channe l ca n b e estimate d a t the receiver . I t i s therefor e reasonabl e t o assum e t h a t th e channe l i s know n t o the receiver . W h e n th e channe l stat e informatio n i s als o know n t o th e trans mitter, w e ca n optimiz e th e transmitte r t o bette r th e syste m performance . Having thi s knowledg e availabl e t o th e transmitte r require s th e receive r t o send bac k th e information , whic h take s time . Fo r wireles s transmission , th e channel varie s rapidly . B y th e tim e th e transmitte r receive s th e channe l pro file, th e channe l ma y hav e changed . Therefore , fo r wireles s application s i t i s often desirabl e t o hav e a transmitte r t h a t i s channel-independent . I n suc h a channel-independen t transmitter , ther e i s n o bit/powe r allocation . Havin g a channel-independen t transmitte r i s als o o f vita l importanc e fo r broadcast ing applications , wher e ther e ar e man y receiver s wit h differen t transmissio n paths. T h e O F D M syste m ha s th e muc h desire d featur e t h a t th e transmitte r is channel-independen t an d furthermor e th e channel-dependen t par t o f th e transceiver i s onl y a se t o f M scalar s a t th e receiver . Moreover , th e mai n processing a t th e transmitte r (receiver ) i s M-poin t I D F T ( D F T ) , whic h ca n be implemente d efficientl y usin g fas t algorithms , an d th e complexit y i s i n th e order o f M log 2 M instea d o f M 2 . T h e discussio n i n Chapte r 6 suggest s t h a t th e O F D M syste m ca n b e severely affecte d b y channe l spectra l nulls . Fo r hig h SN R th e erro r rat e i s usually limite d b y thos e subchannel s t h a t hav e lo w SNRs . On e metho d t h a t prevents th e performanc e bein g dominate d b y a fe w ba d subchannel s i s t o have a precoder . Figur e 7. 1 show s a n O F D M transmissio n syste m wit h a precoder a t th e transmitte r an d a post-code r a t th e receiver . W e hav e see n earlier t h a t whe n th e precode r i s th e D F T matrix , th e precode r D F T ma trix wil l cance l ou t th e I D F T a t th e transmitte r an d th e precode d syste m becomes th e SC-C P syste m show n i n Fig . 6.10 . T h e SC-C P syste m illustrate s t h a t havin g a precode r ca n alte r th e B E R behavio r o f th e system , althoug h the tota l outpu t mea n square d erro r i s unchange d i n thi s case . I n a conven tional single-ban d transmissio n system , B E R i s directl y tie d t o mea n square d error. Fo r multi-subchanne l system s lik e O F D M , SC-CP , o r th e mor e genera l precoded O F D M systems , thi s i s n o longe r true . Transceiver s wit h th e sam e mean square d erro r ca n hav e ver y differen t performance s o f average B E R (ove r all M subchannels) . 193
194
7. Precode d O F D M system s
—J
—►
•• •
— ►
precoder
•• •
— ►
—J OFDM transmission syste m
•• •
M
—►
post-coder
•• •
V\
Figure 7 . 1 . Precode d OFD M system .
In thi s chapter , w e wil l conside r optima l unitar y precoder s i n O F D M sys tems fo r minimu m BER . Th e objectiv e wil l be BER , rathe r t h a n mea n square d error. T o maintai n th e channe l independenc e propert y a t th e transmitter , there i s n o bit/powe r allocation . Transceive r design s wit h bi t an d powe r al location ar e considere d i n Chapte r 8 . Th e presentatio n o f thi s chapte r i s mostly base d o n [78] . Se e Sectio n 7. 6 fo r othe r precode d system s an d a further reading .
7-1 Zero-forcin
g precode d O F D M system s
Let u s choos e th e precode r t o b e a n M x M unitar y matri x P wit h P ^ P = I M and correspondingl y th e post-code r a t th e receive r P+ . Th e resultin g precode d O F D M syste m become s th e on e show n i n Fig . 7.2 . Fo r th e O F D M transmis sion syste m alone , w e kno w fro m th e previou s chapte r t h a t th e overal l trans fer matri x i s th e identit y matri x I M - A S the precode r an d post-code r satisf y P t P = I M , th e precode d syste m remain s zero-forcing . Th e specia l cas e P = I corresponds t o th e conventiona l O F D M system , whil e P = W correspond s to th e SC-C P system . Unles s i t i s mentione d otherwise , a n O F D M syste m i n this chapte r refer s t o th e cyclic-prefixin g case . T h e ne w transmittin g an d receivin g matrice s G an d S (show n i n Fig . 7.2 ) are, respectively ,
G = wtP, s = Pt r 1 w, where T = dia g [C o C\ • • • C M - I ] an d Ck ar e th e D F T coefficient s o f the channe l impuls e response . Th e resultin g transmittin g matri x i s a genera l unitary matrix . A s i n th e previou s chapter , w e assum e t h a t th e transmitte r input symbol s Sk ar e uncorrelate d an d o f th e sam e varianc e £ s. Du e t o th e unitary propert y o f G , th e o u t p u t s o f th e transmittin g matri x als o hav e vari ance £ s. S o th e transmissio n powe r £ s i s th e sam e fo r al l unitar y precoders . As G i s a n arbitrar y unitar y matrix , th e proble m o f finding th e optima l pre coder i s equivalen t t o on e o f designin g a n optima l bloc k transceive r wit h a unitary transmittin g matrix . Le t u s first analyz e th e effec t o f th e precode r o n the subchanne l SNRs . Optima l precoder s t h a t minimiz e th e erro r rat e wil l b e derived i n subsequen t sections .
195
7 . 1 . Zero-forcin g precode d O F D M system s
\q(n) W'
P/S \{M)
cyclic prefix
C(z)
->#receiving matrix S
SM-:
A
/ f\
1
1
1
/
I/O.
transmitting matri x G W
1/C
i/c u
Figure 7.2 . Zero-forcin g precode d OFD M syste m wit h a precoder P an d post-code r pt.
%
q(n)
discard —► S/P(M) prefix
T
>
*• > • • • <1M-\
°„
1/C0
% %
w X M-l
•• •
P+
• •
M-M-l
i/cM_,
eM-i
Figure 7.3 . Illustratio n o f th e nois e pat h o f a zero-forcin g receiver .
Noise analysi s Assume th e channe l nois e q(n) i s comple x circularl y symmetri c AWG N wit h zero-mean an d varianc e A/o - Denot e th e receive r outpu t vecto r b y s a s in dicated i n Fig . 7.2 . T h e n th e outpu t erro r vecto r e = s — § come s entirel y from th e channe l nois e a s th e receive r i s zero-forcing . T h e erro r vecto r e ca n be analyze d b y considerin g th e receive r bloc k diagra m i n Fig . 7.3 , wher e onl y the channe l nois e i s shown . T h e vecto r q i s th e blocke d channe l noise . It s
7. Precode d OFD M system s
196
elements ar e uncorrelate d Gaussia n rando m variable s wit h varianc e A/o - T h e elements o f r = W q continu e t o b e uncorrelate d Gaussia n rando m variable s with th e sam e variance , du e t o th e unitar y propert y o f W . Therefore , th e noise component s jik a s indicate d i n Fig . 7. 3 hav e varianc e a 1= Afo/\Ck\ 2. T h e outpu t subchanne l erro r e ^ i s relate d t o jik b y M-l k=0
where pk,i denote s th e (/c,i)t h elemen t o f P . Th e subchanne l error s ar e als o zero mea n Gaussian , bu t ar e correlate d i n general . A s jik ar e uncorrelated , the i t h subchanne l erro r varianc e i s give n b y M - l
G
X = Yl IPMI
2 7 ,2
* ,Mfc*
fc=0
T h a t is ,
M-l |
al
|
2
Kl
k=0 '
T h e precode r P i s unitar y an d thu s eac h o f it s row s ha s uni t energy . T h a t is , J2i=o \Pk,i\ 2 = 1 , fo r A : = 0 , 1 , . . . , M — 1 . Usin g thi s fact , w e ca n writ e th e average mea n square d erro r M-l
as
^■§Ew- <7 Kr
M-l
k=0 '
fc|
-2)
It i s th e sam e a s th e averag e erro r o f th e O F D M syste m i n (6.8) . Moreover , the averag e erro r i s independen t o f th e precoder . Al l zero-forcin g precode d O F D M transceiver s wit h a unitar y precode r hav e th e sam e averag e erro r give n in (7.2) . Not e t h a t whe n on e D F T coefficien t o f the channe l i s zero, th e averag e error wil l g o t o infinity , a phenomeno n t h a t w e hav e observe d i n th e previou s chapter. Le t S s b e th e powe r o f th e transmitte r input s (^[|s/e| 2 ] = £ s) an d le t /3(i) = £ s/&e- b e th e SN R o f th e i t h subchannel , the n w e hav e
/^)= M J | ^ 7 Z^k=0 |C
P
fc |2
wher
e
^ = ^ / A / ' o . (7.3
T h e subchanne l SNR s depen d o n th e precode r P . Althoug h th e mea squared erro r i s th e sam e regardles s o f P , th e precode r affect s ho w th e sam amount o f nois e i s distribute d amon g th e subchannels . Fo r convenience , w repeat th e result s o f th e O F D M an d SC-C P system s fro m Chapte r 6 i n th following.
) n e e e
• Th e OFD M syste m W h e n P = I M , th e precode d syste m reduce s t o th e conventional O F D M system . Th e i t h subchanne l ha s erro r varianc e G ei,ofdm
■Mo/\Ci\2. (7.4
)
7.1. Zero-forcin g precode d OFD M system s
197
T h e correspondin g subchanne l SN R i s Pofdmii)=l\Cl\2. (7.5
)
For th e zt h subchannel , b o t h th e erro r varianc e an d SN R depen d onl y on th e zt h subchanne l gai n d. The SC-C P syste m W h e n th e unitar y precode r P i s th e normalize d D F T matri x W , th e transmittin g matri x become s G = 1M- T h e post coder p t i s th e I D F T matri x W ^ . T h e resultin g precode d syste m be comes th e SC-C P syste m i n Fig . 6.10 . Al l th e subchannel s hav e th e same erro r varianc e an d th e sam e SNR , ^sc-cp &rri
Psc-cp
C
SJ
G
rr.
Bounds o n subchanne l error s T h e B E R performanc e i s determine d b y the subchanne l SNRs . T h e unitar y propert y o f P allow s u s t o establis h uppe r and lowe r bound s o n th e subchanne l erro r variances . Firs t w e observ e on e interesting propert y o f the erro r variance s t h a t tie s th e precode d syste m nicel y to th e O F D M system . W i t h a n inspectio n o f the erro r variance s give n i n (7.1) , we ca n se e t h a t the y ar e relate d t o thos e o f th e O F D M syste m i n (7.4 ) b y M-l
<=
E IPWI Ve ,ofdm' k
k=0
Each i s a linea r combinatio n o f subchanne l erro r variance s o f th e O F D M system. An d th e linea r combinatio n coefficient s ar e |p/c,i| 2, whic h ad d u p t o unity al l fo r z . T h e linea r combinatio n propert y mean s ™^
) 0 / ( i m
< a\. < max
T h e subchanne l erro r variance s o f th e precode d syste m ar e alway s bounde d between thos e o f th e bes t an d wors t subchannel s o f th e O F D M system . T h e upper an d lowe r bound s hol d fo r an y unitar y precode r P . Fo r differen t choic e of P , th e nois e variance s ar e distribute d differently . Best an d wors t cas e SNR s T h e bound s o n o bounds o n th e subchanne l SNR s /3(i): mm(3ofdrn(k) <
2
ei
lea d t o th e followin g
f3(i) < m a x / 3 o / d m ( / c ) , (7.6
kk
)
where /3 0fdm{k)= 7|C&| 2 i s th e kth subchanne l SN R o f th e O F D M system . For an y unitar y precoder , th e SN R o f the bes t subchanne l i s no bette r t h a n th e best subchanne l o f th e O F D M syste m an d th e SN R o f th e wors t subchanne l is n o wors e t h a n th e wors t subchanne l o f th e O F D M system . T h e invers e o f th e subchanne l SNR , l//5(z) , i s th e nois e t o signa l rati o (NSR) o f th e zt h subchannel . T h e subchanne l NSR s hav e th e sam e linea r combination propert y t h a t hold s fo r th e subchanne l erro r variances , i.e . 1
J_ =
M-
l,
|
V - \Pk,i\
2
=
M-
V I
l1
nI
21
7. Precode d OFD M system s
198
T h e subchanne l NS R l//3(i) i s a linea r combinatio n o f thos e o f th e O F D M system. Th e NS R i s a quantit y t h a t rarel y arise s i n th e discussio n o f com munication systems , bu t w e wil l find i t mor e convenien t t o wor k wit h NS R instead o f SN R i n som e discussion s o f thi s chapter .
7-2 Optima
l precoder s fo r QPS K modulatio n
In thi s section , w e optimiz e th e precode r P t o minimiz e B E R fo r th e zero forcing precode d O F D M syste m give n i n Sectio n 7.1 . Th e computatio n o f error rat e depend s o n th e modulatio n schem e used . W e wil l conside r Q P S K modulation i n thi s section . Th e result s wil l b e extende d t o QA M o f large r constellations an d als o t o othe r modulatio n scheme s i n Sectio n 7.3 . Fo r Q P S K modulation wit h symbo l powe r £ s , Sk = ± \ / £ s / 2 ± jy/£ s/^" Th e B E R wa s derived i n Sectio n 2.3 . Th e B E R o f th e i t h subchanne l i s give n b y V{i) = Q(\fMt))i wher e th e functio n Q(- ) i s a s define d i n (2.11) . Th e averag e B E R
M-l
^ = M E 0 ( V W ) . (7.7 i=0
)
T h e choic e o f precode r determine s th e subchanne l SNR s an d henc e th e error rate . W e woul d lik e t o find th e optima l P suc h t h a t th e averag e B E R i s minimized. Fo r convenience , w e introduc e th e functio n f(y) =
Q(i/y/y), y>o.
(7.8
)
We hav e th e followin g alternativ e expressio n fo r th e subchanne l BERs : (7.9) T h e argumen t o f th e functio n / ( • ) i n th e abov e equatio n i s l//3(i), i.e . th e i t h subchannel NSR . No w w e rewrit e th e subchanne l erro r rate s o f th e O F D M and SC-C P systems , respectively , a s
Vofdm(l) - f
(
7
|Q|2 )
' ^ocp
-
f
(
M
2
^!
T h e bi t erro r rat e performanc e i s directl y relate d t o th e behavio r o f th e func tion f(y). Th e followin g lemm a give s som e importan t propertie s o f f(y). Lemma 7. 1 Th e functio n f{y) = Q(l/y/y) i s monoton e increasing . I t i s convex 1 whe n y < 1/ 3 an d concav e whe n y > 1/3 . ■ Proof. Le t u(y) = 1/^/y fo r y > 0 , the n f(y) = Q(u(y)). Th e lemm a can b e prove d b y computin g th e first an d secon d derivativ e o f f(y). See Appendi x A fo r definition s o f conve x an d concav e functions .
7.2. Optima l precoder s fo r QPS K modulatio n T h e functio n u(y) = tive u'(y) =
199
1/y/y fo r y > 0 i s conve x wit h th e firs t deriva -
— \y~?>^2 an d th e secon d derivativ e u"(y) =
— —|=e_ a ; ^ 2
function Q(x) fo r x > 0 i s als o conve x wit h Q'(x) = ) ? "( x=
^
2TT e
2
\y~ hl2''. T h e an
d
/2
n We hav e /'(?/ ) = Q'(u(y))u'(y) equa l t o /'(?/ ) = ^; e~^y~3/2, w h i c is positiv e fo r y > 0 . Thi s mean s f(y) i s monoton e increasing . W e ca n verify t h a t f /f(y)= Q n\u(y))[u'(y)]2 + Q f(u(y))u/f(y) ca n b e expresse d as
ro/)
4V2TT
e-*7/-7/2(l-37/).
Therefore, / ^ ( y ) > 0 fo r y < 1/ 3 an d f\y) < proves th e lemma . ■
0 fo r y > 1/3 , whic h
/
— f(y ) O (1/3,f(1/3) ) 0
0.2
0.4
0.6
Figure 7.4 . Plo t o f f(y) =
0.8
Q(l/y/y).
A plo t o f f(y) i s shown i n Fig . 7.4 . Eac h subchanne l i s operating i n the conve x or th e concav e regio n o f th e functio n /(• ) dependin g o n th e subchanne l SN R /3(i). I n particula r th e cas e /3(i) > 3 (NS R l//3(i) < 1/3 ) correspond s t o th e convex regio n o f / ( • ) . T h e cas e /3(i) < 3 correspond s t o th e concav e region . Convex an d concav e region s o f /(• ) Fo r a sufficientl y larg e 7 , al l th e subchannels i n th e O F D M syste m wil l b e operatin g i n th e conve x regio n o f / ( • ) . T h a t is , f3 0fdm(i) = 7 | C i | 2 > 3 fo r al l i. Thi s require s 3A
7 > max— — = 71 .
(7.10)
7. Precode d OFD M system s
200
W h e n thi s happen s al l t he subchannels i n t he precoded syste m wil l als o be operating i n t he convex regio n du e t h e SNR bound s i n (7.6) . I n a simila r manner, fo r a sufficiently smal l 7 , al l the subchannel s i n the O F D M syste m wil l be operatin g i n the concav e regio n o f / ( • ) . Thi s i s true i f f3 0fdm{i) = 7|Ci| 2 < 3 for al l i , which require s 7 < m i n - — ^ = 7 0 . (7.11
)
For convenience , w e defin e thre e SN R regions : Klow = {7I7 < 70 } , Timid
= {7|7 0 < 7 < 7 l } J Tlhigh
= {7|7 > 7 l } •
We ar e no w read y t o establish a more precis e connectio n amon g th e B E R performances o f the thre e systems , O F D M , SC-CP , a n d a precode d O F D M system fo r the thre e SN R regions . Le t us first conside r th e cas e Tlhigh- Fo r this SN R rang e w e ca n us e th e convexit y o f / ( •) (Appendi x A ) to obtain an upper boun d fo r the subchanne l B E R s give n i n (7.9) : /M-l -
|
2
l,
\ M-
1
Therefore th e averag e erro r rat e satisfie s M-l M- l
2 ^4EEW / M U t, ' V7|C
fe|
Using th e uni t energ y propert y o f the row s o f P, we hav e 1
M _ 1
/
1
\
^BS'ISF)-^(
"2)
We ca n als o appl y th e convexit y o f / ( •) t o obtai n , M- l / M - l11
. M- l M- l1 1
2\/
2\
V0« T h e inequalit y become s a n equality i f and onl y i f the quantit y 5^/e= o \c I 2 •> which i s the subchanne l NS R l//3(i), i s the sam e fo r all i\ or equivalently all the subchanne l nois e variance s ar e t he same. Again , usin g t h e unit energ y property o f the row s o f P, we hav e /1
M _1
?s/ 5
1
\
( Sw)-^
,713
)
Combining th e tw o inequalitie s i n (7.12) a n d (7.13), w e obtai n V 0fdm >V> Vsc-cp for 7 £ TZhigh- Similarly , whe n 7 G Tliow, w e ca n us e th e concavit y of / ( • ) t o show t h a t V 0fdm <*P< V sc.cv. Summarizing , w e hav e th e followin g theorem.
7.2. Optima l precoder s fo r QPSK modulatio n
201
Theorem 7. 1 Zero-forcin g case . T h e average B E R V o f the precode d O F D M system i n Fig. 7.2 with Q P S K modulatio n satisfie s Vofdm < V < Vsc-cp, for 7
G TZiow,
I of dm i— > ^ _ / sc-cpi
for 7 G ftfcipfc. Each o f t he two inequalities relatin g V an d Vsc-Cp become s a n equalit y i f a nd only i f t h e subchanne l erro r variance s o f t he precode d syste m ar e t he same , i.e. a\. = £ r r , wher e £ rr i s as give n i n (7.2) . ■ For t h e lo w SNR region 7Zi ow1 P = I M is t h e optima l solution ; t h e O F D M system ha s t he smallest erro r rate . B u t for 7 G TZhigh, it i s t he worst solutio n and it s error rat e i s t he largest. However , a s we will se e next, t h e SNR region IZiow correspond s t o a hig h erro r rate , whil e IZhigh correspond s t o a mor e useful rang e o f B E R.
BER behavio r i n 7Zi ow, 7Z mid, an d Hthigh • 7 £ Hiow'- Fo r t he O F DM system , al l t he subchannels hav e f3 0fdm(k) < 3 and henc e
of dm > Q{V3) =
0.0416 , fo
G 1Z lo
r7
In thi s rang e o f SNR t he error rat e V 0fdm i s at leas t 0.0416 , a relativel y large BER . T h e error rat e o f t he precoded syste m i s lower bounde d b y t h a t o f t he O F DM system . S o t he B E Rs o f precoded systems , includin g the SC-C P system , wil l b e eve n higher . • 7 G IZhigh'- Fo r this range , al l t he subchannel s i n t h e precode d syste m have f3(i) > 3 an d subchanne l erro r rat e les s t h a n Q ( \ / 3 ) ; t h e averag e error rat e V < 0.0416 . Not e t h a t whe n 7 = 71 , t he worst subchanne l o f the O F D M syste m ha s an erro r rat e Q ( \ / 3 ) = 0.0416 , an d t he averag e B E R wil l b e at leas t Q(y/3)/M. I n other words , 7 1 is t he minimum SNR for t h e O F DM syste m t o achieve a n error rat e lowe r t h a n Q(\/3)/M. A s an example , whe n 7 = 7 ! , t he error rat e i s at leas t Q ( \ / 3 ) / 1 6 = 0.002 6 for M = 1 6 and at leas t Q ( V 3 ) / 6 4 = 6. 4 x 10~ 4 fo r M = 64 . Therefor e the SN R region IZhigh correspond s t o a mor e usefu l rang e o f B E R. • 7 G Timid'- I f we plot Vsc-cp an d V 0fdm a s function s o f 7, t h e curve s o f Vofdm an d V Sc-cp wil l cros s eac h othe r i n thi s range . Thi s i s becaus e Vofdm i s smaller t h a n V sc-Cp for 7 < 7 o and larger t h a n V sc-Cp for 7 > 7 1 •
Error-equalizing precoder s T h e abov e theore m state s t h a t t h e error rat e o f a precoded O F D M i s t he sam e as t h a t o f t he SC-CP syste m i f and only i f t he subchannel nois e variance s o2e. are equalize d t o t h e averag e valu e £ rr. T h a t is , 2K
M—l 1
r ST P
.
fc
9K
»* f \Ck\2 "
= k=0 ' K | k 0
M M
r
M—l
.
) V^1 ^ \C '
2
k\
K |
'
202
7. Precode d OFD M system s
T h e abov e error-equalizin g propert y ca n be achieve d usin g a unitar y precode r t h a t satisfie s \Pm,n\ =
- i,0 VM
< m,7 l < M - 1 . (7.14
)
In thi s cas e al l t h e subchanne l B E R s ar e t h e same , V(i) = V = V sc-cpTwo well-know n unitar y matrice s satisfyin g t h e equa l magnitud e propert y i n (7.14) ar e t h e D F T matri x an d t h e H a d a m a r d matri x [2] . W h e n P = W , the transmittin g matri x G = I M , an d t he transceive r i n Fig . 7.2 become s t h e SC-CP system . T h e H a d a m a r d matrice s ca n b e generate d recursivel y whe n M i s a powe r o f 2 . T h e 2 x 2 H a d a m a r d matri x i s give n b y
*
-
*
11 1 -
1
T h e 2n x 2n H a d a m a r d matri x ca n be give n i n term s o f t he n x n H a d a m a r d matrix b y
H2
""7i
H„, H
^
Hn—
Hri
Except fo r a scalar , t h e H a d a m a r d matri x i s rea l wit h element s equa l t o ± 1 . T h e resultin g transmittin g matri x G = W ^ H wil l b e complex . T h e implementation o f H a d a m a r d matrice s require s onl y additions . T h e overal l complexity o f t h e transceive r wil l b e mor e t h a n t h a t o f t h e O F D M syste m (or t h e SC-C P system ) du e t o t h e tw o extr a H a d a m a r d matrices , on e a t the transmitte r an d on e a t t h e receiver . W i t h t h e H a d a m a r d precoder , t h e complexity o f t h e transmitte r i s highe r compare d t o t h e O F D M o r SC-C P systems, bu t t h e complexit y o f t h e receive r i s betwee n thos e o f t h e O F D M and SC-C P systems . Havin g a H a d a m a r d precode r als o ha s t h e advantag e of havin g a P A P R smalle r t h a n t h a t o f t h e O F D M syste m [112] , thoug h larger t h a n t h a t o f t h e SC-C P system . W h e n w e hav e a unitar y precode r t h a t ha s t h e equa l magnitud e propert y i n (7.14) , w e ca n us e i t t o generat e other unitar y matrice s satisfyin g t h e equa l magnitud e property . A n exampl e is give n i n Proble m 7.8 . Bot h t h e D F T an d H a d a m a r d matrice s ar e channel independent, s o t he resultin g transmitte r i s als o channel-independent .
7-3 Optima
l precoders : othe r modulation s
T h e computatio n o f erro r rat e i n Sectio n 7. 2 i s carrie d ou t fo r Q P S K mod ulation. W e ca n exten d t h e result s t o PAM , QAM , an d P S K (phas e shif t keying) wit h sligh t modifications . Le t u s tak e Q A M modulation a s a n exam ple. Suppos e t h e input s ar e 26-bi t Q A M symbol s wit h varianc e £ s. T h e zt h subchannel B E R V{i) ca n b e approximate d b y (Sectio n 2.3) ,
V{i) « aQ{^cW)) = af (jjL^J , (7.15 where a
= | M - ^ j
a n d C = 3 / ( 2 2 b - 1) . (7.16
) )
7.4. MMS E precode d OFD M system s
203
As ever y subchanne l carrie s t h e sam e numbe r o f bits , t h e averag e erro r rat e can als o b e obtaine d b y averagin g t h e subchanne l erro r rates ,
*-sS>w-s£>U))- (7
-17)
T h e subchanne l SNR s ar e independen t o f t h e modulatio n schem e use d an d they observ e t he same uppe r an d lower bound s give n i n (7.6) , min ^ f30fdm(k) < P{i) < maxf c Pofdm{k). W h e n 7 i s larg e enoug h suc h t h a t al l t he subchannel s are operatin g i n t h e conve x regio n o f / ( • ) , equalizin g t h e subchanne l nois e variances wil l minimiz e t h e approximate d erro r rat e give n i n (7.17) . T h e same error-equalizin g precoder s give n i n Sectio n 7. 2 wil l b e optima l i n thi s case too. On e sufficient conditio n fo r this i s Cfiofdm{i) > 3 , which i s equivalen t to 7 £ | C i | 2 > 3 , for al l i. Thi s mean s (2 2 6 - 1 ) 7 > 71 , wher e 7 1 = m a x . (7.18 z k \Ck\
)
On t h e othe r hand , whe n Cfiofdm{i) < 3 fo r al l z , t h e conventiona l O F D M system i s t h e optima l transceiver . T h e conditio n fo r thi s i s 7 < 70 , wher
(2 2 6 - 1 ) e7 0= mm. k \<^k\
T h e condition s no w depen d o n t h e Q A M constellation . Fo r a large r constel lation, i.e . large r 6 , b o t h 7 0 an d 7 1 als o becom e larger . A s t h e constellatio n size increases , t h e SC-C P syste m wil l becom e optima l a t a highe r SNR . T h e abov e derivatio n i s vali d fo r an y modulatio n schem e i n whic h t h e subchannel erro r probabilit y ca n b e eithe r approximate d o r expresse d a s
«Q(v/C^))=a/(^y), for som e constant s a an d £ t h a t ar e independen t o f subchannels . Example s of suc h a cas e includ e PAM , QAM , and P S K modulatio n schemes . Onc e t h e error probabilit y i s in suc h a form , w e can invok e t h e convexit y an d concavit y of f(y) t o obtai n t h e SN R range s fo r whic h t h e O F D M syste m o r t h e SC-C P system i s optimal. Simila r t o t he Q P S K case , w e can conclude t h a t t h e SC-C P system i s optima l fo r t h e hig h SN R cas e whil e t h e O F D M syste m i s optima l for t h e lo w SN R case .
7.4 M M S
E precode d O F D M system s
In thi s sectio n w e conside r t h e cas e wher e t h e receive r ha s minimu m mea n squared error . Simila r t o t h e SC-C P system , t h e precode d syste m benefit s considerably fro m MMS E reception . Als o simila r t o t h e SC-C P system , t h e precoded syste m wit h a n MMS E receive r i s mor e robus t t o channe l spectra l nulls whil e havin g t h e sam e implementatio n cos t a s t h e zero-forcin g receiver .
204
7. Precode d O F D M system s
7.4.1 MMS
E receiver s
Let u s star t wit h a receivin g matri x S t h a t ca n b e an y M x M matri x (Fig ure 7.5) . Le t ! 3 be th e receive r outpu t an d le t th e erro r vecto r e = s ^ — s . Fo r a give n receive d vecto r r , w e woul d lik e t o find th e optima l receivin g matri x S t h a t minimize s th e mea n square d erro r i£[e"!"e] .
/
/
/
/
q(n)
cyclic prefix
w SM-\
C(z)
| discard prefix
'
Figure 7.5 . Precode d OFD M syste m wit h a n MMS E receiver .
By th e orthogonalit y principle , th e optima l S i s suc h t h a t th e erro r i s orthogonal t o th e observatio n r , E [(S r — s)r"l" ] = 0 , o r equivalentl y S = i?[srt](£;[rrt])-
1
.
We kno w t h a t th e transfe r matri x fro m x t o r i s C c ^ r c , wher e C c ^ r c i s th e circulant matri x define d i n Definitio n 5.2 . S o r=
CWcW+P s + q = W + r P s + q ,
where w e hav e use d th e D F T decompositio n C circ = W ^ T W (Theore m 5.2) . Assume th e transmitte r input s ar e uncorrelated , i.e . i^ss^ ] = £ S I, an d un c o r r e c t e d wit h th e noise , i.e . ^[sq^ ] = 0 . Usin g th e abov e expressio n fo r r , we hav e E [sr 1"] =
£ s p t r f W an
dE
[IT* ] =
^ W + I T + W + A/" 0 I M .
By direc t substitution , w e ca n obtai n th e optima l receivin g matri x S t h a t minimizes i^e^e ] a s S=
P j A W , (7.19
)
where A i s a diagona l matri x wit h th e kth diagona l element sA & given b y Afc
l+7|C f e | 2 '
T h e optima l S ha s th e structur e o f a diagona l matri x sandwiche d betwee n P t an d W . T o obtai n th e MMS E receive r w e ca n simpl y replac e th e M channel-dependent scalar s 1/Ck i n th e zero-forcin g receive r b y A& .
7.4. M M S E precode d O F D M system s
-> H S/P (M)
discard prefix
W
-o H
-» H
ru-\
Figure 7.6 . MMS E receive r fo r th e precode d OFD M system .
W i t h th e MMS E receiver , th e overal l transfe r matri x T fro m th e inpu t s of th e transmitte r t o th e outpu t s o f th e receive r i s P f A r P . (7.20
T=
)
Let th e (z , j ) th elemen t o f T b e tij. W e ca n writ e th e zt h outpu t a s Si = UiSi + T i , (7-21
)
where Ti consist s o f IS I fro m othe r symbol s an d channe l nois e an d i s give n b y n=
^UjSj +
[Sq]i .
Comparing th e expressio n si = tuSi + r ^ wit h t h a t i n (3.13) , w e realiz e th e two hav e th e sam e form . Fro m th e discussio n i n Sectio n 3.2 , w e kno w t h a t the mea n square d erro r an d th e unbiase d SN R o f a n MMS E receive r are , respectively, give n b y (3.20 ) an d (3.18) . Therefor e w e ge t th e zt h subchanne l error varianc e an d SN R i n term s o f tu a s
<=
( 1 - Ui)£s, P(i)
= W ( l - Ui).
Note t h a t ei = (tu — l)si + TI ha s mea n E[ej\ = 0 , althoug h th e conditiona l mean ^[e^Si ] = (tu — l)si i s not zero . B y directl y multiplyin g ou t th e matrice s in (7.20) , w e ge t +--T\Pk-\2
7
| fc|
(7.22)
fe=0
As eac h ro w o f th e unitar y precode r P ha s uni t energy , th e subchanne l erro r variance ai. ha s th e for m M-l
Ss < = Y:^\\ +IW fe=0
(7.23)
7. Precode d OFD M system s
206
It follow s t h a t th e averag e mea n square d erro r S rr =
-^E[e^e] i s MJ
M-l
£,rr
-MM E j ^ -i + 7iai ' ^ 1 2
T h e averag e erro r i s agai n independen t o f th e precoder , lik e th e zero-forcin g case. Usin g th e expressio n o f tu i n (7.22) , w e ca n expres s /3(i) a s
^ £ ^ K S p - ] / i £ i + w j - ,7
-24)
Recall t h a t fo r th e zero-forcin g receiver , th e outpu t nois e ca n becom e infinitel y large i n th e presenc e o f a channe l spectra l null . I n th e MMS E case , eve n i f the channe l ha s som e D F T coefficient s equa l t o zero , th e subchanne l SNR s are no t zero . Th e syste m wil l b e mor e robus t t o channe l spectra l nulls . Recal l the followin g tw o specia l cases . • Th e OFD M syste m Th e precode r P is th e identit y matrix . Th e i t h subchannel erro r varianc e i s £
It i s smalle r t h a n th e erro r varianc e i n (7.4 ) o f a zero-forcin g receiver . However, th e subchanne l SN R compute d fro m Ss/a^. 0 fdrn wil l b e bi ased, ftofdm,biased(i) =
1
+ 7 ^1 •
From Lemm a 3.1 , w e kno w t h a t th e biase d an d unbiase d SNR s ar e related b y /3(i) = Pbiasedij) — 1- Th e unbiase d SN R o f the i t h subchanne l is therefor e 7 | C i | 2 , th e sam e a s i n th e zero-forcin g cas e (7.5) . • Th e SC-C P syste m Th e precode r P i s the normalize d D F T matri x wit h \pk,i\ = 1 / v M . I n thi s case , th e i t h subchanne l erro r varianc e become s
U-eusc-cp
° —
MM
1 ^ S ^ 7|Cfc|2> ^1 +1
which i s th e sam e a s w e ha d earlie r i n (6.23) . Subchannel erro r variance s I n th e zero-forcin g precode d system , th e subchannel erro r variance s ca n b e expresse d a s linea r combination s o f thos e in th e O F D M system . Her e w e ca n observ e a simila r propert y i n th e MMS E case. Th e expressio n i n (7.23 ) is , i n fact , a linea r combinatio n o f th e erro r variances i n (7.25) . Not e t h a t th e rati o cr^./S s i s th e biase d nois e t o signa l ratio. W e ge t 1
1
Pbiased(i) l + / 3
M-
i
E l ^ | 2 T — ^ ' (7.26 » ^ ' " ' ' 1+7IQ I
)
7.4. MMS E precode d OFD M system s
207
We observ e t h a t -5 — r ^ ca n b e expresse d a s th e sam e linea r combination s Pbiased \1)
of th e biase d subchanne l NSR s i n th e O F D M system , jus t lik e th e subchanne l error variances . W h e n a n MMS E receive r i s used , th e syste m i s no t ISI-free , an d th e out put error s d o no t com e fro m channe l nois e alone . W e ca n us e th e Gaussia n approximation t o comput e th e bi t erro r rat e a s i n Chapte r 6 . Suc h a n ap proximation allow s u s t o hav e a nice closed-for m expressio n o f B E R t h a t ca n be use d fo r finding th e optima l precode r later . T h e computatio n o f bi t erro r rate depend s o n th e modulatio n schem e used . W e loo k int o th e Q P S K cas e next. Othe r modulatio n scheme s ar e considere d i n Sectio n 7.4.3 .
7.4.2 Optima
l precoder s fo r QPS K modulatio n
For Q P S K modulation , th e bi t erro r rat e o f th e zt h subchanne l ha s th e simple expressio n Q(y//3(i)). I n Sectio n 7.2 , w e exploite d th e convexit y an d concavity o f th e functio n /(• ) t o deriv e th e optima l precoder . Unfortunatel y we canno t repea t th e sam e tric k her e becaus e th e unbiase d NS R l//3(z ) is no longe r a linear combinatio n o f l//3 0 /d m (i) ^ e th e zero-forcin g case . To simplify derivations , le t u s defin e a new functio n %) =
QWV'1 -
1) , for
0 < y < l . (7.27
)
T h e introductio n o f h(y) give s th e followin g erro r rat e expression : 1 \
,
/ 1
l + / 3 ( i ) y \Pbia8ed(i) Note th e argumen t o f th e functio n h(-) i s th e biase d zt h subchanne l NSR . Using th e expressio n o f biase d subchanne l NSR s i n (7.26) , th e averag e bi t error rat e i s give n b y M - l / M -l
Substituting P = 1M an d P = W into th e abov e equation , w e get , respec tively, th e B E R s o f MMS E O F D M an d SC-C P systems : / ofdm,mmse =
' of dm =
/ 1 M _ Vscc^mmse = h ^ — g
1
M_ 1
/
1
\
"7 T 2_^ ^ 1 1 i ry\r>.\2 ) 1
1 1 + 7
\ |^|
2
J '
(
' (7.29
)
7 30
'
)
To stud y th e behavio r ofVmm Se, le t u s first loo k int o som e propertie s o f th e function h(y). A plo t o f h(y) i s give n i n Fig . 7.7 . I t i s a monotone increasin g function define d fo r th e interva l 0 < y < 1 . Not e t h a t th e argumen t o f h(-) i n (7.28) i s alway s betwee n 0 an d 1 . I t ca n b e show n t h a t (Proble m 7.11 ) h(y) is strictl y conve x wit h ti(y) > 0 a n d h"(y) >
0,0
< y < 1.
Using th e convexit y o f /i(-) , w e ca n sho w th e followin g theorem .
7. Precode d OFD M system s
208
Qi^V' 1 ~
Figure 7.7 . Plo t o f h(y) =
x
)-
Theorem 7. 2 MMS E case . Conside r th e precode d O F D M system s wit h a n M M S E receiver . Fo r Q P S K modulation , th e bi t erro r rat e Vmm Se satisfie s — I mmse _ ' of dm,'
sc-cp,mmse
T h e first inequalit y become s a n equalit y i f an d onl y i f th e subchanne l erro r variances a^. ar e th e same . ■ Unlike th e zero-forcin g cas e i n Theore m 7.1 , th e inequalitie s i n th e abov e theorem hol d fo r an y valu e o f 7 = £ s/Afo. W i t h a n MMS E receiver , th e SC-CP syste m ha s th e smalles t B E R an d th e O F D M syste m ha s th e larges t BER. Proof o f Theore m 7. 2 A s i n th e proo f i n Theore m 7.1 , w e ca n us e th e convexity o f h(y) t o sho w t h a t 1
ivi M-1M-1 — 1 ivi — 1 /
Vmmse < ^ g
=
g
1
2 fc>i| /l
|p
( ^+
^
h
MT,
o
= r fdm [l+1\C^) ° -
On th e othe r hand , w e ca n als o appl y th e convexit y o f h(y) t o obtain , /.
M-1M-1
1
i=0 k= 0 M-l
h
M
k=0
1
2\
+ l\Ck\
2
sc-cp,mmse'
7.4. MMS E precode d OFD M system s
209
T h e abov e inequalit y become s a n equalit y i f an d onl y i f th e term s i n th e summation o f (7.28 ) ar e equal , i.e . al l th e subchannel s hav e th e sam e SNR. Thi s i s tru e i f an d onl y i f th e subchanne l NSR s 1/( 1 + /3(i)) i n (7.26) ar e equal , whic h mean s t h a t th e subchanne l erro r variance s a 2e. are th e same . ■
Optimal channe l independen t precoder s Theorem 7. 2 state s t h a t th e minimu m erro r rat e V sc-cp,mmse i s achieve d i f and onl y i f th e subchanne l erro r variance s ar e equalized . Examinin g th e expression o f th e subchanne l erro r variance s i n (7.23) , w e se e t h a t suc h a property i s achieve d whe n th e precode r ha s th e equa l magnitud e propert y i n (7.14). Again , th e H a d a m a r d matri x an d th e D F T matri x ar e example s o f such precoders . T h e precode r t h a t equalize s th e subchanne l erro r variance s fo r the zero-forcin g receive r als o equalize s th e subchanne l erro r variance s fo r th e MMSE receiver . Thi s mean s th e transmitte r doe s no t nee d t o kno w whethe r the receive r i s zero-forcin g o r MMSE . T h e sam e transmitte r ca n b e optima l for differen t receivers . W h e n th e subchanne l error s ar e equalized , al l th e sub channel erro r rate s ar e th e same , equa l t o t h a t o f th e MMS E SC-C P system , a n d W e n a v e / rnmse — ' sc-cp,mmse'
7.4.3 Othe
r modulatio n scheme s
For modulation s othe r t h a n QPSK , w e ca n us e approximation s o f B E R a s i n Section 7.3 . T h e result s wil l b e state d withou t proo f [78] . W e wil l us e 26-bi t QAM a s a n example . W e approximat e th e zt h subchanne l erro r rat e a s i n (7.15), Vmmse{i) ~ aQ(y/£&)), wher e f3(i) i s th e SN R give n i n (7.24) , an d the constant s a an d £ ar e a s give n i n (7.15) . Le t u s defin e
g(y) = aQiVCiy-1 - 1)), (7.3i
)
then Vmmse(i) ca n b e convenientl y approximate d b y 1
j
' mmsey
)
■p{i)
where th e argumen t o f g(-) i s th e biase d subchanne l NSR . I t ca n b e show n (Problem 7.13 ) t h a t th e functio n g(y) i s conve x ove r th e interval s So an d 52 , and i t i s concav e ove r S\. T h e interval s 5o , 5 i , an d 5 2 are , respectively , 50 =
,3
where z
n = an
(0 , ZQ], Si =
+ C - \/C
2
(ZQ,
zi), an d 5 2 = [z
- 1Q C + 9 ,
88
d z\
3
=
.
u
1) ,
+ C + y / C 2 -iQC + 9
(7.32) Note t h a t fo r th e O F D M system , whe n 7 i s larg e enoug h suc h t h a t 1/( 1 + Pofdmip)) < £0 ? th e zt h subchanne l i s operatin g i n th e conve x regio n So o f g(-). Thi s require s Zo1 -
7> °
\C%.-|2
1
210
7. Precode d O F D M system s
T h e conditio n fo r al l t h e subchannel s o f t h e O F D M syste m t o b e operatin g in So i s z'1 - 1 7 > 7i > wher e
0
z'1 = mi n \ .
1
(7.34
)
For convenience , w e defin e
n'low = { 7 |7<7o), Kud = bho<j
= bh><}• (7.35
)
W h e n t h e SN R i s sufficientl y hig h s o t h a t 7 G 7 £ ^ ^ , o r whe n t h e SN R is lo w enoug h s o t h a t 7 G ^[ owl al l t h e subchannel s o f t h e conventiona l O F D M syste m wil l operat e i n on e of t he two convex region s o f #(•). I n eithe r case, w e ca n invok e t h e convexit y o f g(-) t o sho w t h a t t h e MMS E SC-C P system i s optimal . However , unlik e t h e Q P S K case , 7 G 7 £ ^i d doe s no t mea n t h a t t h e O F D M syste m i s optimal . I n fact , ther e ma y no t exis t a n SN R region i n whic h al l t h e subchannel s ar e operatin g i n t h e concav e regio n o f g(-) (Proble m 7.16) . Simila r conclusion s ca n be draw n fo r othe r modulatio n schemes whos e subchanne l erro r rate s assum e t h e for m aQ(^/(/3(i)). T h e boundar y point s 7 0 an d 7^ o f VJ low an d VJ hih ar e determine d b y t he channel an d t he two scalars (ZQ — 1 ) an d {z^ — 1), which ca n be compute d from t h e constellatio n siz e usin g (7.32 ) an d (7.15) . Tabl e 7. 1 gives t h e value s of thes e tw o scalar s fo r Q A M constellation s o f size s 4 , 16 , 64 , an d 256 . A s the siz e increases , t h e lef t boundar y o f VJ hih i s pushe d furthe r t o t h e righ t while t h e righ t boundar y o f VJ low i s pushe d t o t h e left . Fo r t he constellatio n of siz e 4 , b o t h scalar s ar e equa l t o one , i.e . t h e functio n g{y) i s conve x fo r all y. Thi s i s consisten t wit h ou r earlie r findings t h a t fo r Q P S K t h e MMS E SC-CP i s optima l fo r al l 7.
zr1 -1 1.00
^1 1.00
16
0.37
13.63
64
0.34
61.66
256
0.34
253.66
QAM constellatio n siz e 4
Table 7 . 1 . Th e two scalars (z 0 x - 1 ) an d (z 1 x - 1 ) fo r differen t QA M constellatio n sizes.
7.5. Simulatio n example s
211
Compared t o t h e B E R behavio r o f t h e MMS E precode d syste m fo r t h e Q P S K scheme , t h e non-QPS K case s ar e different . T h e MMS E SC-C P syste m is n o longe r optima l fo r al l 7 . I t i s optima l fo r lo w an d hig h SN R (VJ low an d ^hiqh)- I n a simulatio n exampl e show n later , w e wil l se e t h a t VJ low corre sponds t o a B E R rang e to o larg e t o b e considere d i n practica l applications . T h e B E R discussio n fo r t h e MMS E syste m i n thi s sectio n is , to som e extent , similar t o t h a t fo r t h e zero-forcin g cas e i n Sectio n 7.2 . T h e solutio n o f t h e optimal precode r depend s o n t he give n SN R and t he SC-C P syste m i s optima l for a highe r SN R range .
7.5 Simulatio
n example s
Example 7. 1 Zero-forcin g receivers . W e will assum e t h a t t h e noise i s AWG N with varianc e A/o - T h e modulatio n symbol s ar e Q P S K wit h value s equa l t o ± \JE SJ2 ± j \JE SJ2 an d 7 = £ s/Afo- T h e number o f subchannels M i s 64. T h e length o f t he cycli c prefi x i s 3 . T h e two four-tap channel s i n Exampl e 6. 1 wil l be used . T h e magnitud e response s o f t he tw o channels c\(n) an d C2(n) ar e a s shown i n Fig . 6.4. For t h e channel c i ( n ) , w e compute t h e values o f 70 an d 71 defined i n (7.11 ) and (7.10) . The y are , respectively , - 0 . 5 1 d B an d 14.7 4 d B . Figur e 7.8(a ) shows Vofdm an d V sc-cp a s function s o f 7 . Fo r comparison , w e als o sho w a n arbitrary exampl e o f B E R whe n t h e transmittin g matri x i s unitary , bu t i t i s neither t h e identit y matri x no r t h e I D F T matrix . T h e transmittin g matri x G t h a t i s chose n her e i s a commonl y know n unitar y matri x calle d a typ e I I D C T matrix. 2 T h e B E R i s denote d a s Vdct i n t h e figure. I n thi s case , t h e precoder P i s W G, whos e element s d o no t hav e t h e uni t magnitud e propert y in (7.14) . Wheneve r 7 = S s/Afo i s larger t h a n 7 1 = 14.7 4 d B, t he erro r rat e o f the SC-C P syste m become s t h e minimu m B E R fo r an y unitar y precode r P . For 7 < 70 , t h e conventiona l O F D M syste m i s t h e optima l solution . W h e n 7 = 70 , w e observ e fro m Fig . 7.8(a ) t h a t V 0fdm ~ 0.2 . T h a t is , t h e O F D M system i s optimal onl y fo r B E R larger t h a n 0.2 . Fo r eithe r SN R range, 7 < 7 0 or 7 > 71 , t he performanc e o f Vdct lie s betwee n V 0fdm an d V sc-cpT h e channe l C2(n) has a zer o aroun d 1.17 T and it s D FT coefficient s aroun d the spectra l nul l ar e ver y small . T h e value s o f 7 0 an d 7 1 are , respectively , 1.4 d B an d 51. 9 d B. Du e t o t h e zer o o f t h e channe l t h a t i s clos e t o t h e uni t circle, 7 1 become s ver y large . Figur e 7.8(b ) show s t h e B E R performanc e o f 'Pofdm, Vdct, an d Vsc-cp • Fo r al l thre e systems , t h e B E R s becom e smal l onl y when SN R i s ver y large . Not e t h a t t h e crossin g o f V sc-cp an d V 0fdm occur s around 7 = 3 7 dB and t he B ER corresponding t o t he crossing is approximatel y 6 x 1 0 - 3 . Fro m thi s exampl e w e se e t h a t , fo r a reasonabl e B E R , t h e SC-C P system outperform s t h e O F D M system . Fo r example , whe n B E R is 1 0 - 3 , t he gains ar e 2 an d 1 0 dB fo r c i ( n ) an d C2(n) , respectively . ■
2
A n M x M typ e I I D C T matri x C J J is given b y [Cjj]mn =
J — co s ( — m(n + 0.5) ) , 0 < m , n < M - 1 .
212
7. Precode d O F D M system s
10"
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2I
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\\ \ \ \\ \ \ \x \ \x
CO
10"'
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,
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,
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5
E s /N 0 (dB)
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v
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03 04 Es/N0(dB)
Figure 7.8 . Exampl e 7.1 . Bi t erro r rat e performance s o f V (a) channel ci(n) and (b) channel C2(n).
Vv
V\
i
sc-cp
10 2
20
V
CQ
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for
7.5. Simulatio n example s
213
Example 7. 2 MMS E receivers . T h e simulatio n environmen t an d parame ters use d i n thi s exampl e ar e th e sam e a s thos e i n Exampl e 7. 1 excep t t h a t the receive r i s no w MMS E rathe r t h a n zero-forcing . Fo r channe l c i ( n ) , Fig ure 7.9(a ) show s th e B E R performance s o f tw o MMS E receivers , Vdct,mmse and V Sc-cp,mmsei wher e Vdct,mmse correspond s t o th e cas e whe n th e transmit ting matri x G i s a D C T matri x a s i n Exampl e 7.1 . W e hav e als o show n th e error rate s o f th e zero-forcin g receivers , Vdct an d V sc-Cp- I n eithe r case , th e B E R o f th e MMS E receive r i s lowe r t h a n t h a t o f th e zero-forcin g receive r fo r all SNR . (Fo r comparison , V 0fdm i s als o show n i n th e sam e plot . Fo r th e O F D M system , a n MMS E receive r doe s no t improv e th e BER , s o th e M M S E case i s no t shown. ) W e ca n se e t h a t th e curv e o f th e MMS E SC-C P syste m is belo w al l th e others . Thi s corroborate s th e resul t t h a t th e MMS E SC-C P system ha s th e smalles t erro r rat e fo r al l 7 . For channe l C2(n) , Fig . 7.9(b ) show s th e five B E R performanc e curve s mentioned above . Du e t o th e zer o o f the channe l t h a t i s close to th e uni t circle , the erro r rate s o f th e thre e zero-forcin g system s ar e considerabl y affected . However, ther e i s n o seriou s performanc e degradatio n i n th e MMS E SC-C P system. Agai n w e ca n observ e t h a t th e curv e o f th e MMS E SC-C P syste m stays belo w al l th e othe r curve s fo r al l 7 . Comparin g Fig . 7.9(a ) an d (b) , w e see t h a t th e gai n o f usin g a n MMS E receive r i s substantiall y mor e significan t when th e channe l ha s a spectra l null . ■
214
7. Precode d O F D M system s
10" TO\.
w
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Q)
O
10"
Q CO —
10" . P
: dc :
ofd
\\ \ \
m
V\ \
t
. ... P . _ _ P dct,mmse
10"
' —1 — p sc-cp | ■ sc-cp,mms
V\
X
e
10 E s /N 0 (dB)
15
20
E s /N 0 (dB)
Figure 7.9 . Exampl e 7.2 . Erro r rat e performance s o f MMS E receiver s fo r (a ) channe l c i ( n ) an d (b ) channe l C2(n) .
7.5. Simulatio n example s 21
5
Example 7. 3 Rando m channel . W e us e a rando m channe l wit h fou r coeffi cients. T h e channe l ha s a n exponentia l powe r dela y profile . T h e coefficient s c(n) ar e obtaine d fro m independen t circula r comple x Gaussia n rando m vari ables wit h zero-mea n an d variance s give n b y 8/15 , 4 / 1 5 , 2/15 , an d 1/1 5 fo r n = 0 , 1 , 2 , 3 , respectively . T h e inpu t symbol s ar e Q P S K . W e comput e th e B E R performance s V 0fdm, V dct,mmse, an d V Sc-cP,mmse an d averag e th e re sults fo r 2000 0 rando m channel s (Figur e 7.10) . Fo r hig h SNR , th e SC-C P system wit h th e MMS E receive r require s a significantl y smalle r transmissio n power t h a n th e O F D M syste m fo r th e sam e BER . T h e averag e performanc e of y Pdct,mmse i s betwee n th e othe r tw o system s fo r al l SNR . ■
10"1
>^ \\( \>
B
00 DC -
-1 0 o
\> N \ 'N . \ \ \N \ \\
2
LU
10"3
i—
■—
10"
P «\ Pofdm \ \
'N
dct,mmse \
P\
sc-cp,mmse \ ,,
4
01
02
N
'
A
.
.
N
N
N
.
. ,
03 E s /N 0 (dB)
v
L
04
^
.
0
Figure 7 . 1 0 . Exampl e 7.3 . Bi t erro r rat e performance s Vofdm, 'Pdct,mmse, an d Vsc-cp,mmse ove r a four-ta p rando m channel .
Example 7. 4 QA M modulation . I n thi s exampl e th e inpu t modulatio n sym bols ar e 4-bi t QAM . T h e tw o channel s i n Exampl e 7. 1 wil l b e use d i n ou r simulations. Fo r channe l c i ( n ) , w e comput e th e value s o f 7 Q an d j[ defined , respectively, i n (7.33 ) an d (7.34) . The y are , respectively , —9. 6 an d 21. 3 d B . For channe l C2(n) , th e value s o f 7 Q an d 7 ^ are , respectively , —7. 7 and 58. 5 d B . T h e B E R performance s ar e show n i n Fig . 7.11 . In Fig . 7.11(a) , fo r b o t h th e SC-C P syste m an d th e syste m wit h th e D C T matrix a s th e transmitter , th e performanc e o f th e zero-forcin g receive r i s al most indistinguishabl e fro m t h a t o f th e MMS E receiver . I n b o t h figures , w e observe t h a t th e curv e o f th e MMS E SC-C P syste m i s n o longe r belo w al l the othe r curves . Fo r c i ( n ) , i t crosse s th e othe r curve s a t aroun d 7 = 1 4 dB , which correspond s t o a n erro r rat e o f aroun d 0.07 . Fro m Sectio n 7.4.3 , w e know th e MMS E SC-C P ha s th e lowes t erro r rat e whe n 7 > 7 ^ = 21.3 . Fo r C2(n), it s curv e crosse s th e other s a t aroun d 7 = 2 3 dB , a valu e muc h smalle r t h a n 7 { = 58. 5 d B . T h e B E R correspondin g t o 7 = 2 3 d B i s aroun d 0.02 . T h e
7. Precode d OFD M system s
216
(a)
■"" —
10 1
-»
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.
r"~^^»
- ^ ^S
■^St
10" r
(b)
— \ ^ ' ^: ^ vv
\
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x
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of dm
P
dct dct,mmse
-
P
sc-cp
psc-cp, mmse
-
Xx
\N
10"
\ \ i
20
30
40 Es/N0(dB)
50
60
Figure 7 . 1 1 . Exampl e 7.4 . Erro r rat e performance s o f 4-bi t QA M modulatio n symbol s for (a ) channe l c i ( n ) an d (b ) channe l C2(n) .
simulations demonstrat e t h a t th e B E R o f th e MMS E SC-C P become s lowe r t h a n othe r system s a t a n SN R muc h smalle r t h a n j[. Althoug h i t i s no t th e optimal solutio n fo r al l SNR , i t outperform s th e other s fo r a mor e usefu l SN R range. ■ Example 7. 5 Code d erro r rates . I n thi s example , th e bit s ar e code d usin g convolutional code s befor e the y ar e m a p p e d t o Q P S K symbols . W e us e th e same four-ta p rando m channe l i n Exampl e 7.3 . Th e B E R s o f th e MMS E SC C P an d O F D M system s wit h cod e rate s 1/ 2 an d 1/ 3 ar e shown , respectively , in Fig . 7.12(a ) an d (b) . I n th e uncode d case , Theore m 7. 2 show s th e MMS E SC-CP syste m achieve s a lowe r erro r rat e t h a n th e O F D M system . Fo r th e two code d case s i n Fig . 7.12(a ) an d (b) , w e se e t h a t th e tw o performanc e curves cros s a t a ver y hig h erro r rate . Th e O F D M syste m perform s slightl y
7.5. Simulatio n example s
217
better fo r ver y lo w SN R o r hig h BER . T h e MMS E SC-C P syste m i s bette r for a mor e usefu l rang e o f erro r rate , an d th e ga p betwee n th e tw o system s widens a s SN R increases . For th e uncode d O F D M system , th e subchannel s nea r th e nul l hav e a larger erro r rat e whil e th e bit s o n othe r subchannel s ar e decode d correctl y with hig h probability . Usin g a n erro r correctin g code , th e bit s o n th e ba d subchannels ar e likel y t o b e correcte d wit h th e ai d o f correc t bit s fro m th e good subchannels . T h e erro r rat e o f th e O F D M syste m ca n b e considerabl y improved whe n a n erro r correctin g cod e i s employed . Fo r example , t o achiev e an erro r o f 1 0 - 4 , th e require d SN R fo r th e uncoded , 1/ 2 code d an d 1/ 3 coded case s ar e respectivel y 37 , 22. 5 an d 1 8 d B . W e shoul d als o not e t h a t the reductio n i n SN R i s achieve d a t th e pric e o f a decreas e i n transmissio n bit rates . T h e transmissio n rate s o f th e tw o code d case s ar e respectivel y half an d one-thir d t h a t o f th e uncode d case . Fo r th e MMS E SC-C P system , the uncode d erro r rate s ar e th e sam e fo r al l subchannels . Fo r channel s wit h a spectra l null , th e effec t wil l b e sprea d t o al l subchannels . Ther e ar e n o good subchannel s t o hel p wit h th e correctio n o f th e ba d subchannels . T h e improvement i n erro r rate s i s no t a s muc h a s th e O F D M system . T o achiev e the sam e erro r rat e 1 0 - 4 , th e require d SN R fo r th e sam e thre e case s ar e respectively 23 , 1 8 an d 1 5 d B . Fo r th e uncode d case , th e O F D M syste m requires 1 4 d B mor e SN R t h a n th e SC-C P system , whil e fo r 1/ 3 code d cas e the ga p narrow s t o aroun d 3 d B . T h e bloc k erro r rat e (o r packe t erro r rat e i f each bloc k i s sent i n on e packet ) is sometimes a mor e relevan t measur e becaus e a n uncorrecte d bi t erro r usuall y means th e whol e bloc k need s t o b e re-transmitted . T h e bloc k erro r rate s o f the tw o system s fo r th e cod e rat e 1/ 3 ar e show n i n Fig . 7.12(c) . T h e M M S E SC-CP syste m ha s a lowe r bloc k erro r rat e t h a n th e O F D M syste m fo r al l SNR. T h e differenc e betwee n th e tw o i s mor e pronounce d t h a n i n th e B E R plots. T h e reaso n i s t h a t error s mostl y occu r whe n th e channe l i s no t good , e.g. channel s wit h spectra l nulls . Bu t thi s i s eve n mor e s o fo r th e MMS E SC C P system . Fo r th e sam e BER , th e error s o f th e MMS E SC-C P syste m ar e even mor e concentrate d i n th e case s o f ba d channel s t h a n th e O F D M system . Therefore, fo r th e sam e B E R th e MMS E SC-C P syste m i s mor e likel y t o hav e error-less block s t h a n th e O F D M syste m (i.e . a smalle r bloc k erro r rate) . ■
218
7. Precode d O F D M system s 1(T ofdm-cp,1/2 coded sc-cp,mmse,1/2 coded
y = E s / N 0 (dB
25
)
icr ofdm-cp,1/3 coded
- P sc-cp^mseJ/S code d
y = E s / N 0 (dB
10
)
20 2
5
ofdm-cp,1/3 coded sc-cp,mmse,1/3 coded
CD 1 0 "
m
10 "
10"
10 1 5 y=E s /N 0 (dB )
25
Figure 7 . 1 2 . Exampl e 7.5 . Performanc e o f th e OFD M an d MMS E SC-C P system s with channe l coding , (a ) BE R fo r 1/ 2 code d symbols ; (b ) BE R fo r 1/ 3 code d symbols ; (c) bloc k erro r rat e fo r 1/ 3 code d symbols .
7.6. Furthe r readin g 21
7.6 Furthe
9
r readin g
Block transceiver s wit h a genera l transmittin g matrix , sometime s referre d t o as precoding , hav e bee n studie d extensivel y i n th e literatur e [5 , 75, 126 , 134 , 135]. Thes e transceiver s ar e optimize d fo r a give n channe l profile , includin g the channe l impuls e respons e an d th e channe l nois e statistics . Th e solution s are henc e generall y channel-dependent . Transceiver s tha t ar e optima l i n th e sense of minimum transmission power or minimum total noise power have been developed [5 , 75, 134, 135]. Fo r the class of zero-padding transceivers, a n opti mal solution that minimize s the total outpu t nois e variance was found i n [135] . The optima l receive r an d zero-padde d transmitte r ca n b e give n in terms o f a n appropriately define d channe l matri x an d th e autocorrelatio n matri x o f th e channel noise. Informatio n rat e optimize d system s were considered i n [5 , 134]. In [75] , optimal transmitter s an d receiver s tha t minimiz e transmissio n powe r for a give n bi t rat e an d erro r rat e wer e derive d unde r a n optima l bi t alloca tion. A fundamentall y differen t usefu l unifie d approac h t o th e join t desig n of transmitter s an d receiver s fo r MIM O channel s usin g conve x optimizatio n was develope d i n [106] . Severa l commonl y use d optimizatio n criteria , suc h a s mean square d erro r an d geometri c mea n o f SINRs , wer e considere d therein . Optimal bloc k transceiver s ca n b e obtaine d unde r thi s unifie d framework . When th e length s o f transmitting an d receivin g filters ar e no t constraine d by th e bloc k size , th e transmitte r an d receive r ar e n o longe r characterize d by constan t matrice s [47 , 72 , 74 , 170] . Du e t o longe r transmittin g an d re ceiving filters, ther e i s mor e desig n freedom . A join t transmitter-receive r optimization schem e usin g filter ban k precodin g wa s give n i n [72] . Optima l ideal transmittin g an d receivin g filters tha t minimiz e th e mea n square d er ror ca n b e obtaine d i n th e frequenc y domain . I n [74] , th e desig n freedo m was use d t o obtai n transmittin g an d receivin g filters wit h bette r frequenc y selectivity. Frequenc y selectivit y i s importan t i n application s wher e ther e i s certain constrain t o n th e transmitte r outpu t an d fo r suppressin g narrowban d noise a t th e receiver . Given th e channe l profile , transceiver s hav e als o bee n optimize d fo r BE R minimization. I n [33 , 34] , base d o n th e zero-forcin g solutio n give n i n [135] , a clas s o f optima l precoder s wit h powe r loadin g wa s considered . Fo r a give n channel profile , th e optima l precode r fo r BPS K modulatio n ca n b e foun d i n a close d for m whe n th e SN R i s sufficientl y large . I n [108] , transceiver s wer e designed fo r subchannel s wit h differen t constellations . Fo r a give n channe l profile an d bi t allocation , th e transceive r wa s optimize d fo r minimu m BE R by usin g th e prima l decomposition . Th e proble m o f minimizin g BE R fo r binary signalin g ove r FI R MIM O channel s wa s considere d i n [51] . A n iter ative algorith m wa s give n i n [51 ] fo r jointl y optimizin g th e FI R transmitte r and receiver . Fo r th e cas e whe n ther e i s oversamplin g a t th e receiver , i.e . fractionally space d equalizer , transceive r optimizatio n wa s developed i n [176] . With oversampling , remarkabl e gai n ca n b e achieve d fo r practica l channels , as demonstrate d i n [176] .
7. Precode d OFD M system s
220
7-7 Problem
s
7.1 Le t M = 4 an d le t th e lengt h o f cycli c prefi x v = 2 . Suppos e th e channe l has tw o coefficient s c(0 ) = c(l ) = 1/ 2 an d Q P S K modulatio n i s used . Compute 7 0 an d 71 . Fin d th e SN R rang e fo r whic h th e O F D M syste m has th e smalles t bi t erro r rat e amon g al l zero-forcin g precode d systems . 7.2 Suppos e th e settin g i s th e sam e a s i n Proble m 7. 1 excep t t h a t 4-bi t QAM i s used . Comput e 7Q , an d j[. Fin d th e SN R rang e fo r whic h th e O F D M syste m ha s th e smalles t bi t erro r rat e amon g (a) al l zero-forcin g precode d systems ; (b) repea t (a ) fo r th e SC-C P system . 7.3 Suppos e th e SN R quantit y 7 =
£ 3/Af0 i s 1 0 d B i n Proble m 7.2 . Fin d
Psc-cp &H Q Psc-cp,mmse' 7.4 Le t M = 4 an d le t th e lengt h o f cycli c prefi x v = 2 . Suppos e th e channe l is C(z) = 1 + 0.52: - 1 . Th e transmitte r inpu t modulatio n symbol s ar e Q P S K . Fin d th e SN R regio n wher e th e SC-C P i s optima l fo r th e case s when (a) th e receive r i s zero-forcing ; (b) th e receive r i s MMSE . 7.5 Repea t Proble m 7. 4 whe n th e transmitte r inpu t modulatio n symbol s are (a) 4-bi t QAM ; (b) 8-bi t QAM . 7.6 Determin e th e subchanne l SN R /3(i) give n i n (7.24 ) whe n P = M = 4 , v = 2 , an d th e channe l i s a s i n Proble m 7.1 .
I M for
7.7 Fo r th e precode d O F D M syste m wit h a zero-forcin g receiver , w e showe d t h a t th e subchanne l SNR s ar e bounde d a s i n (7.6) . Sho w t h a t fo r th e M M S E cas e th e subchanne l SN R f3{i) continue s t o satisf y th e bounds . T h a t is , min7|Cfc|2
(Hint. Us e th e relatio n i n (7.26). ) 7.8 Le t P b e a unitar y matri x wit h th e equa l magnitud e propert y i n (7.14) . Consider a matri x P ' wit h V'm,n =
e
jiem+an)
Pm,m 0
< m ,7 1 < M - 1 ,
for arbitrar y choice s o f rea l 9 m an d a n. Sho w t h a t th e ne w matri x P ' is als o unitar y an d t h a t i t als o ha s th e equa l magnitud e property . 7.9 I n (7.3) , w e comput e th e SN R o f th e i t h subchanne l o f th e precode d O F D M syste m wit h a zero-forcin g receiver . Equatio n (7.24 ) give s th e i t h subchanne l SN R whe n th e receive r i s MMSE . I s th e i t h subchanne l SNR give n i n (7.24 ) alway s large r t h a n th e i t h subchanne l SN R give n in (7.3) ?
221
7.7. Problem s
7.10 Conside r th e MMS E precode d O F D M syste m i n Fig . 7.5 . Suppos e th e inputs o f th e transmitte r ar e symmetri c i n th e sens e t h a t th e rea l an d imaginary part s hav e equa l varianc e £ s/2. T h e channe l nois e q(n) i s circularly symmetri c Gaussian . W h e n th e MMS E receive r i s used , d o the rea l an d imaginar y part s o f the signa l plu s th e nois e t e r m Ti in (7.21 ) also hav e equa l variance ? 7.11 Deriv e th e first derivativ e h!(y) an d th e secon d derivativ e h n(y) o f th e function h(y) define d i n (7.27) . Sho w t h a t h(y) i s conve x wit h h'(y) > 0 and ti'(y) > 0 fo r 0 < y < 1 . 7.12 I n Sectio n 7.3 , w e conside r optima l precoder s fo r modulatio n symbol s other t h a n Q P S K . T h e derivatio n i s vali d fo r an y modulatio n scheme s in whic h th e subchanne l erro r probabilit y ca n b e eithe r approximate d o r expressed a s aQ I \/CM^) J f ° r som e constant s a an d £ t h at ar e indepen dent o f subchannels . P S K modulatio n i s suc h a n example . Determin e the SN R regio n i n whic h th e SC-C P syste m i s optima l fo r th e channe l C(z) = 1 + 0.5Z" 1 . Note: A b-bi t P S K symbo l wit h symbo l powe r S s i s o f th e for m s = ^yS scos (
-z^m I + jySg sin
( -^m J
,0 < m < 2 h.
W h e n i t i s transmitte d ove r a n AWG N channe l wit h nois e varianc e A/o , the symbo l erro r rat e ca n b e approximate d b y VSER
W h e n a Gra y cod e i s used i n the mapping , th e B E R i s well approximate d by VsERJh. 7.13 Sho w t h a t th e functio n g(y) define d i n (7.31 ) i s conve x i n interval s So and c> 2 an d concav e i n S i . 7.14 Sho w t h a t whe n 7 G Whigh i n (7.35) , al l th e subchannel s o f th e O F D M system ar e operatin g i n th e conve x regio n S$ o f g(-) an d al l th e sub channels o f th e MMS E precode d O F D M syste m ar e als o operatin g i n the conve x regio n So o f #(•) . 7.15 Sho w t h a t whe n th e precode r satisfie s th e equa l magnitud e propert y i n (7.14), i t i s optima l fo r 7 G 1Zfhigh i n (7.35) . 7.16 I n th e zero-forcin g precode d O F D M syste m wit h Q P S K modulation , we sa w t h a t th e SC-C P an d O F D M system s ar e optima l fo r differen t SNR regions . T h e SC-C P syste m i s optima l fo r th e SN R regio n IZhigh while th e O F D M syste m i s optima l fo r 7Zi ow. Fo r th e MMS E cas e wit h general QA M modulation , ther e ar e als o SN R region s (7l[ ow an d VJ hih given i n (7.35) ) fo r whic h th e SC-C P syste m i s optimal . Sho w t h a t 7 G T^mid doe s no t mea n t h a t al l th e subchannel s ar e operatin g i n the concav e regio n o f #(•) ; thu s th e optimalit y o f th e O F D M syste m
222
7. Precode d OFD M system s
is no t guarantee d i n thi s SN R region . Giv e a counterexampl e t o sho w that ther e ma y no t exis t a n SN R regio n i n whic h al l subchannel s ar e operating i n th e concav e regio n o f g(-).
8 Transceiver desig n wit h channe l information a t th e transmitte r From Chapte r 6 , w e kno w t h a t a n O F D M syste m convert s a n LT I channe l int o a se t o f paralle l subchannels . W h e n th e channe l i s frequency-selective , thes e subchannels ca n hav e ver y differen t subchanne l gains . T h e syste m perfor mance ca n b e severel y limite d b y a fe w ba d subchannels . On e solutio n t o thi s problem i s t o us e a precode r a s demonstrate d i n Chapte r 7 . I n man y applica tions, th e transmissio n environmen t doe s no t chang e frequently . Thes e appli cations includ e mos t wire d transmissio n schemes , suc h a s ADS L an d VDS L systems, an d man y wireles s transmissio n schemes , suc h a s fixed wireles s ac cess system s an d wireles s LA N syste m wher e th e en d user s ar e no t mobile . Under thes e environments , i f th e channe l stat e informatio n i s know n a t th e transmitter, on e ca n exploi t thi s informatio n t o carr y ou t bi t allocation . B y optimally assignin g th e bit s t o th e subchannels , th e syste m performanc e ca n be substantiall y improved , especiall y whe n th e channe l i s highl y frequency selective. I n thi s chapter , w e wil l deriv e th e optima l zero-forcin g transceiver s when ther e i s bi t allocatio n a t th e transmitter .
8.1 Zero-forcin
g bloc k transceiver s
T h e DFT-base d transceiver s studie d i n Chapte r 6 employ eithe r D F T o r I D F T operations a t th e transmitte r an d receiver . T h e filter ban k formulatio n show s t h a t thei r polyphas e matrice s ar e constan t matrice s relate d t o th e D F T / I D F T operations. I n thi s chapter , w e stud y genera l bloc k transceiver s wher e th e polyphase matrice s ca n b e arbitrar y constan t matrices . Figur e 8. 1 show s a block transceive r system . T h e M x l inpu t vecto r s i s processe d b y th e T V x M transmitting matri x G o t o produc e a n i V x l outpu t vecto r x = G 0s, which i s converte d t o a sequenc e x(n) an d transmitte d ove r th e channel . A t the receiver , th e receive d sequenc e i s blocke d int o vector s o f siz e TV . T h e TV x 1 receive d vecto r r i s the n processe d b y th e M x T V receiving matri x S o to obtai n th e M x l outpu t vecto r s = S 0 r. 223
224 8
. Transceive r desig n w i t h channe l informatio n a t th e transmitte r
/
/
/
/
r
\q(n)
x(n) P/S
C(z)
->• H
SM-\
S/P
o
r
l
So
•• •
rN-\
XjV-1
Figure 8 . 1 . Block-base d transceive r system .
In thi s chapter , w e conside r onl y zero-forcin g bloc k transceiver ; tha t is , th e transfer matri x fro m th e transmitte r inpu t s t o th e receive r outpu t s ^ is a n identity matrix . Becaus e ther e i s n o interbloc k interferenc e (IBI ) i n a zero forcing transceive r an d th e processin g i n a bloc k transceive r i s don e i n a block b y bloc k manner , onl y one-sho t transmissio n i s considered . Therefor e in Fig . 8.1 , we omit th e tim e inde x n o f th e inpu t an d outpu t vectors . Assume tha t th e integer s N an d M satisf y N > M s o tha t zero-forcin g equalization i s possible . Th e numbe r v = N -M represents th e numbe r o f redundant sample s inserte d pe r inpu t block . I n thi s chapter, i t i s assumed tha t th e channe l i s an LT I FI R filte r wit h orde r L < v\ C(z) =
c(0 ) + c ( l ) ^ - 1 + • • • + c{y)z~
v
.
The las t fe w coefficient s ar e zer o whe n L < v. Th e channe l nois e q(n) i s a zero-mea n WS S Gaussia n proces s wit h know n autocorrelatio n coefficients . Unlike i n Chapte r 7 , q{n) i s no t restricte d t o b e whit e an d i t ca n b e colore d as i n mos t wire d environments . Th e channe l impuls e respons e an d th e nois e statistics ar e know n t o bot h th e transmitte r an d receiver . In thi s section , w e wil l deriv e th e necessar y an d sufficien t condition s o n the matrice s G o an d S o suc h tha t th e bloc k transceive r syste m achieve s zero forcing. Recal l fro m Chapte r 5 tha t th e TV-inpu t TV-outpu t syste m fro m x to r i s a n LT I syste m an d th e transfe r matri x i s th e pseudocirculan t C ps(z). The overal l transfe r matri x fro m s t o s i s a n M x M matri x give n b y T(z) = S
0Cps(z)G0.
(8.1)
The block transceiver i s free fro m IB I if T(z) i s a constant matri x independen t of z. A s th e channe l orde r L satisfie s L < v < N, th e N x N matri x C ps(z) has th e specia l for m give n i n (5.5) ; onl y th e v x v submatri x a t th e to p righ t corner o f C ps(z) i s z-dependent. Fro m Chapte r 5 , we know that w e can avoi d or remov e IB I b y settin g eithe r th e las t v row s o f G o t o zer o o r th e firs t
8.1. Zero-forcin g bloc k transceiver s
225
v column s o f S o t o zero . T h e forme r i s equivalen t t o paddin g zero s a t th e transmitter an d th e latte r i s equivalen t t o discardin g o r zer o jamming th e IB I corrupted sample s a t th e receiver . Thu s th e forme r i s referre d t o a s a zero padded (ZP ) syste m an d th e latte r i s calle d a zero-jammin g (ZJ ) syste m [162] (als o know n a s a leadin g zer o syste m [135]) . I n th e following , w e shal l derive th e zero-forcin g solution s fo r th e Z P an d Z J transceivers .
8.1.1 Zero-forcin
g Z P system s
For a Z P system , w e pa d v zero s fo r ever y bloc k o f M d a t a samples . T h e transmitting an d receivin g matrice s are , respectively , o f th e form s G zp 0
Go
and S
o= S
2
where G zp i s o f dimensio n M x M, th e b o t t o m matri x 0 i s of dimensio n v x M , and S zp i s a n M x N constan t matrix . Substitutin g th e abov e relatio n int o (8.1), th e transfe r matri x T(z) become s a constan t matrix : J- ^zp^low^zpi
where C\ ow i s th e N x M lowe r triangula r Toeplit z matri x give n i n (5.13) , which w e reproduc e below :
c(0) 0 c(l) c(0 )
0
ciy) -'low
0 0
c(0)
0
(8.2)
c{y) 0 c{y)
0 0
. c(0 ) c(l)
0
c(u)
Using thi s matri x representation , w e ca n redra w th e transceive r syste m i n Fig. 8. 1 a s Fig . 8.2 , wher e th e nois e vecto r q_ zp i s a n N x 1 vector obtaine d b y blocking q(n). Du e t o th e padde d zeros , ther e i s n o overla p betwee n adjacen t blocks afte r channe l filtering . Processin g ca n b e don e o n a bloc k b y bloc k basis. T h e zero-forcin g conditio n become s ^zp^iow ■-M: which implie s *zp an d G Zp hav e ful l rank .GSinc e th e matri x G zp i s zpi Clow, tMh axt th sS Me, matrice i t i s invertible . Thu s th e zero-forcin g conditio n ca n b e rewritte n a s ^zp^zp^lo
i-M-
This mean s t h a t G ^ S ^ i s a lef t invers e o f Ci ow. Sinc e Gi ow i s N x M , it s left invers e i s no t uniqu e whe n N > M. T o characteriz e al l th e lef t inverse s of Ciow, l e t u s appl y th e singula r valu e decompositio n (SVD ) t o Ci ow: -'low
uz
A 0
zpi
(8.3)
8. Transceive r desig n w i t h channe l informatio n a t th e transmitter
226
s= s +e /
/
/ ■ > s»
G«
SM-\
Xi
J
low
■
>
#
XM-\
-> s M-\
Figure 8 . 2 . Simplifie d representatio n of the zero-padded system .
where XJ zp an d \ z p are , respectively, N x N an d M x M unitar y matrice s and A is an M x M diagona l matri x whos e diagona l entrie s ar e the singula r values o f Ci ow. A s Ci ow ha s full rank , al l M singula r value s ar e positive and hence A i s invertible . Usin g th e SV D of C/ its lef t invers e B^ p ca n be expressed a s (8.4) P ]U| V ^ A - ^ M A,zp\ 'zp ^ zp-> where A zp i s an arbitrar y M x v matrix . I n summary , w e have show n tha t the Z P system i s zero-forcing i f and only if (a) th e M x M matri x G zp i s invertible; (b) fo r a give n transmittin g matri x G zp, th e receivin g matri x i s S zp = ^zp *3zp-
Note that th e matrices XJ zp, V^p , and A are determined by the channel matri x Clow After w e impose th e zero-forcing condition , th e free parameter s avail able i n the transceive r desig n ar e the invertibl e matri x C zp an d the M x v matrix A zpi whic h i s completely arbitrary .
8.1.2 Zero-forcin
g Z J system s
In a ZJ system, th e IBI is removed by dropping the v corrupte d receive d sam ples at the receiver. Th e transmitting an d receiving matrices are, respectively , Go = G Zj, S o = [ 0 S Zj], where G zj, 0 , an d S^ - ar e o f dimension s N x M , M x z/ , an d M x M , respectively. Th e transmitted sequenc e i s obtained b y unblocking th e vecto r x = G^-s . As Gzj i s arbitrary, th e redundant sample s o f a ZJ system ar e not necessarily zero s o r a cycli c prefix . The y ar e embedde d i n th e transmitte d sequence a s a linea r combinatio n o f the inpu t symbol s Si. Usin g th e abov e
8 . 1 . Zero-forcin g bloc k transceiver s
227
expression, w e ca n sho w t h a t th e transfe r matri x T(z) i n (8.1) become s a constant matrix , where C up i s the b o t t o m M x N submatri x o f th e pseudocirculan t channe l matrix C ps(z). I t is a n uppe r triangula r Toeplit z matri x give n b y c(v) c ( z / - l 0 c{y)
) •• • c(0 c{y-\) ••
)0 • c(0
•• ) ••
•0 •0
. (8.5 )
J
up 0 ••
c{y)
•0
c{y-l) ••
• c(0 )
T h e subscrip t "up" indicate s t h a t i t is uppe r triangular . s= s+e
/
/
s
l —
S
M-\ >
• • •
►
/
w
►
r
W
GZj
^up
Ww w
• • •
&zj
► • • • ►
M-l
Xtf-i
Figure 8.3 . Simplifie d representatio n o f th e zero-jammin g system .
T h e bloc k diagra m o f a Z J syste m ca n b e draw n a s Fig . 8.3 . Not e t h a t the nois e vecto r q_ zj i s a n M x 1 vector obtaine d b y blockin g q(n). T h e zero forcing conditio n T = 1M implie s t h a t th e squar e matri x S^ - i s invertible. T h e zero-forcin g conditio n ca n b e rewritte n a s ^up^zj^zj = AM * Thus th e Z J syste m i s zero-forcin g i f an d onl y i f th e produc t G zjSzj i inverse o f C up. Le t th e SV D o f C up b e U*i[A 0 ] V t
s a right
(8.6)
where JJ zj an d \ zj are , respectively , unitar y matrice s o f dimension s M x M and N x N. T h e M x M matri x A is diagona l wit h diagona l entrie s equa l t o the singula r value s o f C up. I t can b e show n (Proble m 8.3 ) t h a t th e singula r values o f C up ar e identica l t o thos e o f Ci ow. T h e righ t inverse s o f C up hav e the for m B,
V,
IM
A ^j
A -
1
^ ,
U)
228 8
. Transceive r desig n wit h channe l informatio n a t th e transmitte r
where A zj i s a n arbitrar y vx M matrix . Simila r t o th e Z P case , on e ca n solv e for th e zero-forcin g solution s fo r th e Z J syste m i n Fig . 8.3 . Th e Z J syste m i s zero-forcing i f an d onl y i f (a) th e M x M matri x S^ - i s invertible ; (b) fo r a give n receivin g matri x S^- , th e transmittin g matri x i s G z= j B
zjSzj •
Note t h a t th e fre e parameter s i n a zero-forcin g Z J syste m ar e th e invertibl e matrix S^ - an d th e arbitrar y v x M matri x A zj.
8.2 Proble
m formulatio n
In a transceive r system , th e performanc e i s characterized b y th e erro r rate , th e t r a n s m i t t e d power , an d th e averag e bi t rate . I n transceive r desig n w e ca n fix two parameter s an d optimiz e th e thir d one . Ther e ar e man y differen t criteri a for transceive r designs . I n thi s chapter , w e conside r th e cas e whe n th e averag e bit rat e an d th e erro r rat e ar e fixed, an d th e zero-forcin g bloc k transceive r i s optimized s o t h a t th e t r a n s m i t t e d powe r £ i s minimized . Belo w w e wil l deriv e the expressio n fo r th e t r a n s m i t t e d powe r fo r th e bloc k transceive r show n i n Fig. 8.1 , whic h include s b o t h th e Z P an d Z J system s a s specia l cases . T h e modulatio n symbol s Sk ca n b e PA M o r QAM , dependin g o n whethe r i t is a baseban d o r passban d communicatio n scheme . W e wil l deriv e th e result s for th e PA M case . Th e derivatio n fo r th e QA M cas e i s ver y similar . Le t Sk be a PA M symbo l carryin g bk bit s o f information . The n th e averag e bi t rat e per symbo l becomes 1 M-l
6 = - ^ 6
f c
. (8.8
)
fc=0
Each inpu t bloc k s carrie s Mb bits . I n th e followin g derivation , w e assum e t h a t th e inpu t symbol s hav e zero-mea n an d the y ar e uncorrelated . T h a t is , the autocorrelatio n matri x o f s i s a diagona l matrix :
£s,o o 0£
Rs
0 ..
..
.o
..
.0
.0
£,
SA
(8.9) s,M-l
T h e assumptio n i s usuall y reasonabl e wit h prope r bi t interleaving . Not e t h a t the /ct h signa l powe r £ s^ ca n b e differen t fo r differen t k. Consider Fig . 8.1 . Th e t r a n s m i t t e d powe r £ = ,E[|x(n)| 2 ] ca n b e compute d by averagin g th e powe r o f xi , an d i t i s give n b y £=
1 N ~1 1
= N^2aL
n=0
^ traced).
1 For th e QA M case , i f th e /ct h symbo l carrie s 2b^ bits , th e averag e bi t rat e pe r symbo l becomes
_. M - l K M ^ k=o
8.3. Optima l bi t allocatio n Using th e fac t t h a t R x =
229
G o R s G j , w e ca n writ e
^ = ~/ V traCe (
G
oR«Go)=
where w e hav e use d trace( AB) = t r a n s m i t t e d powe r a s
T 7 trace ( RSGJG0J ,
t r a c e ( B A ) . Usin g (8.9) , w e ca n writ e th e M-l
£=^£M G ° Go L' (8 k=0
-10)
where th e notatio n [A]kk denote s th e kth diagona l entr y o f th e matri x A . Note t h a t th e kth colum n vecto r o f G o correspond s t o th e kth transmittin g filter. Thu s th e quantit y [GQGO]/C/ C i s simply th e energ y o f the kth transmittin g filter. In th e followin g sections , w e wil l stud y th e desig n o f zero-forcin g Z P an d ZJ transceivers . Fo r a fixed bi t rat e b an d a fixed erro r rate , th e transceive r is optimize d s o t h a t th e transmitte d powe r £ i s minimized . T h e optimizatio n process ca n b e decompose d int o tw o steps . Firstly , th e bit s bk ar e optimall y allocated t o minimiz e th e transmitte d powe r fo r an y give n pai r o f transmittin g and receivin g matrice s G o an d So - Then , unde r th e optima l bi t allocation , we wil l find th e transmittin g an d receivin g matrice s t h a t minimiz e th e trans mitted power . I n th e secon d step , th e fre e parameter s ar e eithe r ( G ^ p , A zp) or (Szj , A zj) dependin g o n whethe r i t i s a Z P o r a Z J system .
8.3 Optima
l bi t allocatio n
In a Z P o r Z J system , th e scala r LT I channe l C(z) i s converte d t o a se t o f M parallel subchannels . I n man y applications , thes e subchannel s ca n hav e ver y different SNRs . I n Sectio n 2.4 , w e hav e learn t ho w t o exploi t thi s t o d o bi t loading whe n th e pea k powe r allowe d o n eac h subchanne l i s constrained . Her e we shal l conside r th e bi t allocatio n proble m fo r a differen t setting . Unde r th e assumption t h a t al l th e subchannel s hav e th e sam e fixed erro r rate , w e wil l show ho w w e ca n emplo y bi t allocatio n t o minimiz e th e transmitte d powe r needed fo r a fixed bi t rate . Le t s ^ b e th e kth outpu t o f th e receiver . T h e n the kth outpu t erro r i s e ^ = s ^ — s^. A s th e transceive r i s zero-forcing , th e error come s entirel y fro m th e channe l nois e q(n). Defin e th e M x l outpu t noise vecto r a s e = [e o e\ • • • e M - i ] ; the n e = Sq , where S = S zp an d q = q_ zp fo r th e Z P cas e an d S = S^ - an d q = q_ zj fo r th e ZJ case . Recall fro m (2.14 ) t h a t fo r a 6^-bi t PA M symbol , th e bi t numbe r && , th e symbol powe r £ s^ an d th e nois e powe r o\ ar e relate d b y ,1 , (, fc^lofe^l +
,
£;k/°l k\ ^ r - j ,
where Tk i s th e SN R ga p fo r PA M symbol s give n i n (2.15) . I t i s a quantit y t h a t depend s onl y o n th e erro r rate , an d th e value s o f T^ ar e liste d fo r som e
230 8
. Transceive r desig n wit h channe l informatio n a t th e transmitte r
typical erro r rate s i n Tabl e 2.1 . A s al l th e subchannel s hav e th e sam e erro r rate, th e SN R ga p i s the sam e fo r al l subchannels , i.e . Tk = T , fo
r al l k.
By rearrangin g th e term s i n th e expressio n fo r bk, we ge t £s,k =
T{2^-l)al
k.
Assume tha t b k i s large enoug h s o tha t 2 2bk — 1 « 2
2bk
, the n w e have 2
£s,k^T22bkai. (8.11
)
The abov e equatio n give s a n approximatio n o f th e require d signa l powe r fo r transmitting bk bit s o n th e kth subchanne l whe n th e nois e varianc e i s G\ . For a give n bk and o\ k, on e ca n allocat e symbo l powe r £ sj~ accordin g t o th e expression give n i n (8.11) . Substitutin g (8.11 ) int o (8.10) , th e transmitte d power become s
^ = i E2 X ^ „ M-l
U2)
kk
k=0
Applying th e arithmeti c mea n (AM ) an d geometri c mea n (GM ) inequalit y (Appendix A) , w e obtai n M-l
' ^ n ( * [ ° H J " = «- • 1/M
U3)
k=0
Equality hold s i f an d onl y i f th e quantit y 2 2bka2k [ G Q G O ] ^ i s th e sam e fo r k = 0 , 1 , . . . , M — 1 . Fro m (8.11) , w e ca n conclud e that , whe n th e bit s ar e allocated optimally , th e quantitie s ^;fc[GjGo]fefc =
a constan t fo
r k = 0 , 1 , ... , M — 1.
U4)
are th e sam e fo r al l k. Not e tha t Eut depend s onl y o n 6 , a^ k1 an d [G^Go]*^ , where a 2k i s determine d onc e th e receive r i s know n an d [GjGo]fcf c i s deter mined onc e th e transmitte r i s given . Therefor e whe n th e transceive r i s give n and the bit rat e per symbo l b is fixed, Eut i s a fixed quantity an d i t i s the lowe r bound o n th e transmitte d power , independen t o f th e bi t allocatio n \bk\k= ®~ The lowe r boun d Eut i s achieve d i f an d onl y i f bit s ar e allocate d suc h tha t 22bkalk[GlGo]kk ar e equalized . Thi s conditio n implie s tha t 22bk
2
GjGo = 2
2b
M-l
T T (al . GjG o )
1/M
,
for
al
l k.
i=0
That is , th e quantit y o n th e lef t i s independen t o f k. Solvin g th e abov e equation fo r th e optima l bk, we hav e bk = b-\log 2 (
< [GSGQ
]
J+
^ l o g 2 (nfj^al
[GJCO
]
J.
(8.15 )
2 For th e 26/g-bi t QA M case , th e signa l powe r S Sik a n < l th e nois e varianc e <J\ als o satisf y the sam e relatio n a s (8.11) , bu t th e SN R ga p fo r QA M i s give n b y th e formul a i n (2.21) .
8.3. Optima l bi t allocatio n
231
Note t h a t i n genera l th e optima l bk i s no t a n integer . Moreove r th e right hand sid e o f (8.15 ) coul d b e negative . W e wil l addres s thi s issu e late r i n thi s section. We ca n quantif y th e performanc e improvemen t o f bi t allocatio n b y com paring Ebu wit h th e transmitte d powe r whe n ther e i s n o bi t allocation . I f we d o no t appl y bi t allocation , ever y subchanne l carrie s b bits. T h e require d t r a n s m i t t e d powe r ca n b e obtaine d b y settin g al l bk = b i n (8.12) , an d i t i s given b y mM-L T2 26 ~ r N
r Efc=0 <
-^no-bit
G
oG0
kk
T h e bi t allocatio n gai n i s thu s give n b y
Gbi;
Z^k=0 a
ek
GjGo
■cf-iMJ
kk
(8.16)
1/M '
In man y cases , th e transmitte r ha s th e sam e [GQGO]/C/ C fo r al l k. Fo r example , the C P - O F D M , Z P - O F D M , an d SC-Z P systems 3 studie d i n Chapte r 6 satisf y such a condition . I n thi s cas e th e optima l bi t allocatio n reduce s t o
^=
^ - ^ og 2 « ) + ^ i o
g 2
(n^
U7)
More bit s ar e assigne d t o subchannel s wit h a smalle r a 2 (good subchannels) and fewe r bit s ar e assigne d t o subchannel s wit h a large r a\ (bad subchannels). Note t h a t i n thi s cas e th e optima l bi t allocatio n i s suc h t h a t £ s^ = T2 2hka2k are equalize d fo r al l k; al l th e subchannel s hav e th e sam e signa l powe r £ s?fc. In othe r words , whe n th e bit s ar e allocate d optimally , no power allocation is needed. T h e bi t allocatio n gai n i s give n b y
Qut
Mj_ k=0 ' < M l/M * M-1 ( k=o (* ;
n
E
^.18)
T h e gai n Q^it i s equa l t o th e rati o o f th e A M ove r th e G M o f th e outpu t nois e variances a 2k. A s th e A M i s alway s large r t h a n equa l t o th e GM , th e gai n can neve r b e smalle r t h a n on e an d i t i s equa l t o on e i f an d onl y i f al l th e subchannels hav e th e sam e nois e variance . T h e gai n come s fro m th e disparit y among o\ . Fo r transmissio n scenario s t h a t hav e a larg e differenc e i n th e o u t p u t nois e variances , thi s gai n ca n b e substantial , a s w e shal l demonstrat e later. Als o observ e t h a t th e bi t allocatio n gai n i s independen t o f th e bi t rat e per symbo l b an d th e SN R ga p T , whic h i s determine d b y th e erro r rate . Moreover i f al l th e variance s o\ k ar e scale d b y a commo n factor , th e gai n remains th e same . 3
Note tha t i n a SC-C P system , th e las t v symbols , SMit, the cycli c prefix . Fo r thi s reason , [GjGo]fef e o f th e firs t (M
...,SM-1J
ar e
duplicate d a s
v) subchannel s ar e differen t from thos e o f th e las t v subchannels . Thi s wil l b e elaborate d later .
232 8
. Transceive r desig n wit h channe l informatio n a t th e transmitte r
OFDM an d single-carrie r system s W e no w analyz e th e syste m perfor mance whe n th e optima l bi t allocatio n i s applie d t o th e previousl y studie d O F D M an d single-carrie r system s (SC-C P an d SC-ZP) . Le t u s assum e t h a t the channe l nois e q(n) i s whit e wit h varianc e A/o - Firstl y le t u s loo k a t th e O F D M system . Fro m Chapte r 6 , w e kno w t h a t th e kth outpu t erro r vari ance o f b o t h th e C P - O F D M syste m an d Z P - O F D M syste m (wit h th e efficien t receiver i n (6.15) ) hav e th e for m 2 -M
(JPl =
a
j27rk M
)
W
where C k = C(e / )i s th e fcth D F T coefficien t o f th e channe l c(n). T h e parameter a i s a constan t equa l t o on e fo r th e C P - O F D M syste m an d equa l to N/M fo r th e Z P - O F D M system . Fro m (8.17) , w e ge t th e optima l bi t allocation a s bk = 6 + log 2 (|C f c |) - ^ l o g
2
( n ^ l Q Q.
T h e las t t e r m i s independen t o f A: . S o mor e bit s ar e allocate d t o subchannel s with a large r gain . Th e correspondin g bi t allocatio n gai n i s yofdm
_ M M 2^k ^k= 0 \C
Uk=0 {\C
k\
k\2j
1/M '
It i s equal t o th e rati o o f AM ove r G M o f l/\Ck\ 2- Th e gai n satisfie s Gofdm > 1 with equalit y i f an d onl y i f \Ck\ ar e th e sam e fo r al l k. W h e n th e channe l i s highly frequency-selective , th e quantit y l/|C/e| 2 ca n var y greatl y wit h respec t to k an d th e bi t allocatio n ca n improv e th e performanc e significantly . Als o note t h a t Qofdm i s no t necessaril y a n increasin g functio n o f M. I t i s no t difficult t o construc t example s wher e th e bi t allocatio n gai n decrease s whe n the numbe r o f subchannels , M , increases . A s M approache s infinity , on e ca n derive th e asymptoti c bi t allocatio n gain . Fo r a ver y larg e M , th e A M ca n b e approximated a s
l^
M - l2
7T
r*
l
duo
l
Similarly, w e ca n writ e th e G M a s M-l/
M-
l
k=0 \
=k 0
/
II (Vl^| 2 )^= exp - ^ l n l / | C
fc|
2
W h e n M i s large , w e hav e th e followin g approximation :
n
M l
~ <
fe=0
i
\
^/
JC*| V *U
exp /
2 \I I n o |CV'
r
If w e defin e th e rati o exp ^J 0 I n | C(eJ-W)|2 2K) 2TT
Jo
1
dcv
| C ( e ^ ) | 2 2T T
dco w
)l227r
233
8.3. Optima l bi t allocatio n
1/|C 0 (e i<0 )| 2
-
-
0.2 0.
4 0. 6 0. Frequency normalize d by7i.
Figure 8.4 . Plot s o f l/\C 0(eJUJ)\2 an Ci(z) = -± E(l + 0.95z- 1).
d l/\d(e JUJ)\2, wher
8
e Co (z) = Kl +
lz' 1) an
d
then th e bi t allocatio n gai n Gofdm asymptotically approache s l/£ . Th e quan tity £ i s a measur e o f th e flatnes s o f th e curv e l/\C(e juJ)\2 [159 , 164] . W e have £ = 1 when l/|C(e j a ; )| 2 i s increasingly fla t an d i t become s smalle r whe n l/|C(e j a ; )| 2 become s nonflat . I n othe r words , a smalle r £ implie s tha t th e channel c(n) i s more frequency-selective . Fo r highl y frequency-selectiv e chan nels, £ ca n b e ver y smal l an d th e asymptoti c gai n ca n b e ver y large . Fo r example, conside r th e tw o channel s C0(z) =
- ( 1 + 2Z' 1) an
1 (1 + 0.95Z- 1 ). 1.95
d d(z)
The flatnes s measur e i s given b y £ = 0.7 5 fo r l/\C 0(ejuJ)\2 an d £ = 0.097 5 fo r l / | C i ( e ^ ) | 2 . W e ca n als o se e fro m Fig . 8. 4 tha t l/\C 0(ejuJ)\2 i s muc h flatte r than l/|Ci(e j a ; )| 2 . A s a result , th e asymptoti c gai n fo r C\(z), whic h i s equa l to 10.2 6 (abou t 1 0 dB), i s muc h large r tha n tha t o f Co(z), whic h i s onl y 1.3 3 (about 1. 2 dB) . Secondly, let u s consider the single-carrier systems . Fo r the SC-C P system , the kth outpu t erro r varianc e o\ i s th e sam e fo r al l k. Th e transmittin g matrix G o o f the SC-C P syste m i s 0\ Go
\M-V
v
0
0L
,
234 8
. Transceive r desig n w i t h channe l informatio n a t th e transmitte r
It ca n b e verifie d t h a t th e quantitie s [GjGo]fcf c satisf y [G0G0]/c/c = < 2
for 0 < k < M - v\ for M - u < k < M.
U9)
Substituting thes e result s int o (8.15) , w e find t h a t whe n th e bit s ar e allocate d optimally, th e first ( M — u) subchannel s ar e assigne d th e sam e numbe r o f bit s and th e las t u subchannel s ar e assigne d th e sam e numbe r o f bits . Moreove r the numbe r o f bit s assigne d t o th e las t u subchannel s ar e 0. 5 bit s fewe r t h a n t h a t assigne d t o th e first ( M — u) subchannels . Substitutin g (8.19 ) an d th e fact t h a t al l a 2 ar e th e sam e int o th e bi t allocatio n gai n formul a i n (8.16) , we hav e 2 - / ^ ( 1 + z//M) . Observe t h a t th e gai n Q sc.cv i s alway s large r t h a n one . Ther e i s bi t allocatio n gain eve n thoug h o\ k i s th e sam e fo r al l k. Th e gai n i s du e t o th e disparit y of [GgGoj/e/ e rathe r t h a n a^. I n Fig . 8.5 , w e plo t th e gai n agains t u/M fo r 0 < u/M < 1 . I t i s see n t h a t th e gai n decrease s monotonicall y t o on e a s u/M decrease s t o zero . Th e gai n i s Gsc-cp = 1.49 , 1.23 , an d 1.11 , respectively , for u/M = 1/4 , 1/8 , an d 1/16 . I n practice , th e rati o u/M i s usuall y a smal l number fo r th e reaso n o f spectra l efficiency . I n thi s case , th e gai n i s negligible . T h u s bi t allocatio n i s no t use d i n th e SC-C P system . Th e sam e i s als o tru e for th e SC-Z P system .
0.4 0.
v/M
6
Figure 8.5 . Bi t allocatio n gai n o f th e SC-C P system .
235
8.3. Optima l bi t allocatio n
Optimal bi t allocatio n subjec t t o positivit y I n th e optima l bi t allocatio n formulas, th e right-han d sid e o f (8.15 ) an d (8.17 ) coul d b e negative . W h e n some bj i s negative , on e ca n se t i t t o zero . Thi s i s consisten t wit h th e K a r u s h Kuhn-Tucker (KKT ) optimalit y condition s an d bk remain s optima l subjec t to positivit y [16] . Belo w w e wil l sho w ho w t o d o thi s fo r th e cas e o f equa l [GQGO]/C/C. Recal l t h a t th e optima l bi t allocatio n formul a i s give n b y (8.17) . We ca n rewrit e th e optima l bi t allocatio n a s bk = D —log2 (Jk for som e constan t D. B y settin g th e negativ e bit s t o zero , w e hav e
K
K
D — log2 cr e/e, i f thi s i s positive ; otherwise.
$.20)
T h e constan t D i s chose n suc h t h a t th e followin g bi t constrain t i s satisfied : M-l
E M k=0
6
^
b.
T h e abov e formula s hav e a beautifu l water-fillin g interpretation . I t i s lik e pouring wate r int o a t a n k wit h a n uneve n floor log 2 cr ek (se e Fig . 8. 6 fo r a n example). T h e numbe r o f bit s bk allocate d t o eac h subchanne l i s suc h t h a t the wate r heigh t D = b+ + log 2 a ek is a constant . Equatio n (8.20 ) say s t h a t whe n log 2 (cr e/e ) i s large , w e "pour " fewer bit s int o t h a t subchanne l an d vic e versa . I f fo r som e subchanne l th e b o t t o m o f th e t a n k log 2 a ek i s to o hig h (log 2 a e3 i n Fig . 8.6) , the n n o bi t i s allocated t o t h a t subchanne l (6 3 = 0 i n th e figure).
a.
water level D
M-\
a, a, oti
ak= l o g
ae
a M-i
7 c 2~ Jt
Figure 8.6 . Water-fillin g interpretatio n o f optima l bi t allocation .
Algorithm 8.1 . Non-negativ e intege r bi t allocatio n T h e optima l bi t allocation formul a i n (8.17 ) ma y no t yiel d a non-negativ e intege r bk- T o obtain a non-negativ e intege r bi t allocatio n fo r PA M symbols , on e ca n follo w the followin g procedure .
236
8. Transceive r desig n wit h channe l informatio n a t th e transmitte r
(1) Se t negativ e bk t o zer o an d roun d eac h bk t o th e neares t integer . Le t the resultin g non-negativ e intege r b e bk. (2) Comput e th e bi t rat e b = 1 / M ^ T ^ " 1 b k. Le t p = M(b - b). (3) I f p = 0, the n bk i s th e solution . (4) I f p < 0 , the n find th e p subchannels correspondin g t o th e p smallest
r22^e2jGjG0l LJ
kk
For eac h o f thes e p subchannels, increas e th e correspondin g bk b y one . T h e ne w bk i s th e solution . (5) I f p > 0 , the n find th e p subchannels correspondin g t o th e p largest T22lk(j;
2
G£G0
Jfcfc
where bk > 1. Fo r eac h o f thes e p subchannels , decreas e th e correspond ing bk b y one . Th e ne w bk i s th e solution . In [167] , th e author s propos e a n efficien t algorith m fo r optima l bi t allocatio n of a non-negativ e integer . Moreover , th e complexit y o f the propose d algorith m is independen t o f th e tota l numbe r o f bits . Note t h a t whe n bi = 0 fo r som e i , n o informatio n bi t i s t r a n s m i t t e d i n th e i t h subchannel . I t is intuitiv e t o se t £ s^ =0. Therefor e th e signa l powe r o f the /ct h subchanne l i s
£sk =
[ T22lk
f ^ 0;
\ 0 , otherwise
.
T h e t r a n s m i t t e d powe r become s Eut — l / ^ 5 ^ i =o ^sA^o^^u an d th e cor responding gai n i s Q^t = £no-bit/£bit- A s th e lowe r boun d S^t may no t b e achievable whe n bk ar e restricte d t o b e non-negativ e integers , i t follows t h a t £>ut ^ £bit a n d Qut < Gut- Fro m previou s discussions , w e kno w t h a t th e theoretical gai n Qut i s independen t o f th e bi t rat e b and th e symbo l erro r rat e SER. Th e sam e i s no t tru e fo r Qut- Se e Proble m 8. 1 fo r a n example . We ca n modif y Algorith m 8. 1 fo r QA M symbols . I n thi s case , th e bi t adjustment i n Step s 4 and 5 is applie d onl y t o p / 2 subchannel s bu t bk ar e increased (o r decreased ) b y 2 bits . Example 8. 1 Conside r a zero-forcing transceive r wit h tw o subchannel s (tha t is, M = 2) . Suppos e t h a t [ G ^ G 0 1= [G^G LJ oo L of th e tw o subchannel s b e 2_
^
2
0
li
= 1 . Le t th e nois e variance s
_ ^
for som e 0 < a < 1 . S o th e rati o an d th e su m o f th e tw o nois e variance s ar e 1/a2 an d 1 , respectively . Th e theoretica l bi t allocatio n gai n i s therefor e give n by l + a2 Qut = —^ = 0.5(a+l/a) .
8.3. Optima l bi t allocatio n
237
Observe t h a t th e gai n Gut ha s a minimu m o f on e whe n a = 1 , i n whic h cas e the tw o subchannel s hav e th e sam e nois e varianc e o f 1/2 . As a decreases , the disparit y o f th e tw o nois e variance s increase s an d th e gai n increases . T o see ho w th e bit s ar e allocated , le t u s conside r fr = 3 . I n Tabl e 8.1 , w e lis t the optima l bi t allocatio n an d th e correspondin g gai n fo r b o t h th e theoretica l optimal cas e an d th e suboptima l cas e o f non-negativ e intege r bi t allocatio n a s obtained b y Algorith m 8.1 . A s w e ca n se e fro m th e table , whe n a decreases , more bit s ar e assigne d t o Subchanne l # 1 . Thi s i s consisten t wit h th e fac t t h a t Subchannel # 1 i s increasingl y bette r t h a n Subchanne l # 0 a s a approache s zero. Not e t h a t , althoug h th e gai n fo r th e suboptima l cas e i s smaller , th e gai n can stil l b e quit e substantia l eve n whe n th e hig h bi t rat e assumptio n i s no t valid (th e averag e bi t rat e fr i s onl y 3 i n thi s example) . ■
a2
fro, h
1
3, 3
Gbu 1
fro, fri 3, 3
Gbu 1
IO- 1
2.17, 3.8 3
1.74
2,4
1.69
2
1.34, 4.6 6
5.05
1,5
4.54
0.51, 5.4 9
15.83
1,5
12.75
ioio-3
Table 8 . 1 . Exampl e 8.1 . Compariso n o f theoretica l an d non-negativ e intege r bi t allocations
Example 8. 2 I n thi s example , w e demonstrat e th e effec t o f bi t allocatio n o n the B E R performanc e o f C P - O F D M system . T h e followin g tw o channel s ar e considered:
C0(z) = d(z)=
l + 2z-\
1 + 0.95Z- 1 .
T h e numbe r o f subchannel s i s M = 6 4 an d th e C P lengt h i s v = 1 . T h e average bi t rat e pe r symbo l i s fr = 4 . T h e modulatio n symbol s Sk ar e QA M symbols wit h Gra y cod e mapping . T h e symbo l Sk satisfie s th e conjugat e symmetric propert y i n (6.33 ) s o t h a t th e transmitte d signa l ha s rea l value . Non-negative intege r bi t allocatio n i s applie d (Algorith m 8.1) . T h e transmit ted powe r i s £ an d th e channe l nois e i s assume d t o b e whit e an d Gaussia n with varianc e A/o - S o th e SN R i s S/Afo. Figur e 8.7(a ) show s th e numbe r o f bits allocate d t o th e subchannels . Du e t o th e conjugat e symmetri c property , the kth an d ( M — k)th subchannel s ar e allocate d th e sam e numbe r o f bits . From th e plot , on e realize s t h a t fo r CQ(Z) al l th e subchannel s hav e th e sam e fr/c becaus e thi s channe l i s fairl y flat. O n th e othe r hand , C\(z) i s a lowpas s filter; it s gai n i n th e high-frequenc y regio n i s muc h smalle r t h a n t h a t i n th e low-frequency region . A s a result , n o bi t i s transmitte d ove r th e 31st , 32nd , and 33r d subchannels , whic h correspon d t o th e frequenc y bin s centere d a t 7T — 7r/32 , 7T , and n + 7r/32 , respectively . Figure s 8.7(b ) an d (c ) sho w th e B E R result s fo r CQ(Z) an d Ci(z), respectively . Fo r th e purpos e o f compari son, w e hav e als o include d th e B E R curv e fo r th e zero-forcin g SC-C P syste m
238 8
. Transceive r desig n wit h channe l informatio n a t th e transmitte r
with n o bi t allocatio n i n thes e figures (a s M = 6 4 is much large r tha n v = 1 , the bit allocatio n gai n is negligible for the SC-C P system) . Fro m these figures, we observ e that , fo r Co(z), th e SC-C P syste m ha s a bette r performanc e fo r a moderat e BER , a s explaine d i n Chapte r 7 . Fo r C\(z), th e bi t allocatio n significantly improve s th e BE R performanc e o f the CP-OFD M system . Fo r a BER o f 1(T 6 , th e gai n i s mor e tha n 1 0 dB. It i s interestin g t o not e tha t fo r highl y frequency-selectiv e channels , the CP-OFDM system outperforms the SC-CP system when bit allocation is applied. Moreover , th e bi t allocatio n gai n ca n b e quit e substantial . ■ Under optima l bi t allocation , th e minimize d transmitte d powe r Eut de pends o n [GjGo]fcf c an d a^ k, wher e th e forme r i s determined b y the transmit ter an d th e latte r i s related t o the receiver . I n the following , w e will show how to choos e th e transmitte r an d receive r s o that Eut i s minimized. Becaus e th e ZP and Z J systems (Figs . 8.2 and 8.3 ) hav e different structures , w e will discuss the optimizatio n o f thes e system s separatel y i n th e followin g tw o sections .
239
8.3. Optima l bi t allocatio n
(a)
'3
0 1
(b)
0 2
0 3 0 4 0 5 tone index
0 6
0
tr
UJ DO
- e — SC-C P - B CP-OFD - V CP-OFD
10"
M M wit h bi t allocatio n
51 01 SNR (dB)
52
0
10u
10" DC
(c)
UJ DO
10" - SC-C P - CP-OFD M - V CP-OFD M wit h bi t allocatio n
10"
10 2 0 SNR (dB)
Figure 8.7 . Exampl e 8.2. (a ) Bi t allocation ; (b ) BE R for C 0(z) = 1 + 2 * - 1 ; (c ) BE R for Ci(z) = 1 + 0.95Z" 1.
240 8
. Transceive r desig n wit h channe l informatio n a t the transmitter
8-4 Optima
l Z P transceiver s
A zero-forcin g Z P system ca n be implemented usin g t h e structure i n Fig. 8.2. T h e fre e parameter s i n thi s syste m ar e a n M x M invertibl e matri x G zp a t the transmitte r a n d a n arbitrar y M x v matri x A zp a t t h e receiver . I n t h e following, w e will sho w ho w to choos e G zp a n d A zp s o t h a t t h e t r a n s m i t t e d power Sbu i n (8.13 ) i s minimized . Fo r a give n A zpi w e deriv e t h e optima l Gzpi base d o n whic h A zp i s optimized .
8.4.1 Optima
lG
zp
We firs t expres s t h e t r a n s m i t t e d powe r Sbu i n (8.13 ) i n term s o f G zp. T write t h e quantit y o\ k i n term s o f G zp, w e define t h e M x l vecto r
o
where t h e matrix B ^ p is t he left invers e o f Ci ow give n i n (8.4) . Usin g t h e fac t t h a t S zp = G~p~B zp, w e can write e = S^ p q^ p = G~ p0. T h e autocorrelatio n matrix o f t he outpu t nois e vecto r e i s give n b y Re =
G~ pR<9G~pT,
where R # i s t he autocorrelatio n matri x o f t he vecto r 6. T h e quantit y o\ k i s the (/c , k)th entr y o f R e : a
lk =
[
G
* p lR 0 G *p]fcfc-
From (8.13) , w e hav e M X
TM (
/M
~ V
\k=0 )
Applying t h e H a d a m a rd inequalit y (Appendi x A ) to t he two positive definit e matrices G ^ p G ^ p a n d G ^ R ^ G " ^ , w e ge t £ut >
™22
b
(de t ( G t p G , p ) de t ( G - ^ Q - t ) ) 1 ^ .
Because t h e matri x G zp i s square, w e can simplify t h e right-han d sid e o f t he above expressio n t o obtai n £ut > ^ 2 2 f e ( d e t R , ) 1 / M ^ £ T h e equalit y hold s i f a n d onl y i f t h e matri x G conditions: (i) Gl pGzp i
s diagonal ;
(ii) G ^ R f l G " ^ i s diagonal .
ut>Gzp.
zp
(8.21
)
satisfie s t h e followin g tw o
241
8.4. Optima l Z P transceiver s
Note t h a t th e lowe r boun d £bit,G zp doe s no t depen d o n th e matri x G zp becaus e 0 i s independen t o f G zp. Thi s lowe r boun d i s achieve d i f an d onl y i f G zp satisfies b o t h condition s (i ) an d (ii) . T h e first conditio n implie s t h a t th e columns o f G zp ar e orthogonal , wherea s th e secon d conditio n mean s t h a t G ^ 1 decorrelate s th e nois e vecto r 0. A s Ho i s positiv e semidefinite , i t i s always diagonalizabl e b y som e unitar y matrix . Le t u s decompos e th e matri x R<9 a s R<9 = QzpEQtp, where Q zp i s a n M x M unitar y matri x whos e column s ar e th e eigenvector s of HQ an d T, i s a diagona l matri x consistin g o f th e correspondin g eigenvalues . Combining condition s (i ) an d (ii) , w e conclud e t h a t G zp i s a matri x o f th e form G zp = Q^pD , wher e D i s a n arbitrar y invertibl e diagona l matrix . Sinc e Gzp = Qzp D achieve s th e lowe r boun d fo r an y invertibl e D , withou t los s o f generality w e ca n choos e D = I an d w e ca n choos e th e matri x G zp a s ^ zp ^Izp'
W i t h thi s choic e o f G zp an d usin g th e fac t t h a t S zp = the receivin g matri x S zp a s Qi pV,PA-1[lM A
S.p =
zp]\Jtp.
G ^ B ^ , w e ca n expres s (8.22
)
Observe t h a t whe n G zp i s chose n optimally , th e autocorrelatio n matri x o f th e o u t p u t nois e vecto r e become s R
e=
G~pIloG~J=
S .
T h a t is , E[eie]:] = 0 fo r i ^ j ; the outpu t nois e component s ar e uncorrelated . In summary , th e optima l transmittin g matri x G zp i s a unitar y matri x whos e inverse decorrelate s th e outpu t nois e vector . Note t h a t th e transmitte d powe r i s equa l t o £bit,G zp whe n th e matri x G zp is chose n a s Q zp an d th e bit s ar e optimall y allocate d accordin g t o th e formul a in (8.17) . T h e quantit y £bit,G zp depend s o n Ho- Recal l t h a t 0 = ~B zpq.zp, where ~B zp depend s o n A zp a s give n i n (8.4) . Nex t w e ar e goin g t o sho w ho w to choos e A zp s o t h a t £bit,G zp i s minimized .
8.4.2 Optima
lA
zp
It follow s fro m (8.21 ) t h a t minimizin g £bit,G z ls equivalen t t o minimizin g d e t ( R ^ ) . T o deriv e th e optima l A zp t h a t minimize s d e t ( R ^ ) , le t u s decompos e the N x N unitar y matri x XJ zp i n (8.3 ) a s [V
Vzp=
0
U i ] , (8.23
)
where U o an d U i consis t o f th e first M an d th e las t v column s o f U z p , respectively. Usin g thi s decomposition , w e ca n redra w th e receive r a s Fig . 8.8, i n whic h w e sho w onl y th e nois e component . Le t th e nois e vector s ^ 0 , /!]_, p 0 , an d r be , respectively , define d a s i n Fig . 8.8 . T h e n th e autocorrelatio n matrices o f 0 an d r ar e relate d b y Re =
V
zpRTVt
242 8
. Transceive r desig n w i t h channe l informatio n a t th e transmitte r ^ v
IJF
W T
v v
zp
e
w ~Q^
Figure 8.8 . Nois e pat h a t th e receive r o f a Z P system .
As V ^ p i s unitary , w e hav e det(R# ) = det(Rr) =
d e t ( R r ) . I t follow s fro m Fig . 8. 8 t h a t
det(A~ 2)det(RPo),
where H po i s th e autocorrelatio n matri x o f p 0 . A s th e matri x A - i s inde pendent o f A zp, minimizin g d e t ( R r ) (an d henc e det(R#) ) i s equivalen t t o minimizing det(R / 0 o ). T o solv e suc h a problem , le t u s defin e th e v x 1 vecto r p l 5 th e ( M + i / ) x l vector s p , an d p , respectively , a s Po Pi
Pi=Pi,P
, an d p
Then, usin g th e fac t t h a t p 0 = p 0 + ^zpPii w IM A
e hav e
Z
0
Po Pi
P,
I,
which implie s t h a t th e autocorrelatio n matrice s o f p an d p ar e relate d b y Hn
J-M
R/y
0
J-M
0
3.24)
Moreover, fro m th e definition s o f th e vector s p , p 0 , an d p l 5 w e kno w t h a t th e autocorrelation matri x o f p ca n als o b e writte n a s H0
HPo R Rt
•poPi
H0
where R PoPl= ^ [ p 0 P i ] i s th e cross-correlatio n matri x o f p 0 an d p x. Usin g the Fische r inequalit y (Appendi x A ) fo r positiv e definit e matrices , w e hav e det(RP0) >
det(Rp) det(RPl)'
with equalit y i f an d onl y i f ~R PoPl= 0 . Usin g (8.24 ) an d th e fac t t h a t p = U £ q ^ , w e hav e det(R / 0 ) = d e t ( R M ) = det(H qzp) sinc e th e matri x
8.4. Optima l Z P transceiver s
243
XJzp i s unitary . Becaus e p 1 = / i 1 ? we have d e t ( R P l ) = relations, t h e abov e equatio n ca n be expresse d a s
d e t ( R / X l ) . Usin g thes e
det(Ra )
Because b o t h q_ zp and \i x ar e independen t o f A z p , t h e abov e lowe r boun d i s also independen t o f A zp. Moreover , thi s lowe r boun d ca n b e obtaine d i f and only i f A zp i s chose n suc h t h a t KP0P1 =
E[(p 0 +
A^MiWi ] =
0 . (8.25
)
Solving t h e abov e equation , t h e optima l A zp i s uniquel y give n b y Azp=
- U j R ^ Ui (u{R
gzpUi)
_1
. (8.26
)
It ca n be show n (Proble m 8.4 ) t h a t whe n A zp i s chose n a s above , t h e matri x — A z p i s t h e optima l estimato r o f p 0 give n t h e observatio n o f \i x. I n fact , the solutio n o f A zp i n (8.26 ) minimize s no t onl y d e t ( R ^ ) (whic h i s als o equa l to d e t ( R e ) becaus e Q zp i s unitary) , b u t als o t h e tota l outpu t nois e powe r given b y X)fc= o a lk o r t race(^e)- T o se e this , w e conside r Fig . 8.8 . Sinc e the vector s e an d r ar e relate d throug h t h e unitar y matri x Q zpVZp, whic h preserves traces , w e hav e trace(Rr) =
trace(R e).
On t h e other hand , becaus e A i s diagonal, t h e diagonal entrie s o f R r an d R p o are relate d a s [KT]kk = [H Po]kk/[A]kk. Thus t h e minimizatio n o f trace ( R r ) ca n be achieve d i f each t e r m [ R ^ J / ^ ca n be individuall y minimized . A s —A zp i s t he bes t estimato r o f ^ 0 , t h e kth ro w of — Azp minimize s [RpJ/c/ c fo r 0 < k < M. Therefore , t h e solutio n o f A zp given i n (8.26 ) i s also optima l fo r minimizing t h e total outpu t nois e trace (Re).
8.4.3 Summar
y an d discussions
In t h e following , w e summarize t h e desig n procedur e fo r t h e optima l Z P sys tem. (1) For m t h e N x M matri x Ci ow i n (8.2 ) an d comput e it s SV D Ci ow= Uz P [A 0 ] T V j p . Partitio n t h e unitar y matri x XJ zp a s (8.23) . (2) Give n t h e noise autocorrelatio n matri x ~R qzp, calculat e t h e optimal M x v matrix A zp i n (8.26) . (3) Comput e t h e matri x ~B zp in (8.4) . Fin d a unitar y matri x Q zp suc h t h a t the matri x Ql p3zpRqzp'3l Q zp i s diagonal . T h e optima l transmittin g matrix i s G zp = Q zp(4) T h e optima l receivin g matri x i s S zp =
Q\
pB2p.
244 8
. Transceive r desig n wit h channe l informatio n a t th e transmitte r
(5) Th e optima l bi t allocatio n i s give n b y (8.17 ) o r b y Algorith m 8.1 . T h e minimu m t r a n s m i t t e d powe r o f th e optima l Z P transceive r i s give n by
f-
^zp,opt
™o2 & d e t ( A ^ ) N
det(R9J
^ ^
-\ 1 /M
—.
det(U t 1 R^ p U 1 )J
(8.27
)
Some remark s o n th e optima l Z P system s ar e liste d below . (1) A s th e optima l G zp= Q zp i s unitary , th e transmittin g matri x G o = [Glp 0 ] T satisfie s G J G 0 = I M 5 i-e. th e column s o f G o ar e orthonor mal. Thi s mean s t h a t th e modulatio n symbol s si ar e t r a n s m i t t e d usin g orthonormal vectors . Th e optima l Z P transceive r ha s a n orthonor ma I transmitter . Ther e i s n o los s o f generalit y i n usin g orthonorma l transmitters. (2) A s [GJGOU/ C = 1 fo r al l k, fro m (8.14 ) w e ca n conclud e t h a t al l modu lation symbol s Sk hav e th e sam e signa l powe r £ sy, n o powe r allocatio n is neede d fo r th e optima l Z P system . (3) Whit e nois e case . W h e n th e channe l nois e q(n) i s white, th e autocorre lation matri x o f th e nois e vecto r q zp become s ~R Qzp = A/oI . I n thi s case , the optima l A zp give n i n (8.26 ) become s
Azp =
-UjU i (uJUi ) ' = 0 ,
where w e hav e use d U Q U I = 0 . Substitutin g A zp= have Bzp= VzpA- 1 [ I M 0 ] Ut p.
0 int o (8.4) , w e
One ca n verif y t h a t B ^ p i s i n fac t equa l t o (Ci owCiow)~1Ciow, th e pseudo-inverse o f Ci ow. Th e nois e vecto r 0 ha s th e autocorrelatio n matrix He=N-0(Clw= Clow)-1 V ^ A t A ) " ^ . As th e optima l G zp i s suc h t h a t G " 1 decorrelate s R# , w e ca n choos e Gzp= V ^ p . Therefore , whe n q(n) i s white , th e optima l transmittin g and receivin g matrice s simplif y t o Gzp =
V zp, S ,
P
=
A - 1 [ I M 0]U t . (8.28
A V | p , w e ca n se e ho w th 0 optimal transceive r works . Th e transmittin g matri x G zp cancel s th effect o f V | i n Ci ow, wherea s th e receivin g matri x S zp nullifie s th effects o f XJ zp an d A i n Ci ow. Th e minimize d t r a n s m i t t e d powe r i given b y
From th e SV D expressio n Ci
ow =
U zp
szp,opt = ^rN22bAfo t
) e e e s
det A
( "
It ca n b e show n t h a t th e diagona l matri x A i s independen t o f th e phas e of C ( e j a ; ) [163] . Thi s implie s t h a t th e minimize d t r a n s m i t t e d powe r i s
8.4. Optima l Z P transceiver s
245
independent o f th e phas e o f C ( e j a ; ) ; th e channe l phas e doe s no t affec t the performanc e o f th e optima l Z P syste m whe n th e nois e i s white . For example , i f C(z) ha s th e facto r ( 1 — 0 . 5 z _ 1 ) an d w e replac e i t b y (0.5 — £- 1 ) , th e performanc e o f th e optima l transceive r remain s th e same. (4) W h e n th e channe l nois e q(n) i s whit e an d th e channe l c(n) i s frequency nonselective, th e optima l transceive r i s no t uniqu e (se e Proble m 8.8) . One o f th e solution s i s give n b y XJ zp = IN an d \ z p = I M - I n thi s case , the optima l Z P transceive r reduce s t o th e SC-Z P syste m wit h G zp = IM and S zp = [I M 0] . Example 8. 3 Optima l Z P systems . I n thi s example , w e evaluat e th e perfor mance o f th e optima l Z P transceive r i n a n ADS L environment . T h e channe l C(z) i s th e equivalen t channe l afte r a time-domai n equalize r (TEQ ) i s use d to shorte n DS L Loo p 6 [7] . T h e T E Q i s a fourt h orde r MSSN R T E Q an d the shortene d channe l ha s orde r v = 4 . T h e nois e i s colore d an d i t include s AWGN, N E X T an d F E X T [7] . T h e magnitud e respons e o f th e channe l C(z) and th e powe r spectru m o f the colore d nois e q(n) use d ar e shown , respectively , in Fig . 8.9(a ) an d (b) . T h e symbo l erro r rat e i s SER = 1 0 - 6 an d th e averag e bit rat e pe r sampl e i s R^ = bM/N = 2 . W e compar e th e transmitte d power s of Z P system s optimize d unde r thre e differen t scenarios : (i ) th e optima l Z P system; (ii ) th e Z P syste m optimize d unde r th e constrain t A zp= 0 ; (hi ) th e ZP syste m optimize d unde r th e assumptio n t h a t th e nois e i s whit e althoug h the actua l nois e i s colored . T h e thre e transmitte d power s ar e respectivel y denoted b y £ zp,0pt, £zp, A= o, an d £ zPiWh- Figur e 8.1 0 show s th e results . A s expected, th e transmitte d powe r o f the optima l Z P syste m £ zPj0pt i s th e small est. B y comparin g £ zPj0pt an d £Z P ,A= O? w e s e e t h a t , b y optimizin g A zp, w e get a n improvemen t o f aroun d 2 d B fo r M = 1 0 an d 0. 5 d B fo r M = 50 . Thi s gain reduce s a s th e numbe r o f subchannel s M increase s because , fo r larg e M , the dimensio n M o f th e signa l subspac e i s almos t a s larg e a s th e dimensio n ( M + v) o f th e receive d signal . Als o observ e fro m th e figure t h a t £ zp,opt i s much smalle r t h a n £ zP:Wh- T h e gai n ca n stil l b e a s larg e a s 5 d B fo r M = 64 . Thus, b y exploitin g th e nois e correlation , w e ca n significantl y improv e th e system performance . ■
246 8
. Transceive r desig n w i t h channe l informatio n a t th e transmitte r
(a)
0.2 0.
4 0. 6 0. Frequency normalize d by7t.
8
(b) Figure 8.9 . Exampl e 8.3 . (a ) Magnitud e respons e \C{e ju)\ o channel fro m a n ADS L environment ; (b ) th e nois e spectrum .
f a TE Q shortene d
8.5. Optima l zero-jammin g (ZJ ) transceiver s
6 5 ^_^ E
wer (d
CD
o Q. "D
rans
"E
111
11
— zp,whit e I zp,A=0 zp,opt
/A
4 "\
w'\A
3
\ / \ ^ . \A
2 1S
CD
-t—< -t—<
11
247
\
0
\
\
\ \
-1
-i—'
-2
^ ^ ^ \ ^^
^
~~ ~ -
-3 -4 10 1
II
III
52
I
02
53
03
I
54
M
04
55
0
Figure 8 . 1 0 . Exampl e 8.3 . Compariso n o f transmitte d powe r fo r thre e differen t Z P systems.
8.5 Optima
l zero-jammin g (ZJ ) transceiver s
In thi s section , w e deriv e th e optima l zero-forcin g Z J system . Th e bloc k diagram o f a Z J syste m wa s show n i n Fig . 8.3 . Recal l fro m Sectio n 8.1. 2 that th e fre e parameter s tha t ca n b e designe d i n a zero-forcing Z J syste m ar e an M x M invertibl e receivin g matri x S zj an d a v x M matri x A zj a t th e transmitter. Fo r a give n A^- , w e will deriv e th e optima l S zj, base d o n whic h Azj i s optimized .
8.5.1 Optima
lS
ZJ
We wil l firs t expres s th e transmitte d powe r Sut i n (8.13 ) i n term s o f S^- . A s e = Szj-qzj , th e nois e varianc e a\ o f the M h subchanne l i s give n b y ^zj^qzjSzj And sinc e G o = G zj =
Jfefc
B^- S • , we hav e
kk
Szj &zj~BzjS zj
kk
248 8
. Transceive r desig n wit h channe l informatio n a t th e transmitte r
Substituting thes e result s int o (8.13) , w e ge t
kk
\k=o
SzjRqzjSzj
Since bot h th e nois e autocorrelatio n matri x R gzj. an d B^-B^ - ar e positiv e semidefinite, w e ca n appl y th e Hadamar d inequalit y t o obtai n £m > ™ 2 2 b ( d e t l S j / B t - B ^ S j / l d e t ^ - ^ S t . ] )
1
^.
Because th e matri x S^ - i s square , th e abov e expressio n ca n b e writte n a s Sbit > ™ 2 2 f e ( d e t p ^ . B ^ ] det[R , j) ^ =
£ bit,sZ3 • (8.29
)
Note tha t th e lowe r boun d £bit,s zj i s independen t o f th e receive r S ^ an d i t is achieved i f an d onl y i f the matri x S^ - simultaneousl y satisfie s th e followin g two conditions : (i) SzjKq^Slj i
s diagonal ;
(ii) S ^ B ^ B ^ S ^ 1 i s diagonal ; tha t is , G^-G^ - i s diagonal . The first conditio n mean s tha t th e receive r S^ - decorrelate s th e nois e vecto r q^j, wherea s th e secon d conditio n implie s tha t th e transmitte r G zj i s or thogonal. T o deriv e th e optima l S^ - tha t satisfie s bot h conditions , w e first decompose th e positiv e definit e matrix 4 R gzj. a s Rfor som e positiv e definit e matri x H
R
1/2
R1/2
1/2 1/
Qzj.
On
1/2
e wa y t o identif y Hq
2 zj
i s th e
Cholesky decomposition . Th e matri x Hq zj i s als o know n a s th e Hermitia n square roo t o f R q • . Usin g th e abov e decomposition , i t ca n b e verifie d b y direct substitutio n tha t Conditio n (i ) is satisfied i f S^- ha s the followin g form : Szj = DQ^-R" 1 / 2 , (8.30 ) where Q zj i s an y M x M unitar y matri x an d D i s an y invertibl e diagona l matrix. Usin g th e abov e expression , w e se e tha t Conditio n (ii ) i s equivalen t to sayin g tha t th e matri x
D-tQ, i RVXB, i H#Q;/D- 1 is diagonal . Thi s i s tru e i f th e unitar y matri x Q zj diagonalize s th e positiv e semidefinite matri x H qzj B ^jBzjTLq-i*. - B ^ R ^ .. Therefor Therefor e Conditio n (ii ) hold s i f th e unitary matri x Q zj satisfie s
4 To avoi d degenerate d cases , w e assum e ^q zj i s positiv e definite . I n practice , thi s i s almost alway s tru e a s Rq ZJ i s th e autocorrelatio n matri x o f a nois e vector .
8.5. Optima l zero-jammin g (ZJ ) transceivers
249
for som e diagona l matri x £ . Therefor e w e conclud e t h a t t h e lowe r boun d £bit,szj give n i n (8.29 ) i s achievable , an d it i s achieve d i f t he receivin g matri x Szj i s chose n a s in (8.30 ) an d t he transmittin g matri x i s selecte d a s Gzj =
B ^ R ^ Q ^ . D - 1 . (8.31
)
Because thi s lowe r boun d i s achieve d fo r an y diagona l D , on e can choos e D such t h a t G'l-Gzj = I M - T h a t is , there is no loss of generality in choosing an orthonormal transmitter, a s in t he Z P case . I t i s shown i n Proble m 8.1 1 t h a t 1 /9
this normalizatio n conditio n i s satisfie d whe n D = 5 ] ' . Note t h a t onc e t h e channe l an d nois e ar e given , t h e lowe r boun d £bit,s zj in (8.29 ) depend s onl y o n B ^ - , whose onl y fre e parameter s ar e t he element s of t h e matri x A zj. I n wha t follows , w e wil l optimiz e A zj s o t h a t thi s lowe r bound i s minimized .
8.5.2 Optima
lA
zj
From t h e expressio n fo r £bit,s zj m
(8.29) , w e see t h at t h e matri x A zj shoul d
be chose n suc h t h a t d e t p j - B ^ - ] i s minimized . Recal l fro m (8.7 ) t h a t B• —V • *->zj — v zj
I A ZJ
A "
1
^ ,
where JJ zj an d \ zj ar e square unitar y matrice s define d i n (8.6) . Substitutin g the abov e expressio n int o det[B ^ B ^ ] , we hav e det[B^.B^] = As A*-A zj i
d e t [ A - 2 ] det[ I + A ^ A ^ - ] .
s positiv e semidefinite , i t ca n be show n (Proble m 8.17 ) t h a t det[I + A ^ . A ^ ] > 1
with equalit y i f an d onl y i f A ZJ- = 0 . Henc e t h e transmitte d powe r satisfie s £bit,szj >
^ 22
f c
[det(A"
2
) det(R,J]1/M =
£ ZJ,opt. (8.32
)
Equality hold s i f an d onl y i f A ZJ- = 0 . Therefore , t h e optima l transmitte r i s given b y G ^ V o A - ^ . S - 1 , where V o i s t he N x M matri x consistin g o f t he first M column s o f \
8.5.3 Summar
zj.
y an d discussions
T h e desig n procedur e fo r t h e optima l zero-forcin g Z J syste m i s summarize d as follows . (1) For m t h e matrix C up i n (8.2) and compute it s SVD Cup =
U zj [ A 0 ] V^ •.
Find it s right invers e ~B zj = V o A U ' • , where V o is t he N xM matri x consisting o f t he first M column s o f V zj.
250 8
. Transceive r desig n wit h channe l informatio n a t th e transmitte r
(2) For m th e matri x * = R^B^-B^R^ autocorrelation matri x o f <3Uj-
2
, wher e K
qzj
i s th e M x M
(3) Decompos e th e matri x \l > a s \l > = Q ^ E Q ^ - , wher e Qr*tz- i s a unitar y matrix consistin g o f th e eigenvector s o f \l / an d X I i s a diagona l matri x consisting o f th e correspondin g eigenvalues . (4) Calculat e th e optima l transmittin g matri x G zj =
T$ zjHqzj Cf
z
-Yt~ ' .
(5) Calculat e th e optima l receivin g matri x S^ - = Tt 1'2QzjTlqz/ . (6) Allocat e th e bit s bk a s i n (8.17 ) o r b y Algorith m 8.1 . T h e n th e minimu m t r a n s m i t t e d powe r £ zj,0pt i s a s give n i n (8.32) . I n th e following, som e remark s o n th e optima l Z J syste m ar e i n order . (1) I n th e optima l Z J system , th e transmittin g matri x G zj ca n b e chose n t o be orthonorma l withou t los s o f generality . A s a result , w e ca n conclud e from (8.14 ) t h a t n o powe r allocatio n i s needed fo r th e optima l Z J syste m and al l Sk hav e th e sam e signa l power . (2) Substitutin g th e expressio n S^ - = Y l1'2QtZj'¥iqzj' int o e = S ^ q ^ - , on e immediately finds t h a t th e autocorrelatio n matri x R e o f the outpu t nois e vector e i s a diagona l matrix . I n th e optima l Z J system , th e outpu t nois e is als o decorrelated . (3) Whit e nois e case . W h e n th e nois e i s white , ~R qzj = A/oI . I t ca n b e shown (Proble m 8.12 ) t h a t th e optima l transmittin g an d receivin g ma trices reduc e t o 1 Gzj=V0, S ^ A " ^ . , (8.33 ) respectively. Fro m th e expressio n C up = U^-[ A 0]V\L- , w e se e t h a t th e receiving matri x cancel s th e effec t o f XJ zj an d A , wherea s th e transmit ting matri x equalize s th e effec t o f V ^ ■. (4) Simila r t o th e Z P case , whe n th e nois e i s whit e an d th e channe l i s frequency-nonselective, th e optima l Z J syste m i s no t unique . On e o f the solution s i s give n b y GZj
IM
0
and S
zj — J-M -
In thi s case , th e optima l Z J syste m become s a Z P syste m an d i t reduce s to th e SC-Z P system . (5) Compariso n wit h th e Z P systems . I n Z J systems , nonzer o prefi x sam ples ar e padded . Ther e i s IB I i n th e receive d signa l du e t o th e channel . To remov e IBI , th e receive r retain s onl y M sample s o f eac h block . Al though th e transmitte r send s ou t N sample s fo r ever y M inpu t symbols , the receive r use s onl y M sample s fo r decoding . O n th e othe r hand , i n Z P systems, zero s ar e padde d a t th e en d o f ever y M samples . Afte r pass ing throug h th e channel , sample s ar e sprea d t o nonoverlappin g block s of lengt h N. A s ther e i s n o IBI , al l th e N sample s ca n b e use d fo r
251
8.5. Optima l zero-jammin g (ZJ ) transceiver s
decoding. Ther e ar e mor e observation s tha n unknowns ; th e dimensio n of th e signa l subspac e i s M , wherea s th e receive d signa l ha s dimensio n TV = M + v. Th e eigenstructur e o f the signa l subspac e ca n b e exploite d to ou r advantag e i n Z P systems . Therefore , th e performanc e o f Z P sys tems i s generally bette r tha n tha t o f Z J system , a s demonstrated i n th e example below . (6) O n th e optima l solutio n o f th e matrice s A zp an d A zj. Fo r Z P sys tems, th e fre e paramete r A zp ca n b e exploite d t o improv e furthe r th e performance. O n th e othe r hand , i t i s foun d that , fo r Z J systems , th e optimal A zj = 0 . T o explain thi s difference , le t u s firs t conside r th e Z P system. Not e tha t th e matri x A zp i s a t th e receiver . Whe n th e channe l noise i s colored , th e matri x A zp a t th e receive r ca n b e use d t o reduc e the nois e varianc e (an d henc e achiev e a smalle r transmissio n power ) when th e nois e component s ar e correlated . Whe n th e nois e i s white , the optima l A zp reduce s t o a zer o matrix . Fo r Z J systems , th e matri x Azj i s at th e transmitter . I f the inpu t symbol s ar e correlated , the n A zj can b e chose n t o deco r relate th e input . However , i n ou r derivatio n th e input symbol s Sk ar e assume d t o b e uncorrelated , thu s A zj canno t b e exploited t o reduc e th e transmissio n powe r further . Optimal cyclic-prefixe d (CP ) transceivers . I n Z J systems , th e redundan t samples ar e implicitl y embedde d i n th e transmissio n block . On e importan t special cas e o f Z J system s i s th e C P syste m studie d i n Sectio n 5.4.2 . Fo r a CP system , th e transmittin g an d receivin g matrice s becom e Go = T? cpGcp an
d S o = S c;p[0 I M ],
where bot h G cp an d S cp ar e M x M matrice s an d F cp i s th eT V x M matri x describing th e actio n o f addin g C P (se e (5.14)) , whic h w e reproduc e below : F
0\
*- cp
\M-V
0
0L
v
,
From Sectio n 5.4.2 , w e kno w tha t th e operation s o f addin g C P a t th e trans mitter an d removin g C P a t th e receive r conver t th e LT I channe l C(z) t o an M x M circulan t matri x C circ o f th e for m i n (5.19) . Therefor e th e C P transceiver i s zero-forcin g i f an d onl y i f ^cp^circ^cp —
*-M-
The abov e conditio n implie s tha t al l thre e matrice s S cp , C c i r c , an d G cp ar e invertible. W e kno w tha t C circ i s invertibl e i f an d onl y i f al l th e DF T coeffi cients o f the channe l ar e nonzero . Le t u s assum e tha t C c ^ rc i s invertible . Fo r a give n invertibl e receivin g matri x S cp , th e zero-forcin g conditio n infer s tha t the transmittin g matri x i s ^cp y^cp^circ)
Therefore, i n a C P system , th e onl y fre e paramete r tha t ca n b e designe d i s the invertibl e matri x S cp. B y repeatin g th e earlie r optimizatio n process , w e
252 8
. Transceive r desig n wit h channe l informatio n a t the transmitter
can obtai n t h e optima l zero-forcin g C P system . Thi s i s lef t a s a n exercis e (Problem 8.18) . T h e desig n procedur e fo r optima l C P system s i s as follows . (1) For m t h e matri x * = ( R g / p 2 C ^ c F t p F c p C ^ c R ^ 2 ) , wher e TL autocorrelation matri x o f t he nois e vecto r q_ zj.
qzj
i s t he
(2) Decompos e t h e matri x \l > a s ^ = Q j p E Q c p , wher e Ql p i s a unitar y matrix consistin g o f t he eigenvector s o f \l/ , a nd X I is a diagona l matri x consisting o f t he correspondin g eigenvalues . (3) Obtai n t h e optimal receivin g matri x S o = [ 0 I M ] S (4) Obtai n t h e optima l transmittin g matri x G o = F
CP
= XI' Q
cpC^rcS^p
1
cp[0
\M\-
.
(5) Allocat e t h e bit s bk as in (8.17 ) o r b y Algorith m 8.1. T h e minimu m t r a n s m i t t e d powe r i s given b y £cP,oPt =
™22
b
(2-detlr-tr- 1] det[R,j) 1/M ,
where r i s an M x M diagona l matri x consistin g o f t he D FT coefficients Ck = C ( e j 2 7 r / e / M ) . Comparin g t h e abov e expressio n wit h (8.32) , on e immediatel y realizes t h a t t h e rati o o f £ cp,opt ove r £ zj,opt i s give n b y &cp,opt &zj,opt
1
cyyjM
.
1
/
M
det[A~2]
T h e facto r 2 VIM i s alway s large r t h a n unit y a n d i t approache s on e when M increases. I t i s foun d numericall y t h a t t h e rati o
[detlT^T-1]/det[A-2]j is als o alway s large r t h a n unit y a n d it decrease s a s M increases . The optimal CP system always needs more transmitted power than the optimal ZJ system. Moreover, t h e difference become s smalle r whe n M increases . Thi s i s consisten t with t h e fact t h a t t h e CP syste m i s a specia l cas e o f t he Z J system . I t ca n be shown (Proble m 8.18 ) t h at t h e optima l transmittin g matri x G o als o satisfie s G Q G O = I M 5 t h e transmitte r i s orthonormal. Furthermore , t h e outpu t nois e vector o f t he optima l C P syste m i s als o decorrelated . T h e C P - O F D M syste m belong s t o t he class of CP systems. A s t he optima l C P syste m achieve s t h e minimu m t r a n s m i t t e d powe r amon g al l zero-forcin g systems wit h a CP , we have £ zj,0pt < £cp,opt < £cp-ofdm- I n fact, £ cp,opt ca n be much smalle r t h a n £ cp-ofdm, a s we shall se e in Exampl e 8.4 . I t i s interestin g to not e t h a t , whe n t h e nois e i s white , t h e optima l C P transceive r become s the C P - O F D M syste m (se e Problem 8.13) . Example 8. 4 Optima l Z J systems . T h e transmission channe l a n d t he trans mission environmen t setting s ar e t h e sam e a s thos e i n Exampl e 8.3 . W e compare t h e t r a n s m i t t e d powe r o f thre e Z J systems : (i ) £ zj,opt, (h ) £cp,opt> and (iii ) £ cp-ofdm' Figur e 8.1 1 show s t h e result s fo r M = 1 0 t o 50 . A s a
253
8.6. Furthe r readin g
6
11
5
\
\
DQ 3
\
V
■
\
o2 £ o
°- 1
\
CD
1°
.
CO
§- 1 i_
-11
V•
N. *
-2
^ • -^
-^
" -^
.-•
— ._.__ -
'• .
"
^
^
j H
#
^ \* ^
1
— cp-ofd m 1 cp,opt H • zj.op t 1 zp.opt h J
^.
* "".* . ^'^--.
"D
"•"'
11
\
4 "D
1
^
•
•
•
•
'
"
"
■
■
■
■
•
-^
■
-3 -4 10 1
11
52
1
0
_i i
11
25
30 M
35
40
1
45
50
Figure 8 . 1 1 . Exampl e 8.4 . Compariso n o f transmitte d power .
comparison, th e transmitte d powe r fo r th e optima l Z P syste m £ zPj0pt i s als o shown i n th e plot . T h e optima l Z J syste m ha s approximatel y 2. 5 d B gai n ( M = 50 ) t o 4 d B gai n ( M = 10 ) ove r th e C P - O F D M systems ; th e gai n o f £zj,opt ove r £ cp,opt i s approximatel y 1. 2 d B fo r M = 1 0 and 0. 3 d B fo r M = 50 . We als o se e i n thi s exampl e t h a t th e optima l Z P syste m perform s bette r t h a n th e optima l Z J system . T h e differenc e ca n b e a s larg e a s 2 d B whe n M = 10 . T h e reaso n i s t h a t i n Z P system s al l th e sample s i n th e receive d block o f siz e N ca n b e employe d a t th e receiver , wherea s i n Z J system s onl y M sample s pe r receive d bloc k ar e available . I n numerica l experiments , i t i s found t h a t i t i s tru e t h a t th e optima l Z P syste m outperform s th e optima l ZJ syste m fo r mos t cases . However , a s Z P an d Z J system s ar e tw o differen t classes o f systems , i t i s possibl e t o construc t to y example s (se e Proble m 8.19 ) such t h a t a Z J syste m i s better . ■
8.6 Furthe
r readin g
There hav e bee n man y report s i n th e literatur e o n th e desig n o f transceiver s t h a t ar e optimize d fo r a variet y o f differen t criteria . T h e desig n method s described i n thi s chapte r wer e adopte d fro m [73 , 75] . On e o f th e earl y design s of optima l multicarrie r transceive r system s wa s give n b y [63] . T h e zero-forcin g ZP bloc k transceive r wa s optimize d unde r th e assumptio n t h a t th e channe l noise i s whit e an d th e resultin g optima l transceive r i s calle d a vecto r codin g
254 8
. Transceive r desig n wit h channe l informatio n a t th e transmitte r
transceiver. I n [135] , zero-forcin g bloc k transceiver s t h a t minimiz e th e mea n squared erro r wer e derived . I n [92] , th e autho r derive d F I R transceiver s wit h minimum mea n square d error , withou t restrictin g th e filte r order s t o b e les s t h a n th e bloc k size . I n [109 ] an d [110] , th e result s presente d i n thi s chapte r were generalize d t o th e multiuse r cas e an d th e multiflo w case , respectively . I n these cases , differen t user s migh t hav e differen t qualit y o f service requirements . It wa s foun d t h a t , eve n thoug h th e bi t allocatio n migh t b e differen t fro m th e single-user case , th e optima l transmittin g an d receivin g matrice s wer e th e same a s thos e presente d i n thi s chapter . I n [186] , unde r th e B E R constraint s the author s derive d th e optima l transceive r t h a t minimize s th e t r a n s m i t t e d power. I n [142] , th e author s investigate d th e conditio n o n th e modulatio n symbols fo r th e optimalit y o f orthonorma l transceiver s whe n th e assumption s of hig h bi t rat e a n d / o r fractiona l bk wer e relaxed . Optima l transceiver s wit h decision feedbac k an d bi t loadin g wer e derive d i n [177] . Reader s ar e referre d to th e abov e reference s fo r furthe r exploration .
8-7 Problem
s
8.1 Suppos e w e hav e a zero-forcin g transceive r wit h thre e subchannels . T h e output nois e variance s are , respectivel y < r e 2 0 = 0 . 1 , ^= 0 . 0 1
, <
= 0.001 .
Suppose t h a t th e inpu t symbol s ar e PA M an d [GjGo]fcf c = 1 fo r al l k. Find th e optima l bk, th e non-negativ e intege r bk, an d th e correspondin g bit allocatio n gain s fo r th e followin g cases : (a) th e bi t rat e pe r symbo l i s b = 3 an d th e symbo l erro r rat e i s SER = 10- 6; (b) th e bi t rat e pe r symbo l i s b = 2 an d th e symbo l erro r rat e i s SER = 10- 6; (c) th e bi t rat e pe r symbo l i s b = 2 an d th e symbo l erro r rat e i s SER = 10- 4. From thi s example , w e se e t h a t fo r th e cas e o f non-negativ e intege r bi t allocation, th e gai n depend s o n b o t h th e bi t rat e an d th e SER . 8.2 Le t Jf c b e th e k x k reversa l matrix . Prov e t h a t th e matrice s Ci Cup ar e relate d b y C up = J MCfowJN. 8.3 Le t th e SVD s o f th e N x M matri x Ci be denoted , respectively , b y ^low ^
VL. C
low
ow
ow
an d
an d th e M x N matri x C
up
up
= Vup[Aup 0]V t
Using th e resul t i n Proble m 8.2 , sho w t h a t (a) th e unitar y matrice s XJiow, Vi owl XJ that U w p=
JMV|*
OW,
V
up,
= up
an d W up ca n b e chose n suc h XJ
1OWJN',
8.7. Problem s
255
(b) whe n t h e unitar y matrice s ar e chose n s o t h a t t h e abov e relation s are satisfied , t h e tw o diagona l matrice s Ai ow an d A up i n t h e tw o SVD expression s ar e t h e same . I n othe r words , sho w t h a t Ci ow and C up hav e identica l singula r values . 8.4 Conside r Fig . 8.8 . Le t u> = —A zp/j,1. Us e t h e orthogonalit y principl e to sho w t h a t , i f A zp i s chose n optimall y a s (8.26) , the n u) i s t h e bes t estimate o f /J, 0 give n t h e observatio n o f \i x. 8.5 Giv e on e exampl e t o sho w t h a t t h e bi t allocatio n gai n Qut o f t h e C P O F D M syste m ca n decreas e whe n t h e numbe r o f subchannel s M in creases. 8.6 Conside r t h e Z P - O F D M system . Suppos e t h a t t h e computationall y ef ficient zero-forcin g receive r i n (6.15 ) i s employe d fo r symbo l recovery . Assume t h a t t h e nois e q(n) i s white wit h varianc e A/o - Comput e t h e bit allocation gai n an d express you r answe r i n terms o f t he D FT coefficient s Ck o f t he channel c(n). Unde r wha t conditio n i s t he gain equa l t o unity ? 8.7 Le t t h e channe l c(n) an d t h e nois e q(n) b o t h b e real . Sho w t h a t t h e o u t p u t nois e variance s o f t he C P - O F D M syste m satisf y 2_
2
for k = l , 2 , . . . , M / 2 ( M even) . Prov e t h a t t h e optima l bi t alloca tion als o satisfie s bk = 6M-/C , fo r k = 1 , 2 , . . . , M / 2 . Thi s implie s t h a t there i s no los s o f optimalit y i n restrictin g t h e symbol s Sk to satisf y t h e conjugate symmetri c propert y i n (6.33 ) whe n t h e syste m i s a baseban d communication system . 8.8 Suppos e t h a t , i n a Z P system , t h e nois e q(n) i s whit e an d t h e channe l has onl y on e nonzer o t a p . Le t t h e SV D o f t h e channe l matri x Ci ow be U z p A \ \ p . Sho w t h a t t h e optima l Y zp ca n b e a n arbitrar y unitar y matrix. Expres s t h e correspondin g XJ zp i n term s o f V zp. 8.9 Conside r a zero-forcin g Z P syste m wit h M = 2 an d v = 1 . Le t t h e channel b e C(z) = \ Jrz~1. T h e noise i s colored wit h t h e autocorrelatio n coefficients give n by rq(k) = A/*o0.5l fel. Fin d t h e optimal transmittin g an d receiving matrices . Suppos e 6 = 4 and A/o = 0 . 1 . W h a t i s t he minimu m transmission powe r neede d fo r a n SE R of 1 0 - 6 ? 8.10 Repea t t h e abov e proble m fo r a zero-forcin g Z J system . 8.11 Sho w t h a t , whe n D =
5 ] 1 ' 2 , t h e transmittin g matri x G zj i n (8.31 ) o f
the optima l Z J syste m satisfie s G ^ -Gzj =
I M-
8.12 Sho w t h a t , whe n t h e nois e i s white, t h e transmittin g an d receivin g ma trices o f t h e optima l Z J syste m reduc e t o thos e give n i n (8.33) . Fin d the expressio n fo r t h e minimu m transmitte d powe r i n thi s case . 8.13 Sho w t h a t , whe n t h e nois e i s white , t h e optima l C P transceive r i s t h e C P - O F D M system .
256 8
. Transceive r desig n w i t h channe l informatio n a t th e transmitte r
.14 Pre-whitening approach. Conside r th e Z P syste m i n Fig . 8.2 . Le t H qzp be th e N x N autocorrelatio n matri x o f th e nois e vecto r q_ zp. Suppos e — 1/2
t h a t w e appl y th e pre-whitenin g matri x Hq zp t o white n th e nois e a s in Fig . P8.14 . Thu s w e hav e a ne w -1/2 channe l matri x C[ ow = R _1/2C^ow, and th e ne w nois e vecto r q ■zp q_ i s white . R qzp zp (a) Fin d th e ne w optima l matrice s G' zp an d S' zp. Ho w ar e th e ne w optimal transmittin g an d receivin g matrice s relate d t o th e optima l Gzp an d S zp give n i n Sectio n 8.4.3 ? (b) Sho w t h a t th e minimu m t r a n s m i t t e d power s o f th e tw o optima l transceivers (wit h an d withou t th e pre-whitenin g matrix ) ar e th e same. I n othe r words , on e ca n emplo y thi s pre-whitenin g approac h for th e desig n o f optima l Z P transceive r fo r th e colore d nois e case . T h e sam e approac h ca n b e adopte d fo r th e desig n o f Z J systems . w
• • •
w
w
w
\* G
'p
• • • X
M-\
w
W ^low
w
►
S'zp
• • •
• • •
►
w
\J
w
Figure P 8 . 1 4 . Z P syste m wit h a pre-whitenin g matrix .
.15 Suppos e t h a t th e setting s ar e identica l t o thos e i n Proble m 8.9 . Us e th e pre-whitening approac h t o desig n th e optima l Z P transceiver . C o m p u t e the minimu m t r a n s m i t t e d powe r an d sho w t h a t i t i s th e sam e a s t h a t obtained i n Proble m 8.9 . .16 Repea t Proble m 8.1 5 fo r th e Z J system . .17 Le t Afc , for 0 < k < M — 1, be th e eigenvalue s o f the matri x ( I + A j - A ^ ) . (a) Prov e t h a tA & > 1 ; the n sho w t h a tA & = 1 fo r al l k i f an d onl y i f AZJ = 0 . (b) Sho w t h a t d e t [ I + A ^ ■ Azj] >
1 , with equalit y i f and onl y i f A zj=
0 .
.18 Deriv e th e expression s fo r S c p , Go , an d £ Cp,opt of the optima l C P system . Also sho w th e following : (a) th e transmittin g matri x G o satisfie s G Q G O = I M 5 (b) th e autocorrelatio n matri x R e o f th e outpu t nois e e = s " — s i s diagonal; t h a t is , th e receivin g matri x decorrelate s th e nois e vector .
257
8.7. Problem s
8.19 Le t th e channe l an d nois e powe r spectru m b e given , respectively , b y C(z) =
1 + 2Z- 1 + z~ 2 an
dS
juJ
q(e
)=
\C(e
juJ 2
)\ .
Suppose tha t M = 2 an d v = 2 . Comput e th e rati o £ Zp,opt/£zj,optVerify tha t thi s rati o i s large r tha n one , provin g tha t th e optima l Z J system outperform s th e optima l Z P syste m i n thi s case .
258 8
. Transceive r desig n wit h channe l informatio n a t th e transmitte r
9 D M T system s wit h improve d frequency characteristic s The topi c o f frequenc y response s o f the transmittin g an d receivin g filter s ha s not come into our earlier discussions of transceiver design . Th e frequency char acteristics o f th e filter s ar e als o a n importan t aspec t o f transceive r designs . The stopban d attenuatio n o f th e transmittin g (receiving ) filter s determine s how well separated th e subchannel s ar e i n the frequenc y domai n a t th e trans mitter (receiver) . Frequenc y separatio n a t th e transmitte r sid e i s importan t for th e contro l o f spectra l leakage , i.e . undesire d out-of-ban d spectra l com ponents. Poo r separatio n wil l lea d t o significan t spectra l leakage . Thi s coul d pose a proble m i n application s wher e th e powe r spectru m o f the transmitte d signal i s require d t o hav e a larg e rollof f i n certai n frequenc y bands . Wire d applications wit h frequenc y divisio n multiplexing , e.g . ADS L an d VDSL , ar e such example s [7 , 8] . Th e powe r spectru m o f th e transmitte d signa l shoul d be properl y attenuate d i n th e transmissio n band s o f th e opposit e directio n to avoi d interference . Th e powe r spectru m shoul d als o b e attenuate d i n am ateur radi o band s t o reduc e interferenc e t o radi o transmission , calle d egress emission [8] . O n th e othe r hand , poo r frequenc y separatio n a t th e receive r side result s i n poo r out-of-ban d rejection . I n ADS L an d VDS L applications , some of the frequenc y band s ar e als o used b y radio transmissio n system s suc h as amplitude-modulatio n station s an d amateu r radio . Th e radi o frequenc y signals ca n b e couple d int o th e wire s an d thi s introduce s radio frequency interference (RFI ) o r ingress [29] . Poo r frequenc y selectivit y o f th e receivin g filters mean s man y neighborin g tone s ca n b e affected . Th e signa l t o interfer ence nois e ratio s o f thes e tone s ar e reduce d an d th e tota l transmissio n rat e decreased. We foun d i n Chapte r 6 that i n the DM T transceive r th e transmittin g an d receiving filter s com e fro m rectangula r windows . Th e spectra l sidelobe s o f these filter s ar e ofte n inadequat e t o provid e sufficien t subchanne l separation . In thi s chapter , w e wil l us e a filte r ban k approac h t o improvin g frequenc y separation amon g subchannels . Base d o n the filte r ban k representatio n o f th e DMT transceiver , w e will introduce wha t w e call subfilters i n the subchannels . We ca n includ e th e subfilter s withou t changin g th e ISI-fre e propert y o f th e DMT syste m b y usin g a cycli c prefi x slightl y large r tha n th e channe l order . For th e transmitte r side , th e subfilter s ca n improv e th e spectra l rollof f o f th e 259
9. D M T system s w i t h improve d frequenc y characteristic s
260
transmitted spectru m whil e having littl e effec t o n the erro r rat e performance . For th e receive r sid e RF I ca n b e suppresse d an d th e transmissio n rat e ca n be improve d considerably . Moreover , whe n th e subfilter s for m a DF T bank , they ca n b e tie d nicel y t o windowing, a very usefu l metho d tha t improve s th e frequency characteristic s o f th e DM T system . Thi s wa y th e window s use d in windowin g ca n als o b e optimize d throug h th e desig n o f subfilters . I n th e literature man y windowin g an d non-windowin g method s hav e bee n propose d to achiev e bette r subchanne l frequenc y separation . Intereste d reader s ar e referred t o Sectio n 9. 8 fo r mor e reference s o n thi s topic .
9-1 Sidelobe
s matter !
The sidelobe s o f th e transmittin g an d receivin g filter s ar e bot h importan t i n transceiver designs . W e wil l firs t examin e th e transmitte d powe r spectru m and se e ho w i t i s affected b y the frequenc y characteristic s o f the transmittin g filters. Figur e 9. 1 show s th e transmitte r wit h a D/ C converter . Recal l tha t the powe r spectru m o f th e transmitte d signa l ca n b e obtaine d b y summin g the powe r spectr a a t th e output s o f the transmittin g filter s (Sectio n 6.5) , i.e .
sx(ejn = ^J2
£
keA
^\F*(eJ^\2i
where A i s the se t o f subchannels tha t ar e actuall y use d fo r transmissio n an d £Sjk i s the powe r o f the kth subchanne l signal . Eac h transmittin g filte r Fk(z) is a bandpass filte r centere d a t 2kir/M. After th e D/ C converte r (Figur e 9.1) ,
r^n
%(») —
►
s
►
:(")
—
fiV
x(n)
-+ J'oCO
D/C
tT
i i
Fx(z)
Pi(t)
-> x
a(t)
ii
W">-
ffF -> FM-\V)
I
transmitter
Figure 9 . 1 . Transmitte r show n wit h a continuous-tim e outpu t x
a(t).
the spectru m o f the continuous-tim e transmitte d signa l x a (t) i s
keA
where T i s th e underlyin g sampl e spacin g an d P\(jQ) i s th e transmittin g pulse. A s Fk(e^) an d Pi(jQ) ar e no t idea l filters , th e spectru m i s nonzer o
9.1. Sidelobe s matter !
261
not onl y i n th e frequenc y bin s o f th e subchannel s t h a t ar e used , bu t als o i n other frequenc y bands ; thi s i s referre d t o a s spectra l leakage . A n exampl e o f Sx(ejuJ) wit h M = 51 2 i s plotte d fo r a ful l perio d [0 , 2TT] i n Fig . 9.2(a) . T h e tones t h a t ar e use d fo r transmissio n ar e 38-89 , an d 111-255 . T h e correspond ing S Xa(jQ) i s show n i n Fig . 9.2(b ) fo r 1/ T = 2.20 8 MHz . T h e transmittin g pulse i s a n ellipti c filter o f orde r 4 an d th e stopban d attenuatio n i s 4 5 d B . We ca n se e fro m Fig . 9.2(a ) t h a t , althoug h tone s 90-11 0 (correspondin g t o uo around 0.47 T i n th e plot ) ar e no t used , S x{e^) i s no t zer o du e t o th e finite stopband attenuatio n o f F k{e^). A s a result , th e spectru m S Xa(jVt) i s no t zero i n th e frequenc y rang e correspondin g t o thes e tone s an d ther e i s spectra l leakage. Ther e i s als o spectra l leakag e fo r Vt > Nyquis t frequenc y (whic h i s equal t o 1.10 4 MH z i n thi s example ) du e t o th e finite stopban d attenuatio n o f Pi(jQ). T h e transmittin g puls e Pi(jQ) wil l hel p t o a t t e n u a t e th e transmit ted powe r spectru m beyon d th e Nyquis t frequency . Fo r th e ban d withi n th e Nyquist frequency , whic h fall s int o th e passban d o f P i ( j D ) , spectru m shap ing relie s o n th e transmittin g filters F k(z). T h e sidelobe s o f th e transmittin g filters wil l directl y affec t th e amoun t o f spectra l leakag e withi n th e Nyquis t frequency. At th e receiver , th e sidelob e o f th e receivin g filters i s als o important , bu t for a differen t reason . I n Sectio n 6.6 , w e discusse d differen t type s o f impair ments i n a D M T system . On e impairmen t i s R F I (radi o frequenc y inter ference). Thes e radi o interferin g signal s hav e a muc h narrowe r bandwidth . To understan d ho w a narrowban d interferenc e ca n affec t th e outpu t o f th e receiver, le t u s conside r a simpl e continuous-tim e exponentia l interferenc e /xexp(j(^ot + 0)) wit h frequenc y D Q and amplitud e \i. Upo n samplin g wit h period T a t th e receiver , th e discrete-tim e exponentia l interferenc e signa l i s v(n) =
\i exp(j(UJQTI + 0)) , wher
e UJQ = QQT.
T h e interference , bein g a par t o f th e receive d signal , i s passe d t o th e receivin g filters. A t th e outpu t o f th e kth receivin g filter (Fig . 9.3) , th e interferenc e t e r m Uk(n) i s als o a n exponential , uk(n)=
ii k exp(j(uj 0n +
0)) , wher
e fi
k
=
/iH
JUJ
k(e
°).
We se e t h a t th e amplitud e i s scale d b y th e receivin g filters. T h e interferenc e cannot b e completel y eliminate d du e t o th e finite stopban d attenuatio n o f the receivin g filters. A highe r stopban d attenuatio n o f th e receivin g filters means bette r suppressio n o f R F I . On e simpl e approac h t o improvin g th e frequency selectivit y o f th e transmittin g an d receivin g filters i s b y includin g short subfilters . W h e n a n additiona l guar d interva l i s available , thes e subfil ters wil l enhanc e th e stopban d o f th e transmitting/receivin g filters withou t incurring additiona l ISI . T h e followin g sectio n discusse s th e overal l transfe r matrix o f th e transceive r an d give s a simpl e observatio n t h a t wil l b e ver y useful i n incorporatin g subfilter s later .
262
9. D M T system s w i t h improve d frequenc y characteristic s
(a)
(a)
1 1. 5 Frequency (MHz)
Figure 9 . 2 . Exampl e of spectral leakage : (a ) th e spectrum S x(e'UJ) o f the discrete-tim e transmitted signa l x(n); (b ) th e spectru m S Xa(jQ.) o f th e continuous-tim e transmitte d signal x a{t).
v(n)
-►I H 0(z) | ►
«„(» )
Hiiz)
*\HM_x{z)\ ►
u
MJn)
Figure 9.3 . Receivin g ban k wit h a n interference-onl y input .
263
9.2. Overal l transfe r matri x
9.2 Overal
l transfe r matri x
Figure 9. 4 show s th e filte r ban k representatio n o f th e D M T transceive r wit h M subchannels . Recal l fro m Sectio n 6.5 , th e firs t transmittin g filte r FQ(Z) i s a rectangular windo w o f lengt h N = M + z/ , wher e v i s th e cycli c prefi x length . All th e othe r transmittin g filter s ar e scale d an d frequency-shifte d version s o f the p r o t o t y p e FQ(Z),
W vkF0(zWk), k
Fk(z)=
= 0,1,..., M - 1 .
(9.1)
For th e receivin g side , th e prototyp e filte r HQ(Z) i s a rectangula r windo w o f length M an d al l th e othe r receivin g filter s ar e scale d an d frequency-shifte d versions o f th e firs t filter , W-" kH0(zWk), k
Hk(z)=
(9.2)
= 0,1,... , M - 1 .
For convenienc e o f notation , w e hav e use d c(n) i n Fig . 9. 4 t o represen t th e equivalent channel , possibl y shortene d wit h tim e domai n equalization . As sume th e channe l c(n) i s a n F I R filte r o f orde r L < v. T h e frequenc y domai n equalizers 1/Af e ar e chose n a sA & = Cfc , wher e C k ar e th e M-poin t D F T o f th e channel impuls e respons e c(n). T h e receive r i s a zero-forcin g receiver . I n th e absence o f channe l noise , th e kth receive r outpu t i s Sk(n) =
Sk(n), k
(9.3)
0,1,...,M-1.
Using Theore m 4.1 , w e kno w t h a t th e syste m fro m th e zt h transmitte r input Si(n) t o th e kth signa l ykiji) a t th e receive r i s a n LT I syste m an d t h a t the transfe r functio n i s Tkl{z)
/
V») 5,(n)
W*)-
Hk(z)C(z)Fz(z)
(9.4)
0
IN
s(«)
y(«)
q(n)
u
x(n)
F0(d
IN
Fx(z)
u
\FM-\(Z)\
C(z)
-
>
#
■
H0(z)
±N
Hx{z)
±N
^\HM_x(z)\
\N
y^n)
-> ►
s Q(n)
y{(n)
yM-l^
T(z)
Figure 9.4 . Filte r ban k representatio n o f th e D M T system .
" ^ ^S
VU>
M-l^>
264
9. D M T system s w i t h improve d frequenc y characteristic s
where th e notatio n [A(Z)]^A T denote s th e TV-fol d decimate d versio n o f A(z) a defined i n (4.8) . I n vie w o f (9.3) , w e ca n se e t h a t th e syste m fro m Si(n) t sk(n) i s LT I wit h transfe r functio n 5(k — i). A s s k(n) differ s fro m y k(n) onl in th e scala r l/A^ , w e ca n conclud e t h a t T ki{z) = X kS(k — i)- Summarizing we ca n obtai n th e followin g lemma .
s o y ,
Lemma 9. 1 Conside r th e syste m i n Fig . 9. 4 wit h filters a s define d a s (9.1 ) and (9.2) . Th e transfe r functio n T ki(z) fro m th e i t h transmitte r inpu t Si(n) to th e kth signa l y k(n) a t th e receive r i s give n b y Tki(z)=
X kS(k - i) , 0
< k, i < M - 1 .
(9.5)
T h e resul t hold s fo r an y F I R filter C(z) o f orde r L < z/ , wher e v = N — M i s the cycli c prefi x length . Th e constan t X k ar e th e M-poin t D F T o f c(n) , i.e . Xk =
C(z)\
z=
e j2nk/M
. ■
So lon g a s th e orde r o f C(z) i s no t large r t h a n z/ , th e syste m i s fre e fro m interblock interferenc e an d inte r subchanne l interference .
(a)
* •
DMT transmitter
x(n)
DMT receiver
p&\
q{n)
(b)
| DM T transmitter
X(ft)
— ►
C(z)
I tin) ^W^
DMT receiver
P(z)
- * • •
Figure 9.5 . Exampl e o f th e D M T syste m wit h a n additiona l filte r P(z) cascade d (a ) before th e channe l an d (b ) afte r th e channel .
Now suppos e t h a t a n additiona l F I R filter P(z) i s cascade d befor e th e channel (Fig . 9.5(a) ) o r afte r th e channe l (Fig . 9.5(b)) . I n eithe r case , th e transfer functio n Tki(z) fro m th e i t h transmitte r inpu t Si(n) t o th e kth signa l Vk(n) i s Tki(z)
Hk(z){P(z)C(z))Fi(z)
IN
It ha s th e sam e expressio n a s (9.4 ) excep t t h a t th e channe l C(z) i s replace d by th e produc t P(z)C(z). Th e lemm a implie s t h a t , a s lon g a s th e produc t P{z)C{z) ha s orde r n o large r t h a n z/ , the overal l syste m remain s ISI-free . T h e idea o f introducin g a n extr a filter i n Fig . 9. 5 wil l b e extende d i n late r section s to includ e a se t o f filters, calle d subfilters , on e fo r eac h subchannel . On e
265
9.3. Transmitter s w i t h subfilter s
can desig n thes e subfilter s fo r variou s purposes . Ou r goa l her e i s t o shap e the frequenc y response s o f th e transmitting/receivin g filter s s o tha t bette r frequency separatio n amon g th e subchannel s ca n b e achieved . s0(n)
\TN
►
Fo
►
P0(z)
s^n)
\fN
►
Fi(z)
►
Pl(z)
Ik
•
• S
M-S")-
\\N
>• w *(" )
►
ik
•
f
► PM-I®
F
M-l(z)
f'(z)
Figure 9.6 . Transmittin g ban k wit h subfilters .
9.3 Transmitter
s wit h subfilter s
Figure 9. 6 show s th e transmitte r wit h additiona l FI R filter s Pk(z) i n eac h subchannel. Suppos e th e order s o f Pk(z) ar e a , Pk(z) =
a
Y,Pk(n)z-
n
.
These additiona l filter s wil l b e calle d subfilters a s the y generall y hav e smal l orders. Wit h th e subfilters , th e kth effectiv e transmittin g filte r i s F'k{z) = F
k(z)Pk(z).
Now th e transfe r functio n fro m th e zt h transmitte r inpu t Si(n) t o th e kth signal yk(n) a t th e receive r (Fig . 9.4 ) become s Tkl{z)
HkizXPiWCizfiFiiz)
IN
It i s th e sam e a s (9.4 ) excep t tha t C(z) i s replace d b y Pi(z)C(z). Usin g th e result i n Lemm a 9.1 , we kno w th e overal l syste m remain s ISI-fre e a s lon g a s the orde r o f th e produc t Pi(z)C(z) i s no t large r tha n v. Th e conditio n fo r this i s a + L < v, (9.6 ) where a i s the order of Pk(z). Also , the transfer functio n i s Tki(z) = XkS(k— z), the sam e for m a s (9.5) . No wA & are th e kth DF T coefficien t o f pk(n) * c(n), Xk = P
k(z)C(z)
= e j2/e7r/M
(9.7)
9. DM T system s wit h improve d frequenc y characteristic s
266
T h e ne w transmittin g filter s F^(z) ar e o f lengt h N + a , a s Fk(z) ar e o f length N. W e kno w t h a t fo r th e reaso n o f bandwidt h efficienc y M i s usuall y much large r t h a n v an d henc e als o muc h large r t h a n a. S o the subfilter s Pk(z) are muc h shorte r t h a n Fk(z). Not e t h a t th e conditio n i n (9.6 ) mean s t h a t th e insertion o f subfilter s require s extr a guar d interva l (y > L). Wheneve r ther e is extr a cycli c prefix , w e ca n inser t a filter o f orde r v — L withou t affectin g the ISI-fre e property . I n th e application s o f D M T systems , th e channe l i s typically shortene d b y a T E Q . T o hav e extr a cycli c prefix , th e channe l need s to b e shortene d s o t h a t th e orde r o f th e equivalen t channe l i s smalle r t h a n the cycli c prefi x length .
9.3.1 Choosin
g th e subfilter s a s a DF T ban k
Let u s conside r th e cas e t h a t th e subfilter s ar e als o shifte d version s o f th e first subfilter, Pk(z) = P 0(zWk). (9.8 ) T h e kth ne w transmittin g filter become s F^(z) = can als o writ e i t a s F,k{z) =
W kF'0{zWk), wher
W ukF0(zWk)Po(zWk). W
e F^z) =
F 0(z)P0(z). (9.9
e )
They ar e als o shifte d version s o f the ne w prototyp e filter FQ(Z) excep t fo r som e scalars. Th e ne w prototyp e i s o f lengt h N + a an d i s n o longe r a rectangula r window. Le t u s denot e it s coefficient s b y cti/y/M, the n 1
N+a-i
F'(z) = —= V a
iZ-\
We cal l di th e transmitte r windo w coefficient s fo r reason s t h a t wil l becom e clear later . Thes e coefficient s com e fro m th e convolutio n o f a rectangula r window an d po(n). A s th e orde r o f P${z) i s muc h smalle r t h a n TV , mos t o f the windo w coefficient s ar e o f th e sam e value . A typica l plo t o f th e windo w is show n i n Fig . 9.7 . W h e n w e choos e th e subfilter s a s a D F T bank , thes e new transmittin g filters agai n for m a D F T ban k an d thu s ca n b e implemente d efficiently, a s w e wil l se e below .
9.3.2 DF
T ban k implementatio n
Using (9.9) , w e hav e
F^Z)
= ^wvk y VM to
N+oc-l
w~
ai
ki
z~\
We ca n writ e i t i n a matri x for m a s follows :
F£(.z) = k
V
- L [ l z-
ML
J
1
■■■
z-
N+1
]A(zN)
yy-k(l-v) W~k(N+a-\-v)
9.3. Transmitter s w i t h subfilter s
267
WM
f0(n) N-l
Po(n)
m• • I
I I i— « « »
0a
N-a coefficient s
. . .■
I ' m
I
N-l+a n
Figure 9.7 . Exampl e o f th e transmitte r window .
where A ( z
V
0A A0 0A i
)
2z-
T h e matrice s A 0 , A i , an d A 2 ar e diagonal , A0 =
dia g [a 0 a
x
A2 = Using W M= as
•• a
•
a-i]
dia g [a N a
,
Ai = N+1
dia g [a a a
•
a+1
•
• • ajv-i ] ,
• • a^v+a-i ] .
1 , th e las t colum n vecto r i n th e abov e expressio n ca n b e writte n
k
w~
yy-k(N+a-l-v
0 0
0
11
yy-k(l-v)
, wher e F i =
-k(M-l)
la
0 0
I-M-V-OL
0 0
la
I, 0 0 I, 0
Therefore, w e hav e
F'k(z)
1 M
1
\1 z-
~N+1] A(z
N
)F1
1 W~k
(9.10)
W-k(M-l)
T h e 1 x M ne w transmittin g ban k f'(;z) , a s indicate d i n Fig . 9.6 , ca n b e obtained b y puttin g F^{z) togethe r i n a ro w vector . I n (9.10) , onl y th e las t column vecto r depend s o n fc; putting thes e colum n vector s togethe r give s ris e to th e I D F T matri x \ / M W ^ . T h e transmittin g ban k i s thu s give n b y f'(z)=[Fl(z) F{(z)
••
• F'
M_1(z)]=
[l
z~N+l] G(z
N
),
(9.11)
9. D M T system s w i t h improve d frequenc y characteristic s
268
t*
s0(n)
>• x(n) i
s^n)
\N
(a)
• • •
W")
(b)
• •
\N
s0(n)
u
-1
^Z
x(n) i
G(z) -+>tJv
••• A
AJL
-+>
-l ^z >z
• • • \
i
_t -+>
s^n)
kZ
►
G(zN)
-1
-1
_r
Figure 9.8 . Polyphas e implementatio n o f th e transmittin g bank .
where G(z) = A(z)F 1W^. Note t h a t G(z) i s th e polyphas e matri x o f th e transmittin g filte r bank . It i s a polynomia l matri x i n £ _ 1 , no t a constan t matri x lik e thos e i n Chap ters 6-8 . W i t h th e polyphas e representation , th e transmittin g ban k ha s th e implementation i n Fig . 9.8(a) . Usin g th e nobl e identit y fo r exchangin g LT I filters an d expander s i n Sectio n 4.1.2 , w e ca n mov e G(z N) t o th e lef t o f th e expanders, a s show n i n th e figure. W e ca n furthe r redra w i t a s i n Fig . 9. 9 using th e expressio n o f G(z) i n (9.11) . Let u s examin e th e implementatio n i n Fig . 9. 9 ste p b y step . Fo r eac h input block , M-poin t I D F T i s applied, followe d b y prefixin g an d suffixing . T h e matrix F i insert s a cycli c prefi x o f lengt h v an d als o a suffi x o f lengt h a t o th e vector u ( n ) . A n illustratio n o f prefixing an d suffixin g i s given i n Fig . 9.10 . T h e prefixed an d suffixe d vecto r t ( n ) , a s indicate d i n Fig . 9.9 , i s o f siz e N-\-a. T h e samples o f the vecto r t ( n ) ar e multiplie d b y th e coefficient s an , a i , . . ., CLN+OL-I as demonstrate d i n Fig . 9.11(a) . (Thi s i s wh y thes e coefficient s ar e calle d window coefficients. ) Th e resultin g vecto r v ( n ) wil l b e th e outpu t du e t o th e n t h inpu t bloc k s ( n ) . The n th e las t a sample s o f v ( n — 1 ) ar e adde d t o th e first a sample s o f v ( n ) (overlap-and-ad d operation) , a s show n i n Fig . 9.11(b) . T h e outpu t sample s du e t o th e n t h an d (n + l ) t h inpu t block s overla p b y a samples . W h e n w e inspec t th e transmitte r output , ther e wil l b e (N — a) samples du e solel y t o th e n t h inpu t bloc k s ( n ) . Th e implementatio n i n Fig. 9. 9 i s th e sam e a s th e transmitte r windowin g give n i n [8] .
269
9.3. Transmitter s w i t h subfilter s
/
s(«) t(«)
u(«)
—J
sM _ *,(») _
W">-
Wt
•• •
-T
•
—M
—w
JTHUL
a
fJV
>—
4^
a lines
x(n)
fJV
->
x Lines fl ►
v(«)
Az A
z
♦z-1
fJV
window
Figure 9.9 . Efficien t DF T implementatio n o f th e transmittin g bank . M sample s (a) outpu
samples of IDF T t vecto r t(k)
■iilli.lillll copy v prefi x
(b) an
samples of prefixe d d suffixed vector t(k)
in In.nlillllliii
a suffi x
copy
j
Figure 9 . 1 0 . Prefixin g an d suffixing : (a ) th e sample s o f th e IDF T outpu t vecto r u ( n ) ; (b) th e sample s o f th e prefixe d an d suffixe d vecto r t ( n ) .
9. DM T system s wit h improve d frequenc y characteristic s
270
a sample s a
samples
window coefficients a %
lL .
N-\+a i
prefixed and suffixed vector t(k)
l.iilli.lillllllll.i ^
output due to the Ath \(k) . input block — _ — L!
11111 ■
11
LLL
(a)
output due to the £th input block
a samples y(k)
IIIIII.
output due to the (£+l)th input block
Ll_L
'<* +1 > . l l l l l l l l . l l
. .ll.llll.ll .,
overlap-and-add (b) Figure 9 . 1 1 . (a ) Applyin g windo w coefficients ; (b ) overlap-and-ad d operatio n t o produce th e transmitte r output .
Window coefficient s Th e windo w i s the convolutio n ofpo(n) an d a rect angular window . Th e middl e (N — a) coefficient s ar e a
cti = ^2po(i), i
= a , a + 1,... , N - 1 .
i=0
They ar e equal t o th e D C valu e o f Po(z), a s shown Fig . 9.7 . W e can writ e th e a coefficient s o n th e tw o end s o f the windo w a s follows : a% = Po(0 ) + p 0 ( l ) H \~Po(i), a>N+i = Po(i + 1 ) -hpo(^ + 2 ) H \-po{a),
(9.12)
9.3. Transmitter s wit h subfilter s
271
for i = 0 , 1 , . . . , a — 1. Not e t h a t ai + ajy+i i s also equa l t o the D C valu e of Po(z). A S a result, th e shift s o f the windo w ad d u p t o a constant , oo
y di+Ni
=
a constant. (9.13
)
This ha s bee n referre d t o a s th e time-domai n Nyquist(N ) propert y i n [97 , 98]. A sequenc e t h a t satisfie s th e tim e domai n Nyquist(A^ ) propert y ha s regula r zero crossing s i n the frequenc y domai n (Proble m 9.4) . W h e n th e D C valu e o f Po(z) i s normalized t o one, a
9 14
5>« = !' (
- )
i=0
the windo w coefficient s i n the middl e ar e equa l t o one an d th e shift s o f the window ad d u p t o one, i.e . Y^L-oo a i+Ni = 1Implementation complexit y Fro m Fig . 9. 9 w e ca n observ e th e complex ity o f the transmittin g bank . I n thi s implementation , onl y th e I D F T operatio n and th e par t o f applying windowin g requir e computations . T h e complexit y for windowin g on e bloc k i s t h at o f the N + a windo w coefficient s plu s a ad ditions du e t o overlap-and-add. W h e n th e D C valu e o f Po(z) i s normalize d to on e a s in (9.14), th e middl e (N — a) coefficient s ar e equa l t o one an d no multiplications ar e needed . Onl y th e 2a coefficient s a t th e tw o end s o f th e window requir e multiplications . T h e complexit y o f each bloc k i s equivalen t to on e I D F T plu s 2a multiplication s an d a additions . Usuall y th e windo w coefficients a>i ar e rea l i n practice . Compare d wit h th e conventiona l case , onl y 2a mor e multiplication s an d a mor e addition s pe r bloc k ar e needed , whic h is a smal l overhead . FEQ coefficient s W e can choose t h e subfilter coefficient s s o t h at t h e F E Q coefficient s 1/A & hav e th e sam e value s afte r th e subfilter s ar e included , i.e. A/ c = Cfc . I n view o f (9.7), th e subfilter s Pk(z) need t o satisf y Pk{e?2*k>M)=
1, k
= 0 , 1 , . . . , M - 1 . (9.15
)
This require s t h a t th e coefficient s o f eac h P k(z) b e normalized . I n the specia l case whe n th e subfilter s ar e shifte d version s o f the first subfilter , a s in (9.8), this conditio n reduce s to Pk{e^k'M)=
P0(e^k/MWk)=
P0(ej0) = 1 , (9.16
)
which i s the sam e a s th e normalizatio n conditio n i n (9.14). I n this cas e ther e is n o nee d t o modify th e F E Q coefficient s i n the origina l D M T receiver . T he F E Q coefficient s remai n th e sam e afte r th e subfilter s ar e included . T h e unit y DC valu e conditio n ca n b e easil y satisfie d b y a simple normalization . W e ca n first desig n PQ(Z) withou t constraint s an d the n normaliz e th e coefficients .
272
9. DM T system s wit h improve d frequenc y characteristic s
Transmitted powe r spectru m Usin g th e filte r ban k representatio n i n Fig. 9.6 , w e ca n expres s th e t r a n s m i t t e d powe r spectru m i n term s o f th e transmitting filter s an d thu s i n term s o f th e subfilter s t o b e optimized . W e know t h a t th e D F T filter s Fk(z) hav e th e conjugate-pai r propert y (fM-k(n)= fk(n)). Suppos e th e subfilter s Pk(z) als o hav e th e conjugate-pai r propert y (not necessaril y th e frequency-shiftin g propert y i n (9.8)) . The n th e equivalen t transmitting filter s continu e t o hav e suc h a property , i.e . ffM-k(n)=
n
f'k(
)'
In th e derivatio n o f th e t r a n s m i t t e d powe r spectru m i n (6.35) , th e onl y as sumption o n th e transmittin g filter s i s t h a t th e filter s ar e i n conjugat e pairs . Therefore, th e t r a n s m i t t e d powe r spectru m wit h subfilter s ca n b e obtaine d directly fro m (6.35) ,
keA
where A i s th e collectio n o f th e subchannel s t h a t ar e use d fo r transmission . T h e spectra l leakag e i s directl y relate d t o th e spectra l rollof f o f th e ne w trans mitting filters . I f th e subfilter s for m a D F T bank , s o d o th e ne w transmittin g filters, an d S
-( e i W ) = Jf E £sMe
j(uJ 2nk/M) 2
-
)\ > (9-18
)
keA which consist s o f th e shift s o f |Fo(e j a ; )| 2 . I n thi s cas e th e spectra l rollof f depends onl y o n th e prototyp e filte r FQ(Z).
9.4 Desig
n o f transmi t subfilter s
T h e additio n o f subfilter s i n Sectio n 9. 3 allow s u s t o modif y th e frequenc y characteristics o f th e transmittin g filters . W e no w desig n th e subfilter s Pk(z) to shap e th e t r a n s m i t t e d powe r spectru m an d minimiz e th e spectra l leakage . We wil l firs t conside r th e genera l cas e o f unconstraine d subfilter s an d the n the specia l cas e o f Pk(z) wit h th e frequency-shiftin g propert y i n (9.8) . Unconstrained cas e W h e n th e subfilter s ar e no t constraine d t o for m a D F T bank , th e t r a n s m i t t e d powe r spectru m i s a s give n i n (9.17) . Th e tota l spectral leakag e i s
S=J S
Jueo ^
u
x(eP
)dLj
4E £ *>* I IW" keA
Jueo
W
)| 2 du, (9.19
where O denote s th e ban d i n whic h leakag e i s undesired . Th e tota l leakag e S ca n b e minimize d i f w e ca n minimiz e th e individua l contributio n Sk fro m each subchannel ,
Sk = f \FUe Jooeo
J
n\2 doj.
T h e filte r F^(z) i s th e produc t o f Fk(z) an d Pk(z); w e ca n writ e it s Fourie r transform a s i^(e*")=Tk(u;)pfc,
)
9.4. Desig n o f transmi t subfilter s
273
where p k i s a n (a + 1 ) x 1 vecto r consistin g o f th e coefficient s o f p k(n) an rk(uj) =
F k(e^)[l e-*»
••
• e-**"]
. (9.20
d )
We ca n the n writ e Sk i n th e followin g quadrati c form : Sfc = p£<&fcPfc , wher
e$
k=
r\{uj)r
k{uj)duj.
(9-21
)
JUJEO
T h e spectra l leakag e du e t o th e kth subchanne l ca n b e minimize d i f w e ca n choose th e kth subfilte r p k t o minimiz e S k. T h e optimizatio n proble m ca n b e cast a s minimize S k, subjec t t o P kPk = 1 . T h e abov e optimizatio n i s th e well-know n eigenfilte r proble m [114 , 159] . A s Sk represent s th e energ y o f th e kth transmittin g filter i n th e frequenc y ban d (9, an d th e F I R transmittin g filters canno t hav e zer o energ y i n (9 , S k i s alway s positive. Therefor e th e matri x $> k is positiv e definite . T o minimiz e S k, w e ca n choose p k a s th e eigenvecto r associate d wit h th e smalles t eigenvalu e o f $> k. If th e subfilter s d o no t for m a D F T bank , neithe r d o th e ne w transmittin g filters (Proble m 9.3) . Constrained cas e W h e n P k(z) satisf y th e frequency-shiftin g propert y in (9.8) , th e transmittin g filters for m a D F T bank . No w w e d o no t hav e the flexibility o f optimizin g eac h transmittin g filter separately . Nonetheless , from th e expressio n i n (9.18) , w e se e t h a t ther e i s n o spectra l leakag e i f th e prototype filter F^{z) i s a n idea l lowpas s filter. T h e undesire d leakag e come s from th e finite stopban d attenuatio n o f th e prototype . T h e tota l leakag e ca n be reduce d b y minimizin g th e stopban d energ y o f th e prototyp e filter, whic h is give n b y
= I \F>(en\
2
diu, (9.22
JUJEOO
where OQ denotes th e stopban d o f th e prototyp e filter. Fo r rea l a^ , a typica l choice o f OQ i s [7r/2M+e , 7r] , wher e e is a smal l number . Followin g a procedur e similar t o t h a t o f derivin g S k, w e ca n writ e th e stopban d energ y (f) a s 0 = p j $ p o , wher
e * =
/ TJ(CJ)T
0 (CJ)GL;,
where TO(CJ ) i s a s give n i n (9.20) . Thu s th e optimizatio n proble m o f th e subfilter coefficient s p o ca n b e formulate d a s minimize (f)
= p j ^ p o , suc
h tha t pjp
o= 1 -
T h e optima l p o i s th e eigenvecto r associate d wit h th e smalles t eigenvalu e o f $. Example 9. 1 Transmitte r subfilter s fo r spectra l leakag e suppression . Th e block siz e i s M = 51 2 an d th e prefi x lengt h i s v = 40 . T h e channe l use d in thi s exampl e i s VDS L l o o p # l (450 0 ft ) [8 ] an d i t i s shortene d b y a T E Q . T h e equivalen t channe l ha s orde r 26 , smalle r t h a n v s o t h a t subfilter s ca n b e
)
274
9. DM T system s wit h improve d frequenc y characteristic s
included. Th e tone s use d ar e 38-9 0 an d 111-255 , an d th e samplin g frequenc y 1/T = 2.20 8 MHz . W e assum e t h a t th e subfilter s ar e shifte d version s o f th e first subfilte r Po(z) 1 an d thu s th e transmittin g filter s F^(z) for m a D F T bank . T h e orde r a o f th e subfilter s i s 14 . W e comput e th e positiv e definit e matri x and th e eigenvecto r correspondin g t o th e smalles t eigenvalu e t o obtai n p 0 . T h e magnitud e respons e o f th e subfilte r Po(e^ UJ) i s show n i n Fig . 9.12(a) . T h e coefficient s o f Po(z) ar e normalize d s o t h a t th e D C valu e i s equa l t o one. Figur e 9.12(b ) show s th e magnitud e respons e o f FQ(Z) normalize d wit h respect t o it s maximum . Fo r comparison , w e hav e als o show n th e magnitud e response o f th e origina l prototyp e F 0(z). Th e ne w prototyp e F${z) ha s a better attenuatio n i n th e stopband . Figur e 9.12(c ) show s th e spectru m o f th e transmitter outpu t wit h an d withou t subfilters . W e se e t h a t th e spectru m of th e windowe d outpu t ha s a muc h smalle r spectra l leakag e i n th e unuse d bands. ■
A not e o n BE R performanc e W h e n w e includ e subfilter s i n th e trans mitter, w e ar e effectivel y changin g th e transmittin g filters . Thi s i n genera l will affec t th e transmissio n power , th e subchanne l SNRs , an d henc e th e B E R performance o f th e overal l system . I t t u r n s ou t t h a t th e subfilter s hav e littl e effect o n an y o f thes e a s w e no w explain . W h e n th e subfilter s ar e normalize d as i n (9.15) , th e subchanne l gain s wil l sta y th e same . A s th e receive r i s no t changed, th e subchanne l SNR s wil l no t b e changed . Als o not e t h a t th e order s of th e subfilter s ar e ver y smal l compare d t o th e numbe r o f subchannel s M. T h e mai n lob e o f Pk(z) i s much wide r t h a n t h a t o f Fk(e^), a s w e ca n se e fro m Figure 9.12(a) . (Th e firs t zer o o f Po(z) i s aroun d 0.17T , whil e th e firs t zer o of th e rectangula r windo w F 0(z) i s aroun d 0.0047T. ) Th e effec t o f subfilter s i s mostly o n th e sidelobe s awa y fro m th e mai n lob e an d th e larg e sidelobe s i n the neighborhoo d o f th e mai n lobes . S o th e subfilter s hav e onl y a mino r effec t on th e transmissio n power . Therefor e th e B E R performanc e i s als o roughl y the same .
275
9.4. Desig n o f transmi t subfilter s
(a)
0.2 0.
4 0.
6 0.
Frequency normalize d byn.
8
(b)
0.2 0.
4 0.
6 0.
8
Frequency normalize d by7i .
I° CQ
wmm mm^mmn
(c)
without subfilter s ■ wit h subfilter s
0.2 0.
4 0.
6 0.
Frequency normalized by7i.
8
Figure 9 . 1 2 . Exampl e 9 . 1 . Transmitte r subfilter s fo r spectra l leakag e suppression : (a) th e magnitud e respons e o f Po(e juJ); (b ) th e magnitud e respons e o f th e prototyp e filter Fo(e JW ) normalize d wit h respec t t o it s maximum ; (c ) th e powe r spectru m o f th e transmitted signa l normalize d wit h respec t t o it s maximum .
9. D M T system s w i t h improve d frequenc y characteristic s
276
9-5 Receiver
s wit h subfilter s
T h e frequenc y characteristic s o f th e receivin g filter s ca n als o b e improve d b y introducing additiona l F I R subfilter s t o th e receiver . Figur e 9.1 3 show s th e receiving bank , wit h a subfilte r include d i n eac h subchannel . Th e /ct h effectiv e receiving filte r i s give n b y H'k{z) =
H
k(z)Qk(z).
Let u s kee p th e transmitte r th e sam e a s i n Fig . 9. 4 (i.e . n o subfilters ) an d us e the receive r i n Fig . 9.13 . No w th e transfe r functio n fro m th e i t h transmitte r input Si(n) t o th e /ct h signa l yk{p) a t th e receive r become s Tu(z)=
[ff
k(Z)(Qfc(z)C(z))Fi(z)]4Ar.
This ha s th e sam e for m a s (9.4 ) excep t t h a t th e channe l C(z) i s replace d b y Qk{z)C{z). Suppos e th e subfilter s hav e orde r equa l t o /? ,
Qk(z) =
n
^2qk(n)z~
.
n=0
From th e resul t i n Lemm a 9.1 , we kno w th e syste m i s fre e fro m IS I a s lon g a s the orde r o f th e produc t Qk(z)C(z) i s no t large r t h a n z/ , i.e . v > /3 + L. T h e difference i s t h at th e subchanne l gain s ar e no w th e M-poin t D F T o f qk(z)*c(n) rather t h a n c(n) , i.e .A & = Qk(z)C(z)\ z= ej2k*/M. A s i n th e cas e o f subfilter s for transmitte r side , extr a cycli c prefi x i s neede d fo r addin g subfilter s withou t affecting th e ISI-fre e property . I f th e channe l ha s bee n shortene d b y a T E Q , the orde r o f th e effectiv e channe l need s t o b e smalle r t h a n th e prefi x length . Similar t o th e transmitte r case , w e ca n choos e th e subfilte r coefficient s s o t h a t th e F E Q coefficient s remai n th e sam e afte r th e subfilter s ar e included . To hav e thi s property , w e ca n normaliz e th e coefficient s o f eac h Qk{z) s o t h a t Qk(ej2*k/M} = 1 , fo r A : = 0 , 1 , . . . , M - 1 .
w
►
Co CO
— ►
H0(z)
1
r
I
w
Qi(z)
w
Hi(z)
•
u (n) ov J u in)
► \N ► \N
• W 17,
, , (~\ J
*| % - l W | I
y,(n)
M fib
1
ATL
▼f
vx0 vxl
• U
^ H QM-&
y0(")
>V-iW
h'(z)
Figure 9 . 1 3 . Receivin g ban k wit h subfilters .
1/u
^
277
9.5. Receiver s with subfilter s
9.5.1 Choosin
g subfilter s a s a DF T ban k
Let u s conside r th e cas e tha t th e subfilter s ar e shifte d version s o f th e firs t subffiter, Qk(z) = Q 0(zWk). The ne w kth receivin g filte r become s H'k(z) =
k W-» kH0(zWk)Q0(zW = )
H k(z)Qk(z)=
W^ kH'0{zWk). (9.23
)
They ar e als o shifte d version s o f H' 0(z) (th e ne w prototyp e filter ) excep t fo r some scalars . A s HQ(Z) is a rectangula r windo w o f lengt h M , an d th e ne w prototype i s the produc t QQ(Z)HQ(Z), th e ne w receiving filters ar e all of length M -\- /3. Suppos e th e subffite r Qo(z) i s causal. Le t th e coefficient s o f HQ(Z) b e bi/y/M, an d writ e i t a s H',{z)
M+/3-1
yv-p
£i=0 kz\
The presenc e o f the advanc e operato r z i s due t o th e origina l prototyp e HQ(Z) (given i n 6.30) , whic h i s noncausal . W e wil l cal l bi th e receive r windo w coefficients. Th e window comes from th e convolution of a rectangular windo w of length M an d a much shorte r qo(n). I n practice , th e subffite r qo(n) usuall y has real coefficients. I n this case, bi will also be real. Simila r to the transmitte r window, most o f the window coefficients ar e of the same value, except fo r thos e at th e tw o end s o f th e window . A plo t o f th e receive r windo w i s ver y simila r to th e plo t o f di i n Fig . 9.7 . Withou t los s o f generality , w e can normaliz e th e DC valu e o f Qo(z) t o unity . The n th e ( M — 0) coefficient s i n th e middl e o f the windo w ar e equa l t o unity . Th e numbe r o f nonunit y coefficient s a t eac h end i s j3. Whe n w e choos e th e subfilter s a s a DF T bank , th e ne w receivin g filters agai n for m a DF T ban k an d thu s ca n b e implemente d efficiently , a s we will se e next .
9.5.2 DF
T ban k implementatio n
Using the relatio n H' k(z) = ing filter a s
W~ vkHfQ(zWk), w
H'k(z) =
?-= Y
e can writ e th e ne w M h receiv -
biW^-ftz*. i=0
We ca n expres s thi s a s
H'k{z)
yV-P
[W~Pk w-W-V
k
...W^-^J
1 z
B
M+/3-1
where B
= dia g [6 0 h
•
•• b
M+(3-i]
•
Note tha t th e /c-dependen t ro w vecto r i n th e abov e equatio n ca n b e writte n as
[W~Pk W-^-^
k
•
• • W k(M-^]=
[ lW
k
•
• • W k(M-^] F
2,
9. D M T system s w i t h improve d frequenc y characteristic s
278
-►—1—H IN ►
p lines
1—413—
r(n)
\ *
H I AT
g(«) »r
7
H^H-
P "™ *
p-i
dOO
-> > -> ►
HI3-
W
1/Jt,
->
|tf
bM |A T ^p
1/X*
lines
window
Figure 9.14 . Efficien t DF T implementatio n o f th e receivin g bank .
where F 2
0
IM-/3
1/3
0
0 1/3
Therefore, w e hav e
H'k(z)
y"-P
[1 ^
...^^-^Fa
B
1 z
(9.24)
M+/3-1
The ne w receivin g ban k W(z) a s indicate d i n Fig . 9.1 3 ca n b e obtaine d b y stacking H' k(z) together . I n (9.24) , only the ro w vector depend s on k; stackin g these ro w vector s togethe r give s ris e t o th e DF T matri x W . Th e receivin g bank i s thus give n b y
h'(s)
H[{z)
#M-l(4
r^WFoB
1 z
(9.25)
M+/3-1
This expression gives us the implementatio n o f the receive r in Fig. 9.14, wher e we hav e move d th e decimator s t o th e lef t b y usin g th e nobl e identit y fo r decimators i n Sectio n 4.1.2 . Not e tha t th e firs t v — f3 samples ar e discarde d due t o th e advanc e operato r z v~$. In th e DF T ban k implementatio n o f Fig . 9.14 , th e receive d sample s ar e first multiplie d b y windo w coefficient s b^. Thi s i s illustrate d i n Fig . 9.15(a) .
279
9.5. Receiver s w i t h subfilter s window coefficients b t I
\L.
■ 11
M-l+p i
received samples of the vector r(k)
LllllL p samples
= (a)
samples of the vector g(k)
iL p samples
.illll.nlli.illlllll
—►
coefficients affecte d b y the windo w
add samples of the vector g(k)
illll.iill p samples
(b)
samples of the vector d(&)
illlll
LLL
p samples
p samples
Figure 9.15 . Receive r windowing , (a ) Multiplyin g windo w coefficients ; (b ) addin g th e first f3 samples t o th e las t ft sample s t o produc e th e sample s o f d ( n ) .
When th e D C valu e o f Qo(z) i s unity, onl y th e coefficient s o n th e tw o end s of the windo w ar e not equa l to on e and multiplication s ar e needed onl y fo r thes e coefficients. The n the matrix F2 in (9.25) performs th e operation of adding th e first P samples t o th e las t j3 samples , a s show n i n Fig . 9.15(b) . Th e resultan t vector d(n) , a s indicate d i n Fig . 9.14 , i s passe d ove r fo r DF T computatio n and FEQ . Thi s i s the sam e a s the usua l receive r windowin g describe d i n [143] . When the window has real coefficients, thes e multiplications an d addition s will also be real. Compare d t o th e conventiona l DM T receiver , th e ne w receiver i n Fig. 9.1 4 need s onl y 2(3 mor e multiplication s an d /3 mor e addition s pe r bloc k of outputs . Th e overhea d i s ver y small . Time domai n Nyquist(M ) propert y A s th e windo w i s th e convolutio n of a rectangula r windo w an d qo(n), th e coefficient s bi ca n b e expresse d i n a form similarl y to those in (9.12) . Simila r t o the transmitter window , th e shift s
9. DM T system s wit h improve d frequenc y characteristic s
280
of th e receive r windo w als o ad d u p t o a constant , oo
2_j bi+Mi
=
constant . (9.26
)
In particular , th e constan t i s on e whe n th e D C valu e o f Qo(z) i s normalize d t o one. A sligh t differenc e i s t h a t th e shift s i n (9.26 ) ar e multiple s o f M , instea d of T V in (9.13) .
9.6 Desig
n o f receive r subfilter s
In earlie r sections , w e sa w t h a t subfilter s a t th e transmitte r ca n hel p t o im prove spectra l rollof f an d reduc e spectra l leakage . A t th e receive r side , w e wil l employ subfilter s fo r a differen t application , namel y R F I suppression . Th e ra dio interferenc e i s know n t o b e o f a narrowban d nature . Fo r th e duratio n o f one D M T symbol , i t ca n b e considere d a s a su m o f sinusoids . Suppos e ther e are J interferenc e sources , occurrin g a t frequencie s w^, fo r H = 0 , 1 , . . . , J — 1 . T h e interferenc e i s modele d a s J-I
v(n) =
2_. I JL£ cos(cj£n + 6i). (9.27
)
£=0
It i s characterize d b y th e amplitude s /j,£, frequencies oj£, and phase s 0£. To analyz e th e effec t o f interference , w e appl y a n interference-onl y signa l v(n) t o th e receive r i n Fig . 9.13 . Th e decimator s d o no t chang e th e amplitude s of th e interferenc e signals . W e ca n conside r th e interferenc e a t th e outpu t o f the /ct h receivin g filte r H k(z), whic h i s 1
Mn)=
J- i
~ Yl »' [H' k(ePUt)eP^tn+et) + £=0
H' k(e-jut)e-j^tn+et)] .
(9.28
)
Minimization o f th e abov e interferenc e t e r m require s th e knowledg e o f J , fi£, out, an d 0%. Dependin g o n th e availabilit y o f thi s knowledge , tw o case s wil l be considere d i n wha t follows . I n th e cas e whe n th e interferenc e source s ar e known, w e wil l se e t h a t a n R F I o f th e simpl e for m i n (9.27 ) ca n b e completel y eliminated i f th e numbe r o f interferin g source s J < /3/2 , wher e /3 is th e orde r the subfilters . Case 1 : interferenc e source s ar e know n I f th e informatio n o f th e inter ference source s i s availabl e t o th e receiver , th e subfilter s ca n b e individuall y optimized. W h e n w e observ e th e expressio n o f th e /ct h interferenc e signa l Uk{n) i n (9.28) , w e se e t h a t it s amplitud e i s a nonlinea r functio n o f th e /ct h subfilter coefficients . T o simplif y th e problem , not e t h a t th e par t o f Uk(n) du e to th e £t h interferenc e sourc e wil l b e smal l i f
tf(\H'k&Ut)\2 +
\ H'k(e-JUt)\2)
is small. Th e /ct h subchanne l interferenc e ca n b e mitigate d b y designin g Q to minimiz e
4 = J2»K\H' k(eJ"e)\2 + \H' k(e-^)\2). (9.29 £=0
k(z)
)
9.6. Desig n o f receive r subfilter s
281
We ca n writ e
k= I \H'
k(e^)\
duj,
(9.30
)
where Ok denote s th e stopban d o f th e kth receivin g filters . Suc h a n approac h also ha s th e advantag e t h a t th e subfilter s nee d t o b e designe d onl y once ; the y need no t b e redesigne d whe n th e interferenc e changes . Following a procedur e simila r t o t h a t fo r designin g transmitte r subfilters , we writ e H k(z) a s H'k(ei")=Tk(u)qk, where r k(uj)= H k(ejuJ) [ l e~ function
juJ
(j)k = q£A f e q f e , wher
j
• • e~
•
eA
k=
/
^] .
I t follow s t h a t th e objectiv e
r\{uj)r
JujEOk
k{uj)duj.
(9.31
)
T h e matri x A & i s positiv e definit e becaus e (f) k represents th e stopban d energ y of th e receivin g filters . T o minimiz e (j) kl w e ca n choos e q ^ a s th e eigenvecto r associated wit h th e smalles t eigenvalu e o f A& . Suppos e th e stopban d O k o f the M h receivin g filte r i s a shif t o f OQ i n frequenc y b y 2/c7r/M . Fro m (9.30) , we hav e 4k= f
\H Jok Jo
2
k(e^)Qk(e^)\
dco
= f \H k
k
0(e^-
^)\2\Qk(e^)\2 dco.
282
9. DM T system s wit h improve d frequenc y characteristic s
W i t h a chang e o f variables , w e observ e t h a t th e abov e equatio n become s 4k = I \Ho(en\
2
\Qk(e^+2k^/M))\2 dr.
T h e abov e expressio n mean s t h a t i f Qo(z) minimize s 0o , the n th e choic e Qk(ej(uj+2h7r/M) = ) Qo(e juJ) wil l als o minimiz e (j> k. In thi s cas e 00 = 0 1 = ' ' ' = 0 M - 1 In othe r words , Qk(z) = Qo(zW k) minimize s 2 J , th e interference canno t b e completel y nullifie d a s th e interferenc e signa l i s not a pur e su m o f sinusoids . • Case 2: Interference sources are not known. W e choos e th e subfilter s Qk(z) t o b e shifte d version s o f Qo(z) an d onl y Qo(z) need s t o b e de signed. Th e subfilte r Qo(z) i s designe d withou t makin g us e o f an y R F I information. I t i s th e solutio n t o th e objectiv e functio n 0 o i n (9.31) . In thi s cas e th e receivin g filter s for m a D F T ban k an d ca n b e imple mented efficientl y a s i n Fig . 9.14 . Th e magnitud e respons e o f Qo(z) i s as show n i n Fig . 9.16(a) . W e sho w th e magnitud e respons e o f HQ(Z) i n Fig. 9.16(b) , normalize d wit h respec t t o it s maximum . Als o show n i n the figur e i s th e magnitud e respons e o f th e rectangula r windo w (HQ(Z)). We ca n se e t h a t wit h subfilte r shaping , HQ(Z) ha s bette r stopban d an d hence bette r frequenc y separation .
9.6. Desig n o f receive r subfilter s 28
3
The SINR s (signa l to nois e interference ratio ) o f the subchannel s ar e as show n in Fig. 9.16(c). Fo r comparison, w e have also shown the subchanne l SINR s fo r the rectangula r window . Th e receiver s wit h subfilter s enjo y highe r SINR s fo r the tone s tha t ar e clos e to th e RF I frequencies . Especiall y whe n th e informa tion o f the RF I source s is known an d th e subfilter s ar e optimized individually , the RF I i s significantly suppressed . A s a result, highe r transmissio n rate s ca n be achieved . A s an example , Tabl e 9. 1 shows the transmissio n rate s usin g th e bit loadin g formul a i n (6.38 ) whe n th e symbo l erro r rat e i s 1 0 - 7 . I t shoul d be note d tha t th e secon d subfilte r desig n attain s a muc h highe r transmissio n rate b y exploitin g th e RF I information . Whe n th e RF I changes , w e nee d t o redesign th e subfilters . Bot h th e desig n an d implementatio n cos t ar e muc h higher tha n th e origina l DM T receiver . O n th e othe r hand , whe n th e subfil ters ar e independen t o f the RFI , a s in cas e 1 , the implementatio n cos t wil l b e much lower . ■ Rectangular windo w Cas 6.84 7.4
e 1 Cas e 2 4 8.5
4
Table 9 . 1 . Transmissio n rate s give n i n meg a bit s pe r second .
9. D M T system s w i t h improve d frequenc y characteristic s
284
(a)
-20
-40
0.2 0. 4 0. 6 0. Frequency normalized byn.
8
(b)
0.2 0. 4 0. 6 0. Frequency normalized byrc.
8
50 r 40 30 (c)
20
10
■ subfilter(firs t case ) ■ subfilter(secon d case ) Rectangular windo w
50
100 15
tone index
0
200
250
Figure 9 . 1 6 . Exampl e 9.2 . Receive r subfilter s fo r RF I reduction : (a ) th e magni tude respons e o f Qo(e juJ)\ (b ) th e magnitud e respons e o f th e prototyp e filte r HQ(e JUJ) normalized wit h respect t o it s maximum ; (c ) th e subchanne l SINRs .
9.7. Zero-padde d transceiver s
9.7 Zero-padde
285
d transceiver s
In previou s sections , th e discussio n refer s t o prefixe d transceivers . W e ca n als o apply subfilter s t o th e zero-padde d syste m i n Sectio n 6.2 . T h e zero-padde d transceiver als o ha s th e filter ban k representatio n i n Fig . 9.4 . T h e receive r for th e zero-padde d cas e i s no t unique . Le t u s conside r th e efficien t receive r in (6.15) . T h e transmittin g (receiving ) filters ar e frequency-shifte d version s of th e first transmittin g (receiving ) filter excep t fo r som e scalars . Bu t no w the transmittin g prototyp e i s a rectangula r windo w o f lengt h M whil e th e receiving prototyp e i s a length- N rectangula r window . W h e n w e inspec t th e filter ban k representatio n Fig . 9. 4 fo r th e zero-padde d transceiver, th e ISI-fre e propert y continue s t o hol d fo r an y channe l C(z) wit h order les s t h a n o r equa l t o th e numbe r o f padde d zero s v. Therefor e th e results i n Lemm a 9. 1 ca n b e appropriatel y modifie d fo r th e zero-padde d case . Subfilters ca n b e employe d a t th e transmitte r t o shap e th e transmitte d powe r spectrum o r a t th e receive r t o mitigat e th e effec t o f R F I . I n particular , w e can hav e transmittin g subfilter s a s i n Fig . 9.6 . A s lon g a s th e orde r a o f th e subfilters Pk(z) an d th e channe l orde r L satisf y L + a < v, th e overal l syste m still enjoy s th e ISI-fre e property . Similarly , w e ca n includ e receivin g subfilter s as i n Fig . 9.1 3 i f th e orde r /3 o f th e subfilter s Qk(z) satisfie s L + /3 < v. Therefore t o ad d subfilter s o f orde r /3 , the channe l need s t o b e shortene d t o an orde r L < v — (3. [68, 89 , 102 , 113 , 141 , 155 ]
9.8 Furthe
r readin g
Many method s hav e bee n develope d i n th e literatur e t o enhanc e th e frequenc y separation amon g th e subchannels . Fo r example , t o improv e th e spectra l rolloff o f th e transmitte d signal , a numbe r o f continuous-tim e puls e shapin g filters hav e bee n propose d [68 , 89, 102 , 113 , 141 , 155]. Usuall y continuous-tim e pulse shape s ar e designe d base d o n a n analo g implementatio n o f th e trans mitter, an d i n genera l i t i s no t possibl e t o hav e a digita l implementatio n [79] . Discrete-time window s hav e bee n considere d i n [15 , 86 , 138] . T h e desig n o f overlapping window s fo r O F D M wit h offse t QA M ove r frequency-nonselectiv e channels wa s studie d full y i n [15 , 138] . W h e n th e channe l i s distortionless , orthogonality amon g th e subchannel s i s preserve d b y th e windo w [15 , 138 ] and a bette r spectra l efficienc y i s achieved . Fo r IS I channels , additiona l post processing ca n b e use d t o remov e interference . Mor e recently , transmittin g windows wit h th e cyclic-prefi x propert y [31 , 80 ] hav e bee n considere d fo r egress control . Window s t h a t ar e th e invers e o f a raise d cosin e functio n wer e optimized i n [31 ] to minimiz e egres s emission . T o compensat e fo r th e transmit ter window , th e correspondin g receive r require s post-processin g equalizatio n [31, 80] . A join t consideratio n o f spectra l rollof f an d SN R degradatio n du e to post-processin g wa s give n i n [80] . Transmitte r windowin g fo r zero-padde d DFT-based transceiver s ha s als o bee n addresse d therein . Per-ton e window s were propose d i n [20 ] fo r shapin g th e transmitte d spectrum . T h e shapin g o f spectra allow s mor e tone s t o b e use d fo r transmission . Windowing i s als o ofte n applie d a t th e receive r side . T o improv e R F I suppression i n DS L applications , receive r windowin g ha s bee n propose d i n
9. D M T system s w i t h improve d frequenc y characteristic s
286
the literatur e [29 , 143] . Th e cleve r ide a o f usin g extr a redundan t sample s and foldin g t o reduc e R F I wa s give n i n [143] . Fo r th e suppressio n o f sidelobe s without usin g extr a redundan t samples , i t wa s propose d i n [61 ] to us e window s t h a t introduc e controlle d IBI , remove d late r usin g decisio n feedback . A join t minimization o f R F I an d channe l nois e wa s considere d i n [123] . Th e optima l window ca n b e foun d usin g th e statistic s o f th e R F I an d channe l nois e [123] . Also usin g th e statistic s o f th e channe l nois e an d RFI , a join t desig n o f th e T E Q an d th e receivin g windo w fo r maximizin g bi t rate s wa s give n i n [188] . Minimization o f R F I usin g subfilter s wa s propose d i n [83] . A nove l combina tion o f raise d cosin e windowin g an d per-ton e equalizatio n withou t usin g extr a redundant sample s wa s give n i n [32] . Per-ton e windowin g fo r R F I suppressio n without usin g th e informatio n o f R F I an d channe l nois e statistic s ha s bee n suggested i n [180] . Receive r windowin g ha s als o bee n considere d fo r O F D M applications. T o improv e th e receptio n o f O F D M systems , i t wa s propose d i n [98] t o us e Nyquis t windows . Optima l Nyquis t window s wer e considere d i n [97] t o mitigat e th e effec t o f additiv e nois e an d carrie r frequenc y offsets . In th e contex t o f R F I mitigation , interferenc e cancellatio n ha s bee n sug gested i n [13] . Usin g th e receive r o u t p u t s o f unuse d R F I tone s o r neighborin g tones, th e parameter s o f th e R F I signa l ca n b e estimate d an d use d t o cance l interference o n th e tone s t h a t ar e affected . P a r a m e t e r estimatio n an d can cellation fo r th e cas e whe n th e R F I sourc e i s a singl e sideban d signa l wa s developed i n [56] . Base d o n th e approac h i n [13] , a mor e genera l framewor k for R F I cancellatio n wa s give n i n [140] . Th e metho d i s applicabl e t o R F I o f all analo g modulatio n schemes , e.g . amplitud e modulation , singl e sideband , double sideband , an d s o forth .
9.9 Problem
s
9.1 Trapezoidal window. Suppos e th e transmitte r windo w ai i s th e discrete time Trapezoida l functio n o f lengt h N + a a s show n i n Fig . P 9 . 1 . T h e coefficients satisf y a a = a a + i = • • • = a^-i= 1 . Th e tw o end s o f th e window dro p linearl y t o zero . Th e numbe r o f nonunit y coefficient s o n each en d i s a. window / coefficients a i /
1k \ 0a
V
\ N
\ , 7V+
af
Figure P 9 . 1 . Trapezoida l windo w function .
(a) Doe s th e trapezoida l functio n satisf y th e tim e domai n Nyquis t con dition i n (9.13) ? (b) Suppos e w e use the trapezoida l windo w i n the D F T implementatio n in Fig . 9.9 . Sho w t h a t th e syste m i n Fig . 9. 9 ha s th e equivalen t subfilter receivin g ban k structur e i n Fig . 9.6 .
9.9. Problem s 28
7
(c) Fin d th e correspondin g subfilte r PQ(Z) i n (b) . 9.2 W e ar e give n a transmitte r windo w functio n a>i t h a t ha s lengt h N + a and satisfie s th e tim e domai n Nyquis t conditio n i n (9.13) . I n thi s cas e does th e D F T implementatio n i n Fig . 9.1 4 alway s hav e th e equivalen t subfilter receivin g ban k structur e i n Fig . 9.13 ? I f so , ho w ar e th e coef ficients o f Po(z) relate d t o ap. 9.3 T h e transmittin g filters Fk(z) i n th e conventiona l D M T syste m for m a D F T bank . Fro m Sectio n 9. 3 w e als o kno w t h a t th e ne w transmittin g filters F^(z) agai n for m a D F T ban k whe n th e subfilter s ar e frequency shifted version s o f th e first subfilter . Sho w t h a t i f th e subfilter s d o no t form a D F T bank , neithe r d o th e ne w transmittin g filters. 9.4 Le t g(n) b e a sequenc e satisfyin g th e tim e domai n Nyquis t conditio n J2i9^ + Nl) = 1 . Le t G(e j a ; ) b e th e Fourie r transfor m o f g(n). Sho w that G{e?2*klN= ) 0 , fo r k = 1 , 2 , . . ., N - 1 . In othe r words , th e Fourie r transfor m o f g(n) ha s th e zero-crossin g prop erty. 9.5 Le t h(n) b e a rectangula r windo w o f lengt h N an d le t g(n) b e a n arbi t r a r y sequenc e o f lengt h P. Sho w t h a t th e lengt h P + N — 1 sequenc e h(n) * g(n) satisfie s th e tim e domai n Nyquist(iV ) property . 9.6 Conside r th e transmitte r i n Fig . 9.9 . Sho w t h a t i f th e subfilte r po(n) n is normalize d suc h t h a t Y^= oPo( = ) 1 ? the n th e complexit y ca n b e reduced t o on e I D F T plu s 2a addition s an d a multiplication s pe r block . 9.7 W e kno w th e receive r i n Fig . 9.1 4 need s onl y 2(3 more multiplication s and (3 more addition s pe r bloc k o f output s whe n i t i s compare d wit h th e conventional D M T receiver . Sho w t h a t whe n th e windo w satisfie s th e Nyquist property , th e numbe r o f multiplication s ca n b e furthe r reduce d to (3. 9.8 Transmitter window not satisfying the time domain Nyquist condition. Suppose th e transmitte r windo w coefficient s a>i i n Fig . 9. 9 d o no t satisf y the tim e domai n Nyquis t conditio n i n (9.13) . Doe s th e D F T implemen tation i n Fig . 9. 9 stil l hav e th e equivalen t receivin g ban k structur e i n Fig. 9.6 ? 9.9 I n Sectio n 9.6 , i t wa s mentione d t h a t th e receive r subfilter s ca n b e in dividually optimize d whe n th e informatio n o f th e interferenc e source s i s available. Sho w t h a t th e objectiv e functio n /c - Fin d th e matri x 4>/c - I s 4>/ c alway s positiv e definite ? 9.10 Transmitter windowing without extra redundancy [80] . T h e transmitte r window discusse d i n thi s chapte r require s th e prefi x lengt h v > L + a, where a i s the orde r o f the transmitte r subfilter s an d L i s the orde r o f th e F I R channel . Thi s mean s w e nee d t o hav e a mor e redundan t sample s t h a n th e D M T syste m i n Chapte r 6 . I n thi s problem , w e conside r th e
9. D M T system s w i t h improve d frequenc y characteristic s
288
case whe n ther e ar e n o extr a redundan t samples , i.e . v = L an d N = M + L. Th e transmitte r i s a s show n i n Fig . P9.10(a) , wher e F cp i s the N x M cyclic-prefi x insertio n matri x define d i n (5.14) . Suppos e th e transmitter windo w ha s th e cyclic-prefi x property . T h a t is , th e firs t v window coefficient s ar e th e sam e a s th e las t v coefficients , :
&M+i
:0,1,...,I/-1.
5
window
'„(«) (a)
^
*,(«)
W")
•• •
"0
w+
ai
P/S (N)
F
* cp
•••
x(ri)
Vi
transmitting matrix G
—>—H
->—H
(b)
discard prefix
S/P (M)
W
H>—H
—>—H
W
—«>—H
w
—»—H
Figure P 9 . 1 0 . (a)Transmitte r windo w withou t extr a redundancy ; (b ) ISI-fre e receive r for th e transmitte r i n (a) .
(a) Deriv e th e receive r s o t h a t th e overal l syste m i s ISI-free . (b) Sho w t h a t th e ISI-fre e receive r i n (a ) ca n b e implemente d a s th e structure i n Fig . P9.10(b) . Th e coefficient s Xk ar e th e reciprocal s of th e channe l D F T coefficients , Xk = 1/C/e , wher e Ck ar e th e Mpoint D F T o f th e channe l impuls e response . Th e post-processin g matrix P a s indicate d i n Fig . P9.10(b ) depend s onl y o n th e windo w but no t th e channel . W h e n ther e ar e n o extr a redundan t samples , applyin g windowin g a t th e transmitter result s i n a receive r t h a t generall y depend s heavil y o n th e
9.9. Problem s 28
9
channel. T h e resul t i n thi s proble m show s t h a t i f th e windo w ha s th e cyclic-prefix property , the n th e channe l dependen t par t o f th e receive r continues t o b e th e F E Q coefficient s only . 9.11 Fin d a windo w functio n o f lengt h M + v t h a t satisfie s b o t h th e tim e domain N y q u i s t ( M ) propert y an d th e cyclic-prefixe d propert y describe d in Proble m 9.10 . Giv e a sketc h o f th e window . 9.12 I n Proble m 9.10 , th e windo w ha s th e cyclic-prefi x property , i.e . ai = dM+i, f ° r * = 0 , 1 , . . . , z / — 1. Suppos e no w t h a t th e windo w doe s no t have thi s property . (a) Deriv e th e receive r s o t h a t th e overal l syste m ha s th e ISI-fre e prop erty. (b) Sho w t h a t th e ISI-fre e receive r i n (a ) ca n b e implemente d a s i n Fig. P9.12 . T h e post-processin g matri x P i s o f th e for m W
Ai 0 0A
2
Wf+ A W
0 C 00
0
(A0-A2)
wf
where A Q , A I, an d A 2 ar e diagona l matrice s wit h diagona l ele ments consistin g of , respectively , th e first v windo w coefficients , th e following M — v coefficients , an d th e las t v coefficients . T h e matri x A i s a s i n Proble m 9.1 0 an d C o i s a n L x L lowe r triangula r Toeplit z matrix wit h th e first colum n give n b y [c o c\ . . . CL-I\ • I s the post-processin g matri x P stil l channel-independen t a s i n P r o b lem 9.10 ?
discard prefix
S/P
m
w
* i
Figure P 9 . 1 2 . Structur e o f th e ISI-fre e receiver .
9.13 Effect of subfilters on the subchannel noises. T o mitigat e interference , we include d subfilter s i n th e receivin g bank . Thes e additiona l subfilter s will affec t th e subchanne l outpu t noise . Le t th e channe l nois e b e AWG N with varianc e A/o - Le t Qo(z) b e th e subfilte r i n th e zerot h subchanne l with Q 0 ( e j 0 ) = 1 . (a) W h a t i s th e zerot h subchanne l nois e varianc e o\ du e t o channe l noise?
290 9
. DM T system s wit h improve d frequenc y characteristic s
(b) Suppos e Qo(z) ha s magnitud e respons e a s show n i n Fig . 9.16(a) . Do you expect th e nois e variance in (a ) to b e larger o r smaller tha n the cas e withou t a subfilter ? (c) Suppos e th e subfilter s ar e frequency-shifte d version s o f th e firs t subfilter Qo(z). Wha t i s th e kth subchanne l nois e varianc e o\ k due t o channe l noise ? I s i t th e sam e a s tha t i n (a) ? 9.14 Zero-padded transceivers. Conside r th e zero-padde d transceive r i n Sec tion 6. 2 wit h th e efficien t receive r i n (6.15) . (a) Appl y subfilter s t o th e transmitte r a s i n Fig . 9.6 . Suppos e th e subfilters Pk{z) ar e shifte d version s o f th e zerot h subfilte r Po(z). Show tha t th e ne w transmittin g filter s for m a DF T ban k an d ca n be implemente d efficiently . Dra w th e DF T implementation . (b) Suppos e th e efficien t receive r give n i n (6.15 ) i s used. Appl y subfil ters t o th e receive r a s i n Fig . 9.13 . Agai n assum e th e subfilter s ar e shifted version s o f th e firs t subfilter . Sho w tha t th e ne w receivin g filters als o for m a DF T ban k an d ca n b e implemente d efficiently . Draw th e DF T implementation .
10 Minimum redundanc y FI R transceivers
In earlie r chapter s w e sa w tha t th e us e o f redundanc y i n bloc k transceiver s allows us to remove ISI completely without usin g IIR filters. Whe n the numbe r of redundan t sample s pe r bloc k v i s mor e tha n th e channe l orde r L , ther e i s no IBI, an d w e can furthe r achiev e zero ISI using a constant receivin g matrix . The mos t notabl e exampl e i s th e OFD M syste m studie d i n Chapte r 6 . Bu t the us e o f redundan t sample s als o decrease s th e transmissio n rate . Fo r ever y M inpu t symbols , th e transmitte r send s ou t N = M + v samples . Th e actua l transmission rat e i s decrease d b y a facto r o f N/M. Ther e ar e v redundan t samples i n ever y N sample s transmitted . Reducin g redundanc y lead s t o a higher transmissio n rat e an d henc e bette r bandwidt h efficiency . A t th e sam e time, we would like the redundancy t o be large enough so that th e zero-forcin g condition ca n stil l be satisfie d withou t usin g IIR filters . A natural questio n t o ask is : fo r a give n channe l an d AT" , wha t i s the smalles t redundanc y suc h tha t FIR transceiver s exist ? I n othe r words , i f w e ar e t o us e a n FI R transceive r that achieve s zer o ISI , wha t i s th e larges t numbe r o f symbol s tha t ca n b e transmitted ou t o f every N samples ? Thi s chapter aim s to answer the questio n of minimu m redundanc y fo r th e existenc e o f FI R zero-forcin g transceivers . We wil l conside r genera l FI R transceiver s (Fig . 10.1 ) i n whic h th e filter s are no t constraine d t o b e DF T filter s a s i n th e OFD M system . Moreove r the lengt h o f th e filter s ca n b e longe r tha n th e bloc k siz e N. I n thi s cas e the transmittin g an d receivin g matrice s ar e allowe d t o hav e memories , rathe r than constan t matrice s a s i n th e OFD M case . W e will se e that th e minimu m redundancy depend s o n th e underlyin g channe l C(z), an d i t ca n b e easil y determined fro m th e locatio n o f th e zero s o f th e channe l C(z) directl y b y inspection. Th e topi c o f minimu m redundanc y fo r FI R transceiver s wa s firs t addressed i n [182] . Th e proble m ha s bee n studie d usin g differen t approache s [62, 76 , 124 , 135] . Fo r consistenc y wit h th e res t o f th e book , w e wil l follo w the approac h give n i n [76] . 291
292
10. M i n i m u m redundanc y FI R transceiver s q(n)
U
11
s0(n) -► ! fN
>\ ^ 0 W | ,
s^n) -+J fN
11
u
H ^l W wZ
7 /_ \ *
n(")
wE
7
/-
wZ
*9
,N
—M
r
c:hannel
i (70
*| V i W |
_1 A fw
s (n)
►•» 0 w
- * | /^(z ) |-* | + # )—►*> ) •• •• •• A
\
^
T f_ \ w
n o yz) —► ! + 7V*
>r
>
• • •
W ^ ft f
4
x(n)
L+\HM_I(Z}\^\IN\—►^
transmitting filters
receiving filters
Figure 1 0 . 1 . M-subchanne l filte r ban k transceiver .
transmitting polyphase matrix
receiving polyphase matrix
s0(n)
J
oW
q(»)
s{(n)
G(z)
1(H)
Cps(z)
i
Wn>
r(«)
s{(n)
S(z)
Ww>
T(z) Figure 1 0 . 2 . Polyphas e representatio n o f th e transceiver .
10-1 Polyphas
e representatio n
In th e followin g discussion , i t i s ofte n mor e convenien t t o wor k wit h th e polyphase representatio n (Chapte r 5 ) o f th e transceiver , whic h allow s u s t o redraw th e filte r ban k transceive r a s i n Fig . 10.2 . I n th e figure , th e N x M matrix G{z) i s the polyphas e matri x o f the transmittin g ban k an d S(z) i s th e M x N polyphas e matri x o f the receivin g bank . Th e matrice s G(z) an d S(z) are no t constraine d t o b e constan t matrice s a s i n Chapter s 6-8 . Th eT V x 1 vector q(n ) i s the blocke d versio n o f the channe l nois e q(n). Throughou t thi s chapter, th e channe l nois e wil l no t appea r i n ou r discussio n a s i t doe s no t affect th e existenc e o f FI R zero-forcin g filte r ban k transceivers . Th e blocke d channel matri x C ps(z) i s a pseudocirculant, whic h ha s been introduce d earlie r in Chapte r 5 . Som e o f its propertie s ar e exploite d i n Sectio n 5. 4 fo r eliminat -
10.2. Propertie s o f pseudocirculant s
293
ing interbloc k interference . T o b e mor e specific , le t th e channe l b e a n FI R filter o f orde r L: c(0 ) + c ( l ) ^ - 1 + • • • + c(L)z~
C(z) =
L
.
To avoi d degenerat e cases , w e wil l conside r C(z) wit h c(0),c(L ) 7 ^ 0 , an d L > 1 . Th e polyphas e representatio n o f C(z) i s C(z) =
C 0{zN) + C 1{zN)z~1 +
• • • + CW-i(* N )*-7V+1
(10.1)
In thi s chapter , th e channe l orde r L ca n b e longe r tha n N. Al l the polyphas e components Ci(z) ar e in general polynomials in z - 1 . The n the blocked versio n Cps(2:) i s give n b y
Qjps(z)
C0(z) C\(z)
z-^izj z-YC2{z)
C'o(z)
CN-i(z) C
(10.2)
C0(z)
N-2(z)
The stud y o f pseudocirculan t matrice s turn s ou t t o b e o f vita l importanc e in determinin g minimu m redundanc y o f FI R transceivers . Som e usefu l prop erties wil l b e give n i n Sectio n 10.2 . Fro m Fig . 10.2 , w e kno w th e conditio n for zer o IS I ca n b e give n i n term s o f polyphas e representation . Ther e i s zer o inter-subchannel interferenc e i f S(z)Cps(z)G(z) =
A(z),
(10.3)
for som e diagona l matri x A(z). Th e transceive r i s ISI-fre e (zer o inter - an d intra-subchannel ISI ) if , i n addition, th e diagona l element s o f A(z) ar e merel y delays, i.e . [A(z)] ^ = A^ _ n % wit h A ^ 7^ 0. I n thi s cas e w e ca n alway s scal e and dela y th e receive r output s s o tha t S(z)Cps(z)G(z)
>-M-
(10.4)
Thus, without los s of generality we can consider this ISI-free conditio n becaus e delays an d scalin g d o no t chang e th e FI R propert y o f the system .
10.2 Propertie
s o f pseudocirculant s
In thi s sectio n w e introduc e som e propertie s o f pseudocirculant s tha t wil l b e useful fo r ou r subsequen t discussion . W e wil l diagonaliz e pseudocirculant s using tw o type s o f decomposition . Th e firs t one , Smit h for m decomposition , uses unimodula r matrice s t o diagonaliz e FI R pseudocirculants . Th e secon d one diagonalizes C ps(zN), rathe r tha n C ps(z), usin g DF T matrice s an d som e simple diagona l matrice s [157 , 159]. 1 1 For a mor e in-dept h treatmen t o f thes e decompositions , th e reader s ar e referre d t o Chapter 1 3 of [159 ] o r [58] .
294
10. Minimu m redundanc y FI R transceiver s
10.2.1 Smit
h for m decompositio n
A polynomia l matri x A(z) i n z~ x ca n b e diagonalize d b y th e so-calle d uni modular matrices . W e sa y a n N x N polynomia l matri x XJ(z) i s unimodula r if det U(z) = c , a nonzer o constan t [159] . Whe n XJ(z) i s causa l FI R unimodular , s o i s it s inverse \J~ 1(z)1 du e t o th e constan t determinan t property . An N x N polynomia l matri x A(z) i n z~ x ca n b e represente d usin g th e Smith for m decompositio n A(z) =
U(z)T s(z)V(z), (10.5
)
where al l thre e matrice s i n th e decompositio n ar e matri x polynomial s i n th e variable z~ x. Bot h XJ(z) an d V(z ) ar e unimodula r matrices , an d T s(z) i s a diagonal matrix :
r*(*)
jo(z) 0 07
.. i (z) 0
00
..
.0
.
-i(4
7JV
Moreover th e unimodula r matrice s U(z ) an d V(z ) ca n b e s o chose n tha t th e polynomials jk(z) ar e monic (i.e . th e highes t powe r ha s unity coefficient ) an d jk(z) i s a facto r o f j k+i(z) (i.e . j k(z) divide s jk+i(z) fo r 0 < k < N — 2) . In thi s cas e th e matri x T s(z), calle d th e Smit h for m o f A(z) , i s unique. Th e diagonal element s ~f k(z) ar e give n b y
k+1(z)/Ak(z),
where A k(z)yk= 1,2 , ...,iV , i s th e greates t commo n diviso r (gcd ) o f al l theA : xA : minors o f A(z) , an d A 0(z)= 1 . Althoug h T s(z) i s unique , th e unimodular matrice s U(z ) an d V(z ) ar e not . Th e decompositio n ca n b e obtained usin g a finit e numbe r o f elementar y ro w an d colum n operation s [159]. Return t o the blocke d version of the transceiver syste m i n Fig. 10.2 . Whe n the scala r channe l C{z) i s causal an d FIR , th e blocke d channe l matri x C ps(z) is als o causa l an d FIR . Le t u s appl y Smit h for m decompositio n t o C p s (z), Cps(z) =
U(z)T s(z)V(z), (10.6
)
where T s(z) i s th e Smit h for m o f C ps(z). Le t PN denot e th e numbe r o f nonunity term s i n the diagonal s o f the Smit h form . Thi s numbe r turn s ou t t o be relate d t o th e minimu m redundanc y fo r th e existenc e o f FI R transceivers , as w e will se e later . Wit h th e definitio n o f p^v , we ca n expres s T s(z) a s r s ( z ) = diag[ l 1
.. . 1
1N-
PN(Z)
..
. lN
-i(z)]. (10.7
)
Example 10. 1 I n thi s example , w e comput e th e Smit h for m o f th e pseudo circulant associate d wit h th e scala r filter C(z) = 1 — 2z~ x + 2z~ 2 — z~ 3. Th e polyphase component s o f C(z) wit h respec t t o N = 3 are, respectively , CQ(Z)
=
1 - z~\ d(z)
=
- 2 , an
dC
2(z)
=
2.
10.2. Propertie s o f pseudocirculant s
295
T h e associate d pseudocirculan t i s
ri-;
(3pS{z)
1
- 1 2Z-
-2Z"1
1
2z~l
1-z-21
2 -
This ha s th e following Smit h for m decomposition : (1
-1 1 0
-1
1" 0 0
10 01 00
o
+ z-
0 1
(i
- )
2 0 0
-2 -1 0
uw
1-z
1 l 2
V(*)
Both XJ(z) an d V ( z ) hav e determinant s equa l t o —1 . T h e Smit h for m ha s tw o nontrivial diagona l terms , 71(2; ) = 1 + z~ x an d 72(^ ) = ( 1 + z~ 1)(\ — z - 1 ) , where 71(2; ) divide s 72(2;) . Therefor e w e hav e p^ = 2 . ■ For a n arbitrar y F I R polynomia l matrix , ste p b y ste p ro w an d colum n operations ar e require d t o comput e it s Smit h form . Bu t fo r pseudocirculant s it i s muc h earlie r t o obtai n it s Smit h form . W e wil l se e i n Sectio n 10. 5 t h a t , given th e zero s o f th e underlyin g scala r filte r C(z), th e Smit h for m ca n b e determined easil y b y inspection .
10.2.2 DF
T decompositio n
It i s known t h a t pseudocirculant s ca n als o b e diagonalize d usin g D F T matrice s [159]. A n arbitrar y pseudocirculan t C ps(z), no t necessaril y FIR , assume s th e following decomposition : Cps(zN= )
1
)WE(z)WJfT>(2
T>(z-
(lOi
where D ( z ) an d T,(z) ar e diagona l matrice s given , respectively , b y
D(*)
"1 0 0 z' 00
H(z)
1
..
.,
0 0
'C(z) 0 0 C(zW) 0
-7V+1
0
C(zW N-l\
with W = e -j27r/N. T h e matri x W i s th e N x N normalize d D F T matri x with [ W ] m n = -^= e-32rnnir/N^ 0<m,n
10. Minimu m redundanc y FI R transceiver s
296
10.2.3 Propertie
s derive d fro m th e tw o decomposition s
Property 10.1 . Determinan t o f C ps(z) Th e determinan t o f C ps(z) i s directly relate d t o tha t o f it s Smit h for m T s (z) an d t o tha t o f th e diagona l matrix 5](z ) i n th e DF T decomposition . Th e tw o matrice s U(z ) an d V(z ) in th e Smit h for m decompositio n ar e unimodula r an d thei r determinant s ar e constants, s o we hav e det C ps(z) =
cdetT s(z)=
070(^)71(2? ) • • •j N-1(z)y (10.10
)
where c = de t XJ(z) det V(z). No w conside r th e determinant s o f the tw o side s of (10.8) . W e ca n obtai n det C ps(zN) =
N-l
de t (E(z) ) = J J C(zW k). (10.11
)
The abov e expressio n implie s i n particula r tha t de t C ps (z) i s a dela y i f an d only i f C{z) i s a delay . Thi s expressio n als o lead s t o th e followin g propert y that relate s th e zero s o f C ps(z) t o thos e o f th e scala r filte r C(z). Property 10.2 . Zero s of det C ps(z) Suppos FIR filte r o f orde r L an d th e zero s ar e
e the channel C(z) i s a causal
#1, 02,..., 0L.
Then de t C ps (z) i s als o a causa l FI R filte r o f orde r L an d th e zero s ar e nN nN nN U1 , U 2 , . . . , t/ L .
Proof. Let C{Z) = C(0 ) [ 1 - 0^] [
1 - 02Z- 1] . . . [ 1 - 0LZ- 1] .
Property 10. 1 tells us that de t C ps(zN)= to
E L Jo C(zW k), whic h is equal
N-l
Ho)f I I t 1 " Oii^)- 1} [ 1 - e 2(zwkr1] • • • [1 - OLizw")- 1] ■ k=0
It ca n b e verifie d tha t (Proble m 10. £ N-l
11 [l-^(^)-1 ] = 1-0
N
z~N.
k=0
It follow s fro m th e abov e expressio n tha t th e determinan t o f C ps (z) ca n be rearrange d a s follows : N■
L
detC ps (z) = [ c ( 0 ) ] A r n [ 1 - ^ " 1 ] i=l
This mean s that de t C ps (z) i s also an FI R filte r o f order L an d th e zero s
a r e < , 0?,...,0». ■
10.2. Propertie s of pseudocirculants
297
P r o p e r t y 1 0 . 3 . Zero s o f jk(z) T h e determinant expressio n i n (10.10 ) means t h a t th e zero s of det C ps(z) ar e distribute d amon g 7^(2:) . T h e previou s property state s t h a t t h e zeros o f detC ps(z) ar e Of. Therefor e t h e zeros of 7^(2:) ar e also Of. A s detC ps(z) ha s only L zeros , ther e ar e at mos t L nontrivial 7^(2:) . T h e numbe r o f nonunity term s o n th e diagona l o f the Smit h form satisfie s pN < nii n [L , N\. W h e n N > L , the first (N — L) diagona l term s of th e Smit h for m ar e equa l t o one. Property 10.4 . Ran k of Cps(z) Fo r a causal F I R pseudocirculan t C ps(z), a matri x polynomia l o f £ - 1 , t h e rank i s a functio n o f z. I n t he Smith for m decomposition t h e unimodular matrice s XJ(z) a n d V ( z) ar e nonsingular fo r all z. Therefor e th e ran k o f Cps(z) i s t he same a s the ran k o f its Smit h for m Ts(z). T h e ran k o f Ts(z) depend s o n the numbe r o f nontrivial term s o n t h e diagonal, i.e . pjy. I f r i s a zer o o f ^N- PN(Z), then , du e t o t he property t h a t ji(z) divide s 7^1(2:) , w e hav e 1N-PN= (T)
-yN-
= PN+i(r)
• • •= 7 T V - I ( V ) =
0 .
T h e ran k o f Ts(r) i s thus N — pjy. Thi s i s also th e smalles t ran k o f Ts(z), i.e . m i n r a n k ( r s ( 2 : ) ) = N — pjy. z
On th e othe r hand , fro m th e D F T decompositio n o f Cps(z) w e kno w t h a t the ran k o f C ps(zN) i s t he same a s t he rank o f 5](z ) a s W a n d D ( z) ar e nonsingular. Therefore , w e hav e iank(Cp8(zN= ))
r a n k ( £ ( ») =
r a n k ( r a ( ^ ) ) ; (10.12
)
the smalles t rank s o f t he three matrice s C ps(z), T, (2:), a n d Ts(z) ar e t h e same. W e already kno w t h a t t h e smallest ran k o f T s(z) i s N — PN fro m (10.12). Therefore , m i n r a n k ( r s ( 2 : ) ) = minrank(CL s (2:)) = minrank(5](2:) ) = N — pN, zz
z
where pN i s t he number o f nonunit y term s i n t he diagonals o f t he Smit h form T s(z). Thi s equatio n tie s t h e smallest ran k o f Cps(z) t o t he number of nontrivial diagona l term s i n its Smit h form . Fo r a given F I R channe l C(z) an d interpolation rati o AT" , t he rank o f C ps(z) ca n actually b e determined mor e explicitly, a s discussed next .
10.2.4 Congruou
s zero s
T h e smalles t ran k o f Cps(z) ca n als o b e obtained b y inspecting th e so-calle d c o n g r u o u s zero s o f C(z). Le t us first factoriz e C(z) i n terms o f its distinc t zeros. Suppos e C(z) ha s p distinc t zeros , c^i , 0 ^ 2 , . . . , ^, o f multiplicities , respectively, 7711 , 777,2 , • • • , Tn p. T h e n mi + 777, 2 + • • • + m p i s equal t o L, t h e order o f C(z), a nd C(z) =
c(0) [ 1 - a.z- 1]"11 [
1 - a.z- 1]"12 •
• • [l - a ^ "
1
]^.
W h e n al l th e zero s ar e distinct , w e hav e p = L an d 777 4 = 777, 2 = • • • =TTT, ^ = 1.
10. Minimu m redundanc y FI R transceiver s
298
Definition 10. 1 Congruous zeros. A se t o f distinc t zero s { a i , a.^ . . . a q} o f C(z) ar e congruous with respect to N i f
T h e zero s t h a t ar e congruou s ar e distinc t bu t thei r magnitude s ar e th e sam e and thei r angle s diffe r b y a n intege r multipl e o f 2TT/N. The y ca n b e considere d as rotation s o f eac h other . W e expres s eac h a s a rotatio n o f a\\ a3 =
OL XW-ni, 0
< rij < N. (10.13
)
Because th e congruou s zero s ar e distinc t b y definition , th e integer s rij ar e also distinct . I n fact , fro m P r o p e r t y 10.2 , w e kno w t h a t a^,a^,...,a^ correspond t o th e zero s o f de t C ps(z). W e ca n sa y t h a t a se t o f q congruou s zeros ar e distinc t zero s o f C(z) t h a t merg e a s a zer o o f multiplicit y q o f detCps(2;). Definition 10. 2 Th e notatio n n?q denote s th e cardinalit y o f th e larges t se t o f congruous zero s o f C(z) wit h respec t t o N. ■ T h e numbe r ji^ represent s th e larges t numbe r o f distinc t zero s t h a t hav e th e same magnitud e an d thei r difference s i n angle s ar e intege r multiple s o f 2TT/N. W h e n C(z) ha s p distinc t zeros , th e numbe r ji^ i s bounde d b y 1 < JIN < PIn th e earlie r example , C(z) = 1 — 2z~x + 2z~ 2 — z~3, th e zero s ar e e J 7 r / 3 , -J7r 3 e / , an d 1 , a s show n i n Fig . 10.3(a) . Fo r N = 2 , w e rais e th e zero s t o the powe r o f 2 an d sho w thei r location s i n Fig . 10.3(b) . N o tw o zero s ar e congruous an d fi2 = 1 . Th e cas e fo r N = 3 i s show n i n Fig . 10.3(c) . Th e tw o zeros, e J 7 r / 3 an d e _ J 7 r / 3 , merg e afte r the y ar e raise d t o th e thir d power . The y are congruou s wit h respec t t o N = 3 , an d w e hav e [13 = 2 . W h e n N = 6 (Fig. 10.3(d)) , th e thre e zero s ar e congruou s an d w e hav e /JLQ = 3 . As anothe r example , conside r C(z) = ( l + z _ 1 ) ( l — z- 1 ) 2 . Thi s filter ha s double zero s a t z = 1 an d on e zer o a t z = — 1. Ther e ar e tw o distinc t zeros , — 1 an d 1 , a s show n i n Fig . 10.4(a) . Th e secon d an d thir d power s o f thes e zeros ar e show n respectivel y i n Fig . 10.4(b ) an d Fig . 10.4(c) . Fo r N = 2 , these tw o zero s merg e an d w e hav e [12 = 2 . Fo r N = 3 , thes e tw o zero s d o not merg e an d w e hav e [13 = 1 . T h e definitio n o f [IN i s base d o n th e congruou s zero s o f th e scala r filter C(z). I t t u r n s ou t t h a t JIN i s actuall y equa l t o p^v , th e numbe r o f th e non trivial diagona l term s o n th e Smit h for m o f th e associate d pseudocirculan t Lemma 10. 1 Fo r a n F I R N x N pseudocirculan t C ps(z), th e numbe r o f non unity term s o n th e diagona l o f its Smit h for m i s the sam e a s the larges t numbe r of congruou s zero s o f C(z), i.e . PN
=
MTV -
g
Proof. Recal l t h a t th e diagona l matri x 5](z ) i n th e D F T decompositio n of C ps(zN) i s F I R i f C{z) i s FIR . Th e numbe r o f term s o n th e diagona l of 5](z ) t h a t hav e commo n zero s wil l determin e th e smalles t ran k o f
299
10.2. Propertie s o f pseudocirculant s
r-plane
Aim N=2
(a)
(b)
N=6 (d)
(c)
fR e
Figure 1 0 . 3 . (a ) Th e zero s a * o f C(z); (b)-(d ) a f fo r N =
C(z) =
1 - 2z~ x + 2z" 2 - z~
3
.
z-plane
1R e
(a)
2,3,6 . Th e channe l i s
N=2
tf=3
1R e
1R e
(b)
Figure 10.4 . (a ) Th e distinc t zero s a ; o f C(z); (b ) a f fo r N = The channe l i s C(z) = ( 1 + z - 1 ) ^ ~ ^ _ 1 ) 2 -
(c)
2 ; (c ) a f fo r J V = 3 .
XI(2:). Observ e t h a t th e zero s o f C(zW k) ar e thos e o f C(z) rotate d b y 2kn/N. I f C(z) an d C(zW k) hav e a commo n zer o a , the n
C(aO = C(c^W fc) = 0 . This mean s t h a t b o t h a an d aV^ fc ar e zero s o f C(z). B y Definitio n 10. 1 these zero s ar e congruou s wit h respec t t o N. T h e larges t numbe r o f terms o n th e diagona l o f 5](z ) t h a t hav e commo n zero s i s th e sam e a s the larges t numbe r o f congruou s zero s /ijy. Therefor e th e smalles t ran k of 5](z ) i s N — /ijy. Bu t w e kno w alread y t h a t th e smalles t ran k o f Cps(z) i s N — PN. Therefore , w e hav e pN = PN- B Example 10. 2 Conside r th e filter C(z) = 1 with L = 3 . I t ha s thre e distinct zero s a t z = e J 7 r / 3 , e J 7 r / 3 , an d — 1 (Fig . 10.5(a)) . Conside r first
10. M i n i m u m redundanc y FI R transceiver s
300
N = 2 . Th e secon d power s o f th e zero s ar e show n i n Fig . 10.5(b) . A s n o tw o zeros ar e congruou s wit h respec t t o N = 2 , w e hav e p2 = 1 . O n th e othe r hand, th e tw o polyphas e component s ofC(z) ar e Co(z) = 1 , an d Ci(z) = z~ x. T h e channe l matri x i s " 1 z-~ CpS[z) l1
s~
Its Smit h for m i s
from whic h w e ca n als o obtai n p^ z-plane
N=2
\ (a)
r
(b)
|lm N=3
( fifth fe \fR e
\
(c)
Figure 1 0 . 5 . (a ) Zero s on o f C{z)\ (b ) af; (c ) a f . Th e channe l i s C{z) =
1 +;
Now suppos e A ^ = 3 . Th e polyphas e component s o f C(z) wit h respec t t o N = 3 ar e Co(z) = 1 + £ _ 1 , C\(z) = 0 , an d Ci{z) = 0 . Th e channe l matri x Cps (z) i s give n b y Qsps[z)
1 + z0 0
1
0
1 + z- 1 0
0 0
1-hz-1
T h e determinan t o f thi s matri x ha s a zer o o f multiplicit y 3 , a s show n i n Fig. 10.5 . I n thi s specia l case , C ps(z) i s th e sam e a s it s Smit h form . T h e number o f nonunit y term s p% on th e diagona l i s three . Thi s resul t i s als o consistent wit h th e fac t t h a t th e thre e zero s o f C(z) ar e congruou s wit h respec t to A f = 3 , an d henc e [13 = 3 . ■ Example 10. 3 Conside r th e second-orde r channe l C(z) = l-\-2z~ 1-\-z~2. T h e channel ha s doubl e zero s a t z = — 1. Th e numbe r o f zero s o n th e uni t circl e i s two bu t th e numbe r o f distinc t zero s i s one . W e hav e [IN = 1 for al l A f i n thi s case. Fo r instance , i f A f = 2 , th e polyphas e component s o f C(z) wit h respec t to N = 2 ar e Co(z) = 1 + z~ x an d C\(z) = 2 . Th e channe l matri x C ps(z) i s given b y 1 [ l - h ^ - 1 2Z1 CpS[z) 21 + z- 1 " T h e correspondin g Smit h for m T s(z) i s
T,(z)
1 01
■1z-x+z-2
10.3. Transceiver s wit h n o redundanc y
301
T h e numbe r o f nonidentit y term s o n th e diagona l i s p2 = 1 . Thi s numbe r i s one fo r al l N i n thi s case . ■ Lemma 10. 1 shows t h a t pN i s equal t o /IN, th e larges t numbe r o f congruou s zeros, a quantit y t h a t ca n b e determine d directl y b y inspectin g th e zero s o f C(z). I n Sectio n 10. 4 w e wil l se e ho w thi s numbe r i s tie d t o th e minimu m redundancy fo r designin g F I R transceivers .
10.3 Transceiver
s wit h n o redundanc y
T h e filte r ban k transceive r i n Fig . 10. 1 i s calle d minima l i f th e interpolatio n ratio N = M an d ther e i s n o redundancy . I n thi s sectio n w e conside r th e properties o f suc h systems . W e wil l sho w t h a t fo r F I R minima l transceivers , we ca n achiev e onl y zer o inter-subchanne l ISI , bu t no t zer o ISI . Moreover , fo r channels t h a t d o no t hav e minimu m phase , ther e doe s no t exis t a n ISI-fre e minimal transceive r t h a t i s causal and stable.
10.3.1 FI
R minima l transceiver s
In th e Smit h for m decompositio n o f C ps(z), ther e ar e tw o unimodula r F I R matrices U ( z ) an d V ( z ) . Thei r inverses , \J~ 1(z) an d V _ 1 0 ) > ar e als o causa l F I R unimodula r matrices . Suppos e w e choos e th e transmittin g an d receivin g matrices a s G ( » = V _ 1 ( ; z ) an d S ( » = U~\z), then b o t h ar e F I R matrices . T h e resultan t overal l transfe r matri x T(z) i s T(z) =
S(z)C ps= (z)G(z) T
s(z).
It i s a diagona l matrix , i.e . ther e i s n o inter-subchanne l ISI . Thus , fo r F I R minimal filte r ban k transceiver s w e ca n achiev e zer o inter-subchanne l ISI . Although inter-subchanne l IS I i s canceled , intra-subchanne l IS I canno t b e removed completel y b y F I R filtering . I n othe r words , i n th e no-redundanc y case, ther e i s n o F I R transceive r t h a t ca n b e ISI-fre e fo r frequency-selectiv e channels. T o se e this , w e ca n conside r th e determinan t o f th e overal l system ,
det(TO)) = det(S0)G0))det(C^0)) As C(z) i s FIR , detCp S(z) i s als o FIR , an d i t i s a dela y i f an d onl y i f C(z) is a delay . Bot h determinant s o n th e right-han d sid e o f th e abov e equatio n are FIR . T h e zero-IS I propert y require s d e t ( T ( z ) ) t o b e a delay . Thi s mean s t h a t b o t h det(S(z)G(z)) an d det(C ps(z)) ar e merel y delays , whic h i n t u r n implies t h a t C(z) i s a delay . S o i t i s no t possibl e t o achiev e zer o IS I usin g F I R transmitter s an d receiver s fo r frequency-selectiv e channels .
10.3.2 M
R minima l transceiver s
If w e ar e allowe d t o us e II R filters , on e possibl e ISI-fre e solutio n i s G O ) = V _ 1 0 ) an d S O ) = ^ 1(z)^~1(z)- Cautio n mus t b e take n i n doin g so . T h e t e r m T ^ O ) m a y n ° t b e stable . I n fact , i f th e channe l C(z) doe s no t hav e
302
10. Minimu m redundanc y FI R transceiver s
minimum phase , ther e doe s no t exis t an y ISI-fre e transceive r t h a t i s b o t h causal an d stable. Lemma 10. 2 Ther e exist s a causa l an d stabl e ISI-fre e minima l filte r ban k transceiver i f an d onl y i f C(z) i s a minimum-phas e filter . W h e n th e conditio n is satisfied , on e suc h solutio n i s give n b y G(z) =
V " 1 ^ ) , an
d S(z)
r j ^ U-
=
1
^ ) , (10.U)
where U ( z ) an d V ( z ) ar e th e unimodula r matrice s i n (10.6) . ■ Proof. Sufficiency of minimum phase C(z). I f C(z) ha s zero s a t #g , fo r £ = 1 , 2 , . . . , L , w e kno w t h a t de t G ps{z) ha s zero s a t Of an d t h a t thes e are als o th e zero s o f jk(z). I f C(z) ha s minimu m phase , th e zero s satisf y \6i\ < 1 . Th e zero s Of o f jk(z) wil l als o b e insid e th e uni t circle . I n thi s case, w e ca n choos e G(z) an d S(z) a s i n (10.14 ) fo r a causa l an d stabl e transceiver solution . Necessity of minimum phase C(z). I f C(z) doe s no t hav e minimu m phase, the n a t leas t on e zero , sa y #i , i s outside th e uni t circle . W h e n th e filter ban k transceive r i s ISI-free , th e overal l transfe r matri x i s diagona l and th e diagona l element s ar e merel y delays . W e hav e det (S(z)C ps(z)G(z)) =
cz-
mo
,
for som e constan t c an d mo. Becaus e detC p s (2;) contain s th e facto r (1 — Of z~ x), eithe r de t S(z) o r de t G(z) shoul d contai n th e facto r
i/(i -efz-
1
).
Therefore S(z) an d G(z) canno t b o t h b e stable . I n othe r words , i f C(z) does no t hav e minimu m phas e ther e doe s no t exis t an y causa l an d stabl e transceiver wit h th e ISI-fre e property . ■ Whenever ther e exist s a causa l an d stabl e transceive r pai r [G(z) , S(z)] , w e ca n use i t t o generat e man y othe r solutions . Fo r example , fo r an y S(z) t h a t i s an M x M causa l an d stabl e transfe r matri x wit h a causa l an d stabl e inverse , the pai r [G(z)&(z),&-1(z)S(z)] also forms a causa l an d stabl e transceiver . Fo r minima l filte r ban k transceivers , the existenc e o f a causa l an d stabl e solutio n depend s o n whethe r th e give n channel C(z) ha s minimu m phase . Th e stabilit y proble m als o explain s wh y non-minimum-phase channel s ar e difficul t t o equaliz e usin g minima l filte r bank transceivers . However , a s w e hav e discusse d i n Chapte r 5 , thi s i s no t the cas e i f a certai n redundanc y i s allowed , i.e . N > M. I n fact , fo r mos t cases, F I R ISI-fre e filte r ban k transceiver s exis t wit h redundanc y a s smal l a s 1 (Sectio n 10.4) . For a single-inpu t single-outpu t (SISO ) system , i t i s wel l know n t h a t th e inverse o f a n F I R syste m i s alway s IIR . Th e II R invers e i s (i ) causa l an d stable i f th e origina l syste m ha s minimu m phase ; an d (ii ) stabl e an d possibl y noncausal i f th e origina l syste m ha s n o zero s o n th e uni t circle . Th e resul t i n Lemma 10. 2 fo r th e minima l transceiver s ca n b e viewe d a s th e blocke d versio n of th e SIS O result .
10.4. Minimu m redundanc y
10.4 Minimu
303
m redundanc y
In th e filte r ban k transmitte r show n i n Fig . 10.1 , the interpolatio n rati o i s TV while th e numbe r o f subchannel s i s M. T h e transmitte r send s ou t T V samples for ever y M inputs . Ther e ar e T V — M redundan t sample s i n ever y T V samples transmitted. T h e followin g theore m give s th e smalles t redundanc y t h a t allow s the existenc e o f F I R transceivers . Theorem 10. 1 Conside r th e filte r ban k transceive r i n Fig . 10.1 . Ther e exis t F I R G(z) an d S(z) suc h t h a t th e transceive r i s ISI-fre e i f an d onl y i f th e redundancy K = T V — M satisfie s K> p
N,
where PN i s th e larges t numbe r o f congruou s zero s o f C(z) a s give n i n Defi nition 10.2 . W h e n a n F I R transceive r doe s exist , th e solutio n i s no t unique . One choic e o f ISI-fre e F I R transceive r i s a s follows : G(^) =
V-
1
^)
IM
0
S(z)=[lM O J U -
1
^ ) , (10.15
)
where U ( z ) an d V ( z ) ar e th e T V x T V unimodula r matrice s i n th e Smit h form decompositio n o f G ps(z) i n (10.6) . T h e minimu m redundanc y fo r F I R transceiver solution s i s pjy, whic h i s als o equa l t o pjy. ■ Proof. W e first sho w t h a t th e conditio n K > pN i s a sufficien t one . T h a t is , i f the conditio n i s satisfie d th e transceive r give n i n (10.15 ) i s on e solution. T h e n w e sho w t h a t i t i s als o a necessar y conditio n b y provin g t h a t i f K < PTV , ther e doe s no t exis t an y ISI-fre e F I R transceiver . Sufficiency. Conside r th e choic e o f F I R transmitte r G(z) give n i n (10.15). T h e n C ps(z)G(z) i s equa l t o
u^r^v^v- 1 ^)
IM
0
U(*)
IM
0
In th e abov e equality , w e hav e use d th e fac t t h a t whe n K > p^ th e first M element s o n th e diagona l o f T s(z) ar e equa l t o unity . Therefore , when K > p^, th e receive r i n (10.15 ) yield s a n F I R transceive r wit h zero ISI . Necessity. Suppos e K < p^ an d ther e exis t F I R G(z) an d S(z) suc h t h a t th e syste m i s ISI-free . Usin g Smit h for m decompositio n C ps(z)= U(z)T8(z)V(z), w e hav e [S(z)U(z)} T
s(z)
[V(z)G(z)} =
I M. (10.16
)
T h e smalles t ran k o f T s(z) i s N — p^, bu t th e ran k o f th e right-han d side o f (10.16 ) i s alway s equa l t o M = N — K, whic h i s greate r t h a n TV — pN whe n K < p^. S o w e hav e a contradictio n i n thi s case , an d a n F I R transceive r wit h th e ISI-fre e propert y doe s no t exist . ■
304
10. Minimu m redundanc y FI R transceiver s
In th e solutio n give n i n (10.15 ) th e transmitte r pad s K zero s t o th e trans mitted input s whil e th e receive r discard s th e las t K receive r outputs . Thi s solution depend s o n th e unimodula r matrice s i n th e Smit h for m decompo sition bu t no t explicitl y o n th e Smit h for m T(z). Not e that , a s lon g a s the redundanc y i s large enough , th e channe l ca n b e equalize d an d IS I ca n b e eliminated, irrespectiv e o f th e location s o f th e channe l zeros . Sinc e th e uni modular matrice s U(z ) an d V(z ) i n th e Smit h for m decompositio n ar e no t unique, G(z) an d S(z) ar e no t unique . Usin g th e solutio n give n i n (10.15) , we ca n deriv e infinitel y man y solutions . Fo r example , suppos e [G(z) , S(z)] i s an ISI-fre e solution , an d A(z ) i s a n arbitrar y FI R unimodula r matrix . The n [G(z)A(z), A-^zWz)] is also a n FI R ISI-fre e solution . Mor e generally , A(z) ca n b e an y FI R matri x with a n FI R inverse . I t i s know n tha t suc h a matri x ca n b e factorize d a s a product o f a n FI R paraunitar y matri x an d a n FI R unimodula r matrix . Se e [158] fo r mor e detail s o n th e factorizatio n o f suc h matrices . Example 10. 4 Conside r th e second-orde r channe l i n Exampl e 10.3 . Fo N = 3 , on e se t o f choices fo r th e unimodula r matrice s U(z ) an d V(z ) i s U(s)
1 0 2 1-2Z-1 1 2-z-1
0 2 1
r
2Z-1 '
U2-Z-1)
V(s)
We kno w p 3 = 1 and th e minimu m redundanc y i s 1 . Le t u s choos e M = and G(z) an d S(z) accordin g t o (10.15) :
2
-z 1 0
G(z)=V-1(z)
00 -1/3 2/ 3
S(z)=[h 0]V-\z) The overal l transfe r functio n i s 00 -1/3 2/ 3
S(z)Cps(z)G(z)
1 z-' 21 12
2z~r z-1 1
-z 1
I2.
0
In fact, w e have seen earlier i n Example 10. 3 that pjy = 1 for al l N. Therefor e the minimu m redundanc y i s 1 for al l N. ■ Example 10. 5 Conside r th e channe l i n Exampl e 10.2 . Fo r N = 2 , n o tw o zeros ar e congruous . W e hav e p 2 = 1 and thu s th e minimu m redundanc y i s 1. On e se t o f choices fo r th e unimodula r matrice s U(z ) an d V(z ) i s V(z)
1
and V(z)
1 z01
2
'
We ca n choos e G(z) an d S(z) accordin g t o (10.15) : G(z)=V-l(z)
, S(z ) = [ 1 0 ] U " 1 ^ ) = [ 1 0 ] .
10.4. Minimu m redundanc y
305
We onl y nee d K = 1 fo r th e existenc e o f F I R transceiver s fo r N = 2 . No w suppose w e us e N = 3 . W e hav e compute d earlie r i n Exampl e 10. 2 t h a t ps = 3 . I n thi s cas e F I R transceive r solution s d o no t exist . ■ From Exampl e 10. 5 w e observ e th e followin g interestin g properties . • W h e n M = 1 , K = 1 , an d N = 2 , w e hav e p 2 = 1 an d F I R solution s exist. However , whe n w e increas e K t o 2 , keepin g M = 1 , i.e . i V = 3 , we hav e ps = 3 an d ther e ar e n o F I R solution s i n thi s case . This demonstrates that if solutions of FIR transceivers exist for a given K, a solution does not necessarily exist if we increase redundancy from K to K + 1 and keep M fixed. • W e als o se e fro m thi s exampl e t h a t th e interpolatio n rati o N ca n b e smaller t h a n th e channe l orde r L (N = 2 an d L = 3) .
z-plane
N=2
(b)
(a)
1R e
Figure 10.6 . (a ) Zero s o f C(z); (b ) th e zero s o f detC
ps(z).
Example 10. 6 A channel with almost congruous zeros. Conside r th e second order channe l C(z) = 1 + 2 s i n e 2 : _ 1 + z~ 2. T h e zero s ar e e iW2+e) a n
as show n i n Fig . 10.6(a) . Le t N = Qjps{z)
de
-j(,r/2+ £ )
2 . T h e channe l matri x i s give n b y _1 1 2z 2 sine
1
sine
1 + z- 1
W h e n e = 0 , th e Smit h for m o f C ps(z) i s th e sam e a s C ps(z) an d F I R transceivers d o no t exist . Fo r a smal l e , th e tw o zero s o f detC ps(z) ar e distinct bu t clustered . T h a t is , th e zero s ar e almos t congruous . T h e zero s of de t Cp S(z) ar e show n i n Fig . 10.6(b) . W h e n sin e ^ 0 , i t ha s th e followin g Smith for m decomposition : 1 + z- 1 2 sine
uw
2 sin e
0
10 0 l + 2z- 1cos(2e)
W
11 0
1 V(*)
If e ^ 0 , w e hav e C ps(—1) « 0 . However , r a n k ( r s ( — 1 ) ) > 1 a s lon g a s sine 7 ^ 0 . Therefor e eithe r U(—1 ) o r V(—1 ) i s a n ill-conditione d matrix ,
306
10. Minimu m redundanc y FI R transceiver s
although the y ar e unimodula r an d hav e constan t determinants . T o b e mor e specific, on e ca n comput e th e conditio n number . Fo r a n A ^ x i V matri x A , the conditio n numbe r i s define d a s | | A | | | | A _ 1 | | , wher e | | • || denote s a matri x norm. Le t u s us e th e matri x nor m t h a t i s define d a s th e maximu m o f th e absolute colum n sum , i.e . | | A | | = max j X^= o l l ^ k i ! - W e ca n verif y t h a t th e condition numbe r o f V(—1 ) i s 1 , wherea s th e conditio n numbe r o f U(—1 ) i s
[2sine]2 + [l/( 2 sine ) , which goe s t o infinit y a s e approaches zero . I n thi s case , th e receivin g matri x in (10.15 ) i s S ( » = [ 1 1/(2sine) ] . This ha s a ver y larg e coefficien t 1/(2sine) , whic h wil l amplif y channe l nois e although th e overal l signa l gai n i s stil l unit y (S(z)C ps(z)G(z) = 1) . T o avoi d this, w e ca n increas e K b y 1 and ge t aroun d th e proble m o f noise amplification .
■
Example 10. 7 Conside r th e channe l C(z) an d th e powe r spectru m o f th e colored channe l nois e show n i n Fig . 10.7 . Th e coefficient s o f th e channe l c(n) ar e [0.165 9 0.304 5 - 0 . 1 1 5 9 - 0 . 0 7 3 3 - 0 . 0 0 1 5 ] . Th e channe l C(z) has orde r 4 . Th e channe l an d th e channe l nois e ar e draw n fro m a n ADS L environment. Th e channe l i s obtaine d b y shortenin g loo p 6 i n [7 ] t o five t a p s and th e channe l nois e i s a combinatio n o f F E X T crosstal k an d AWG N noise . For N = 5 , th e minimu m redundanc y i s 1 . W e choos e K = 1 and M = 4 . T h e F I R transmittin g an d receivin g matrice s ar e a s give n i n (10.15) . Th e input s are B P S K symbols , renderin g a bi t rat e o f 0. 8 bits/sample . Th e plo t o f bi t error rat e versu s transmissio n powe r i s give n i n Fig . 10.8 . Fo r comparison , we als o plo t th e bi t erro r rat e performanc e o f a cyclic-prefixe d DFT-base d transceiver wit h th e sam e bi t rat e an d relativ e redundancy , i.e . sam e K/N or sam e K/M. Th e DFT-base d transceive r need s a t leas t fou r redundan t samples pe r bloc k a s th e channe l orde r i s 4 . W e choos e M = 16 , K = 4 fo r the DFT-base d system . Th e syste m wit h minimu m redundanc y require s a much smalle r transmissio n powe r fo r th e sam e bi t erro r rate . In thi s exampl e th e minimu m redundanc y i s 1 wherea s th e redundanc y is 4 fo r th e DFT-base d system . I n mos t case s th e minimu m redundanc y i s less t h a n th e usua l redundanc y L. A t th e sam e relativ e redundancy , th e system wit h minimu m redundanc y ha s a smalle r M , i.e . a shorte r bloc k length. W e shoul d not e t h a t a s th e syste m i s no t DFT-based , th e filter ban k transceiver ha s mor e channel-dependen t element s i n th e desig n an d implemen tation phases . Fo r example , th e transmitte r an d receive r ca n onl y b e designe d after th e channe l i s known . Th e transmittin g an d receivin g matrice s wil l b o t h be channel-dependent . W e shoul d als o not e t h a t th e desig n o f th e filter ban k transceiver i s no t a s straightforwar d a s th e DFT-base d system . Th e unimod ular matrice s V ( z ) an d U ( z ) ca n b e ill-conditione d a s w e hav e demonstrate d in Exampl e 10.6 . Th e computatio n o f th e unimodula r matrice s ca n als o b e sensitive t o numerica l accurac y an d channe l estimatio n error . Thi s i s eve n more s o fo r a larg e N. ■
10.4. M i n i m u m redundanc y
307
(a)
0.2 0. 4 0. 6 O.i Frequency normalize d by7u.
(b) w -7 0
0.2 0. 4 0. 6 O.i Frequency normalize d by7u.
Figure 1 0 . 7 . (a ) Magnitud e respons e o f th e channe l C{z)\ (b ) powe r spectru m o f th e additive nois e q(n). 10"
i-jn
- B - DF T —I— minimu m redundanc y
1
10"
\J
10"
10"
10"
10
15 2 02 53 Transmission power (dB)
0
Figure 1 0 . 8 . Bi t erro r rat e fo r th e transceive r wit h minimu m redundanc y an d th e DFT-based syste m wit h th e sam e relativ e redundancy .
10. Minimu m redundanc y FI R transceiver s
308
For a give n F I R channe l C(z) o f orde r L < N ", th e minimu m redundanc y PN i s betwee n 1 an d L. I t woul d b e interestin g t o kno w whe n PN i s a t it s smallest an d whe n i t i s equa l t o it s larges t possibl e valu e L. • W h e n PN = 1 , w e onl y nee d t o us e redundanc y K = 1 , whic h i s the low est redundanc y possibl e fo r a frequency-selectiv e channel . Thi s occur s when C(z) doe s no t hav e congruou s zero s wit h respec t t o N. W h e n th e order o f channe l i s finite, w e ca n alway s find N suc h t h a t thi s i s true . This i s becaus e ther e ar e a t mos t L set s o f congruou s zeros . Fo r thos e zeros t h a t hav e th e sam e magnitude , w e ca n calculat e thei r difference s in angle . Collec t al l th e angl e difference s togethe r an d denot e t h e m b y 27rp£/q£, fo r som e coprim e integer s p£ an d qn. A s lon g a s N i s no t a multiple o f q^ n o tw o zero s ar e congruous . I n practica l cases , th e prob ability t h a t PN = 1 i s almos t unity . Therefor e redundanc y o f K = 1 is almos t alway s sufficien t fo r th e existenc e o f F I R ISI-fre e transceivers . However, thi s statemen t shoul d b e treate d wit h caution . Althoug h fo r an F I R C(z) w e ca n alway s find N suc h t h a t th e minimu m redundanc y is 1 , th e zero s ca n b e almos t congruou s wit h respec t t o N a s w e hav e observed i n Exampl e 10.6 . Nois e amplificatio n ma y b e a n issu e fo r th e resulting transceive r whe n th e channe l nois e i s take n int o consideration . • When is PN equal to L? Fo r L < N, th e maximu m o f PN i s L. T h e minimum redundanc y PN i s equa l t o L i f an d onl y i f al l th e zero s o f C(z) ar e congruous . Thi s happen s i f an d onl y i f C(z) ha s distinc t zero s and thes e zero s li e o n th e sam e circl e wit h angle s difference s t h a t ar e integer multiple s o f 2TT/N. A n exampl e i s show n i n Fig . 10.9 .
Urn
-plane
InIN
0
0» O
.0
w,
Re
Figure 10.9 . A n example of C(z) wit h p N = L.
10.5 Smit
h for m o f FI R pseudocirculant s
A pseudocirculan t matri x C ps(z) i s completel y determine d b y it s underlyin g scalar filte r an d henc e s o i s it s Smit h form . I t t u r n s ou t t h a t w e ca n ver y easily obtai n th e Smit h for m o f C ps (z) i n a close d for m fro m th e scala r filte r C(z). Th e diagona l term s i n th e Smit h for m ca n b e determine d directl y b y inspection. I n th e previou s sectio n w e gav e on e ISI-fre e F I R solution , whic h depends onl y o n th e numbe r o f nontrivia l diagona l term s o f th e Smit h for m
10.5. Smit h for m o f FI R pseudocirculant s
309
but no t o n th e Smit h for m explicitly . Althoug h th e Smit h for m itsel f i s no t of importanc e i n th e desig n o f FI R transceivers , i t i s academicall y satisfyin g to kno w tha t i t i s closel y relate d t o congruen t zero s o f the scala r filte r C(z). From th e propertie s derive d i n Sectio n 10.2.3 , w e alread y kno w tha t th e zeros o f detC ps(z) ar e distribute d amon g th e Smit h form' s diagona l term s 7^(2:). Th e followin g lemm a provide s u s a mor e explici t lin k betwee n congru ous zero s o f the scala r filte r C(z) an d 7^(2:) . Lemma 10. 3 Give n a n FI R channe l C(z) an d it s blocke d N x N channe l matrix C ps(z), le t {ai , a^, • • •, OLq} b e a se t o f congruou s zero s o f C(z) wit h respect t o N. Suppos e n o othe r zero s ca n b e include d i n th e se t t o for m a larger congruou s set . Then , (1) r a n k ( C
ps
«))= iV-g;
(2) onl y th e las t q functions ^N- q(z), 1 tor ( 1 - a^z' 1). M
N-q+i(z), •
• •, 1N-I(Z) hav
e the fac -
Proof. We expres s th e congruou s zero s a s OLJ = aiW~ nj. Conside (N — rij)th diagona l ter m o f £(z ) i n (10.8) . Observ e tha t n C(zWN-n*)\z=ai=C(a= 1W- *)
C(a j)= 0, fo
r th e
r j = 1 , 2 , . . ., q.
That is , q terms o n th e diagona l o f 5](z ) ar e equa l t o zer o when z = OL\. No othe r ter m ha s a zer o a t OL\ because thi s i s the larges t congruou s se t containing OL\. Therefore, w e have rank(5](ai) ) = N — q, which implie s that rank(C p s (a?)) = rank(T s (a?)) = N - q. This, i n turn , mean s tha t exactl y q diagona l term s o f th e Smit h for m contain th e facto r ( 1 — a^z~ x). Thes e ar e th e las t q term s a s 7^(2: ) divides 7^+1(2:) . ■ To obtai n a n exac t expressio n o f 7^(2:) , le t u s partitio n al l th e zero s o f C(z) int o set s Bj. Eac h Bj contain s on e se t o f congruou s zero s an d n o othe r zeros ca n b e include d i n th e se t t o for m a large r congruou s set . I n addition , when C(z) ha s multipl e zeros , identica l zero s ar e groupe d int o th e sam e set . Suppose ther e ar e a tota l o f s suc h sets , Si , 6 2 , . . ., B s. Suppos e Bj contain s £j congruou s zero s an d the y ar e denote d b y 0 ^ 1 , 0 ^ 2 , . • - , ^ j , ^ , (10.17
)
where w e have , fo r convenience , renumbere d th e zero s o f C(z) usin g doubl e subscripts. (Th e numbe r o f element s i n Bj ca n b e large r tha n £j du e t o identical zeros. ) Le t th e multiplicit y o f OLJ^ b e rrij^ an d m
j,l ^
m
j,2 —
' '' —
m
3,lj '
Then w e ca n writ e C(z) a s C(z) =
c(0 ) f [ (l - a^z- 1)^ (
1 - a^z- 1)™*" •
• • (1 " " ^ " T ^ •
310
10. M i n i m u m redundanc y FI R transceiver s
Without los s of generality, w e assume £\ > £2 • • • > £ s- B y definition, £\ = fijy. Lemma 10. 3 says that th e las t £j terms hav e the facto r ( 1 — a^z-1). I n othe r words, jN-k(z) contain s ever y facto r ( 1 — a^z -1) tha t satisfie s £j > k. Fo r k > £1 , w e hav e jN-k(z)= 1 . Th e followin g theore m give s a close d for m expression fo r th e Smit h for m o f C ps(z). A proo f i s given i n Sectio n 10.6 . Theorem 10. 2 Th e Smit h for m T s(z) o f th e pseudocirculan t C ps(z) ha s di agonal term s give n b y /s lN-k{z)={ ,7>
J ] (l-a^*fe
1
)™'.*, k
= l,2,...,£
1,
(10-18
)
^ 1, otherwise . For example , suppos e C(z) ha s n o multipl e zero s an d assum e th e zero s ar e divided int o thre e set s £>i , £>2, and B3 with, respectively , £\ = 4 , £2 = 3 , an d i3 = 2 a s give n i n Tabl e 10.1 . Th e zero s i n th e sam e colum n wil l g o t o th e Sets Bi
B2 B3
Zeros an
OL\2
«13
OL21
OL22
«23
«31
«32
i
I
lN-l{z)
7N-2(Z)
ai4
I
1
7JV-3 (* )
1N-A{Z)
Table 1 0 . 1 . A n exampl e o f zer o partitio n an d th e respectiv e 7*5(2 )
same function jk(z)- Th e resultant JN-4(Z) n a s o n e z e r o <^n > 7TV-3(^) ha s two zeros a^L,^^ , an d bot h 7.^-2(2 ) an d JN-I(Z) hav e thre e zero s a^ , a ^ , a ^ . All th e othe r diagona l term s ar e equa l t o unit y i f N > 4 . I n summary , w e have lk(z)= 1N-A(Z) =
1 , 1 -
for a^z
fc< N-1
1 77V-3(^) = ( 1 - aZz'11 )^
1N-2(Z) =
JN-
a^z-l)(l
(1
,N ,
■^(l
- * "
1
) ,
3 Example 10. 8 Conside r th e scala r filter C(z) = 1 i n Exampl e 10.2 . I t J7r 3 _J7r 3 has zero s a t z = e / , e / , an d — 1. Whe nTV 2, w e hav e B 1 = {e j7r<3} B2 = {e~ i 7 r / 3 }, an d £ 3 = {-!} • Ther e i s only one nontrivial diagona l elemen t 71(2). Th e zero s o f ■ji(z) ar e thos e o f C(z) raise d t o th e powe r o f 2 :
7i(*) = ( l ■ ^ V ) ( i
e
-i2./3z-l)(1
(-1)2,"1)
10.6. Proo f o f Theore m 10. 2
311
We immediatel y ge t th e Smit h for m a s 10 01
- z~
3
Now suppos e N = 3 ; th e thre e zero s o f C(z) ar e congruou s wit h respec t t o 3 . We hav e onl y on e B se t {e J 7 r / 3 , e _ J 7 r / 3 , —1} . Eac h ^i(z) i s o f orde r 1 , 7o(*) = 7 i ( * 0= 12(2) We ca n conclud e t h a t l-z0 0
rs(z)
0 1-z-1 0
0 0 1-z'1
One ca n readil y determin e th e Smit h for m o f th e F I R pseudocirculan matrix C ps(z) b y inspectio n onc e th e zero s o f th e associate d scala r filte r C(z) are known . T h e theore m implie s t h a t th e numbe r o f nontrivia l term s i the Smit h for m i s l\, i.e . th e cardinalit y o f th e larges t congruou s set . Thi number, a s show n i n th e previou s section , i s exactl y th e minimu m redundanc for th e existenc e o f F I R transceivers .
10.6 Proo
t n s y
f o f Theore m 10. 2
W h e n th e zero s o f C(z) ar e distinct , Theore m 10. 2 follow s directl y fro m Lemma 10.3 . However , thi s i s no t th e cas e whe n C(z) ha s multipl e zeros . For example , suppos e C(z) ha s congruou s zero s OL\ an d a^ , wit h multiplici ties 77 i 1 and 77i2 , respectively. W e kno w t h a t detT s (2:) ha s (m\ +777,2 ) zero s a t a^ an d w e kno w t h a t thes e zero s wil l b e distribute d t o th e las t tw o diagona l terms. Bu t th e exac t distributio n canno t b e determine d fro m Lemm a 10.3 . It i s intuitiv e t o conjectur e t h a t 771 2 zeros g o t o 7 ^ - 2 ( 2 ) an d 777 4 zero s g o t o 7 J V - I ( ^ ) i f mi > 7712 - An d i t i s indee d ho w th e zero s ar e distributed . T h e proof fo r C(z) wit h multipl e zero s turn s ou t t o b e mor e involved . I n practica l cases, th e probabilit y t h a t th e zero s C(z) ar e distinc t i s almos t unity . How ever, th e zero s ca n b e almos t congruou s (Exampl e 10.6) , an d th e probabilit y for thi s i s no t zero . Fo r completenes s w e presen t th e proo f fo r genera l channel s t h a t ca n hav e multipl e zeros . We wil l se e t h a t whe n w e comput e th e Smit h for m o f C ps(z), w e ca n con sider zero s fro m eac h B{ separately . T h a t is , zero s fro m differen t Bi decoupl e in th e computatio n o f Smit h form . A s a resul t w e ca n conside r C(z) wit h only on e se t o f congruou s zeros , i.e . C(z) o f th e for m C(z) =
( 1 - aiz' 1)1711 (
1 a • ( 1 - OLqZ- l\m )
1 - a 2z~1)m2 ••
(10.19)
, aq ar e congruou s wit h respec t t o N wit h otj = aiW nj an d ■ • > Tn q. W e wil l sho w t h a t th e Smit h for m o f th e associate d 777i > W12 > Cps(z) i s give n b y
where c^i , 0^2,
diag [l
1 (l-afz-
1
)™* (l-afz-
1
)™*-1
(1
■ a^z
-l\mil^
(10.20)
312
10. Minimu m redundanc y FI R transceiver s
The proo f i s organized a s follows: W e will first sho w in Sectio n 10.6. 1 tha t the Smit h for m o f C ps(zN) i s the sam e a s tha t o f 5](z) , th e diagona l matri x in th e DF T decomposition . Therefor e w e ca n find th e Smit h for m o f 5](z ) instead. Thi s i s tru e regardles s o f whethe r th e zero s o f C(z) ar e distinc t o r not. I n Sectio n 10.6.2 , w e argu e wh y w e onl y nee d t o conside r C(z) wit h a single se t o f congruou s zeros . The n w e give a n exampl e fo r finding th e Smit h form o f 5](z ) (Sectio n 10.6.3) . Thi s exampl e wil l motivat e th e notatio n an d the approac h tha t w e us e t o find th e Smit h for m o f 5](z ) i n Sectio n 10.6.4 .
10.6.1 Identica
l Smit h form s
From th e DF T decompositio n o f C ps(z) i n (10.8) , w e know tha t T>(z)Cps(zN)T>(z-1= )
W S f z j W 1 . (10.21
)
_1
The left-han d side , althoug h havin g advance s i n D ( z ) , i s a polynomia l matrix i n z - 1 , a s th e right-han d sid e W S ( z ) W ^ i s causal . I t turn s ou t tha t the left-han d sid e ha s th e sam e Smit h for m a s C ps(zN) an d als o th e sam e Smith for m a s 5](z) . W e wil l prov e thi s wit h a mor e genera l setting . Let E(z ) b e a polynomia l matri x i n z - 1 suc h tha t B(z) =
D(z)E(z)D(z - 1 )
is als o a polynomia l matri x i n z~ x. W e wil l sho w tha t B(z ) an d E(z ) hav e the sam e Smit h form . Le t Ai[z) b e th e gc d o f all the ixi minor s o f E(z) an d A[(z) b e th e gc d o f al l i x i minor s o f B(z) . Fo r convenience , w e defin e th e set Si a s th e collectio n o f al l 1 x i inde x vectors , [fci k
2
•••
ki],
0
< fci < k 2 < •• • < h < N- 1 . (10.22
)
For k, j G Si, w e defin e Ekj(z ) a s th e ixi submatri x o f E(z ) obtaine d b y keeping th e row s in the inde x vecto r k an d th e column s i n the inde x vecto r j . For example , i f k = [fc i k
2
•••
ki],
j
= [j!
j
2
•••
ji]
,
then Ekj(z ) i s th e ixi submatri x o f E(z ) obtaine d b y keepin g row s #/ci , # / c 2 , . . . , #ki an d column s # j i , # j 2 , • • •, #ji- W e ca n writ e Al(z)= 9cd
d e t E k j ^ ) , A'
= i(z)
$
cd
detB
kJ(*).
Note tha t th e (k, j)th elemen t o f B(z ) i s relate d t o E(z ) b y Bkj(z) = Ekj(z)zi~k, s o we hav e A[(z) =
9
cd
\zZin = iUm-k
m)
de t E k j(2?)
We see that A[{z) ca n diffe r fro m Ai(z) a t mos t b y a dela y term. I n fact , thi s term i s equa l t o unit y (Proble m 10.16) , an d A[{z) = Ai(z). Becaus e th e it h diagonal element o f the Smit h for m o f E(z) i s Ai(z)/Ai-i(z), w e can conclud e that th e Smit h form s o f E(z) an d B(z ) ar e the same . Therefor e th e left-han d side o f (10.21 ) ha s th e sam e Smit h for m a s C ps(zN). Th e right-han d sid e o f (10.21) i s 5](z ) sandwiche d b y unitar y matrice s W an d W"!" , whic h d o no t change th e Smit h form . W e ca n conclud e tha t th e Smit h for m o f C ps(zN) i s the sam e a s tha t o f 5](z) .
10.6. Proo f o f Theore m 10. 2
10.6.2 Zero
313
s fro m differen t Bi decoupl e
For simplicity , suppos e C(z) ha s tw o set s o f congruou s zeros . Th e zero s ar e partitioned int o tw o set s B\ an d B 2 (th e definitio n o f Bi i s give n i n Sec tion 10.5) . W e writ e C(z) a s Ci(z)C2(z), wher e Ci(z) contain s al l th e zero s in B^ , for i = 1 , 2. Le t Ai(z) b e th e gc d o f al l th e i x i minor s o f 5](z) , wher e 5](z) i s th e diagona l matri x i n th e DF T decompositio n o f C pz(zN) give n i n (10.8). The n w e hav e Ai(z)= $
cd k e
k
f[C(zW
* n=
-),
i
where Si i s th e collectio n o f al l 1 x i inde x vector s i n (10.22) . Observ e tha t Ci(zWrn) an d C2(zW n) d o not hav e any commo n facto r fo r an y m , n becaus e the zero s i n B\ ar e no t congruou s wit h th e zero s i n B2 . A s a resul t gcd
Ai(z)
YlC^zW^l
3cd
k
Y[C
n=l
2(zW
-)
n=l
This mean s tha t i n finding th e Smit h for m o f 5](z) , w e ca n conside r zero s from eac h Bi separately . W e ca n comput e th e Smit h form s correspondin g t o C\(z) an d C^iz). Takin g thei r product , w e ca n the n obtai n th e Smit h for m corresponding t o C(z). Therefor e w e only nee d t o conside r C(z) wit h a singl e set o f congruou s zeros , i.e . C(z) o f th e for m i n (10.19) . Showin g tha t th e Smith for m o f C ps(z) i s give n b y (10.20 ) i s equivalen t t o showin g tha t th e Smith for m o f 5](z ) i s diag[l ••
•1
(l-a?z-
N m
) * ••
• (l-a?z-
N mi
) ], (10.23
i.e. th e diagona l matri x i n (10.20 ) wit h z replace d b y z
10.6.3 A
)
N
.
n exampl e o f derivin g th e Smit h for m o f E(z )
We wil l us e on e exampl e t o brin g forwar d som e propertie s o f the Smit h for m of 5](z ) an d motivat e th e constructio n use d i n Subsectio n 10.6.4 . Conside r the scala r filter C(z) o f the for m i n (10.19 ) wit h q = 3 . Le t C(z) =
( 1 - ouz- 1)™1^ -
a 2z-1)m2(l -
asz-
1
)™*,
where 0^1,0^2,0^ 3 ar e congruou s wit h respec t t o AT" , OLJ = a\W~ nj^ an d m\ > ^2 > ^^3 - Not e tha t th e zt h diagona l elemen t o f 5](z ) i s C(zW l). Th e zero s of C(zW l) ar e thos e of C(z) rotated , i.e . aiW~ l,a2W~l,asW~l. The y ca n also b e expresse d a s
each a differen t rotatio n o f OL\. A S a n example , w e us e A^ = 6 , 771 1 = 4 , 771 2 = 2 , 771 3 = 1 , 77 2 = 1 , ^ 3 =
3 .
Table 10. 2 list s th e zero s o f C(zW l) i n thi s case . A s eac h zer o o f C(zW l) can b e expresse d a s a\W~ k° fo r som e intege r /CQ , w e onl y sho w th e intege r
314
10. M i n i m u m redundanc y FI R transceiver s
ko i n th e table . Th e i t h ro w o f th e tabl e list s th e zero s o f C(zW l). Suc ha table wil l b e calle d a rotatio n table . Eac h colum n i n th e tabl e contain s th e set { 0 , 1 , . . . , N — 1} , whic h wil l b e referre d t o a s a complet e rotation . T h e table contain s a tota l o f (m i -\-rri2-\- m^) complet e rotations . Mor e generally , the rotatio n tabl e contain s (ra i + 777, 2 + • • • + m q) complet e rotations . i
0 0 1
2
Zeros du e t o a\
Zeros du e t o OL
Zeros du e t o a%
(ko shown )
(ko shown )
(ko shown )
(rai columns )
(777,2 columns )
(7773 columns )
1 1 1
2 2 2
3
3 3 3
5
5 5 5
4 4
0 0 0
1
1 2
2
3
4 4 4
5
1 3
2 4
3
3 5
4
4 0
5
5 1
0
0 2
Table 1 0 . 2 . Rotatio n table : zero s o f C(zW l) expresse ko wit h onl y th e intege r ko show n i n th e table .
d a s a±W
k
° fo r som e intege r
W h e n w e collec t al l th e entrie s i n th e rotatio n table , w e obtai n a se t A o f TV (rai +777,2 + 777,3 ) elements. Le t Ak b e a subse t o f A obtaine d b y keepin g onl y those row s o f th e rotatio n tabl e i n th e inde x vecto r k . W e us e th e notatio n k to denot e th e complemen t o f k . Fo r example , i f N = 6 an d k = [ l 2 5] , then k = [ 0 3 4 ] . Therefore A^ i s th e subse t o f A obtaine d b y deletin g th e rows i n th e rotatio n tabl e correspondin g t o th e inde x vecto r k . A s a n example , Table 10. 3 show s th e element s o f A^ fo r k = [ l 2 5] . I t i s obtaine d b y deleting row s # 1 , # 2 , an d # 5 fro m th e rotatio n table . W e als o divid e A^ into thre e group s accordin g t o th e thre e distinc t zero s a i , 012, and 013. T h e ith grou p contain s th e entrie s o f A^ du e t o an a s show n i n Tabl e 10.3 . Ther e are (N — i)rri£ element s i n th e ith grou p whe n k i s o f dimensio n 1 x i. I n the followin g w e sho w ho w th e rotatio n tabl e i s relate d t o th e computatio n o f
\(z).
Now le t u s comput e th e gc d o f al l th e i x i minor s o f 5](z ) t o obtai n it s Smith form . Denot e th e gc d b y Ai(z). Th e N x N mino r o f 5](z ) i s de t 5](z) , and w e hav e A 6(z)= Yl i= ^ C(zW l). W e ma y not e t h a t suc h a produc t can als o b e obtaine d usin g th e rotatio n table . Observ e t h a t th e zero s o f th e product ar e exactl y thos e liste d i n th e rotatio n table ! Havin g on e complet e rotation give s ris e t o th e facto r (1 - a i ^ X l - aiz^W- 1)... (
1- ai*-
1
^-^-1))=
( 1 - a?z~
N
T h e tabl e contain s (ra i +777 2 +777,3) complet e rotations . I t follow s t h a t A
). 6(z)
10.6. Proo
315
f o f Theore m 10. 2
i Zero
s du e t o QL\ Zero (mi columns ) (777,
0
00
0
0
s du e t o a 2 Zero
s du e t o a^
2 columns) (777,
3 columns )
11
3
-±
-4- -4 - -4 - -4 -
-2-
-£ ~2r
3
33
3
3
44
0
4
44
4
4
55
1
-5-
-A
-2- -2 -
s- -a -
~2r ~2r
-5-
-4- -4 - -4 - -4 -
-0- -0 -
-2-
(group 1 )
(group 2 )
(group 3 )
Table 1 0 . 3 . Element s o f A^ wit h k = [ 1 2 5], obtaine d b y deletin g row s # 1 , # 2, an d # 5 fro m th e rotatio n tabl e i n Tabl e 10.2 .
is A
6 (z) =
N-i l Y[ C(zW = )
( 1- a f
z
- ^ ) ^ i + ^ 2 + m 3=
^ _
0,^-6 ^
T h e t e r m A 5 ( z ) i s th e gc d o f al l possibl e combination s o f five term s o n the diagona l o f 5](z) . W e ca n obtai n A$(z) b y examinin g th e rotatio n table . Considering th e zero s o f five arbitraril y chose n diagona l term s o f 5](z ) i s the sam e a s considerin g five arbitraril y chose n row s o f th e rotatio n table , or equivalentl y th e rotatio n tabl e wit h on e ro w deleted . Therefor e w e ca n consider th e subse t o f A wit h on e ro w i n th e rotatio n tabl e deleted , i.e . A^, where 0 < k < 5 . W e ca n obtai n A 5 ( z ) b y finding th e intersectio n
k=0
Note t h a t w e onl y nee d t o conside r complet e rotation s becaus e th e non identity diagona l term s o f th e Smit h for m o f £ ( z ) ha s onl y factor s o f th e for m (1 — a^z~ N). Le t u s loo k int o eac h Aj,. Observ e t h a t th e se t AQ i s obtaine d by deletin g ro w # 0 fro m th e rotatio n table . W e se e fro m Tabl e 10. 2 t h a t t o have complet e rotations , th e number s missin g i n grou p 2 are tw o I s an d grou p 3 lack s th e numbe r 3 . However , w e ca n find thes e missin g number s i n grou p 1. Therefor e AQ contain s thre e complet e rotations . I n a simila r manner , w e can verif y t h a t eac h A^, fo r 0 < k < 5 contain s thre e complet e rotations . In fact , th e intersectio n f \ = o ^k * s exactly { 0 , 1 , . . . , 5 , 0 , 1 , . . ., 5 , 0 , 1 , . . ., 5} , i.e. thre e complet e rotation s an d n o othe r elements . Suppos e thi s i s no t tru e and ther e i s on e mor e entry , sa y £Q, i n th e intersectio n i n additio n t o th e three complet e rotations . Fo r eac h A^, al l th e element s i n group s 2 an d 3 ar e already containe d i n th e thre e complet e rotations . Thi s additiona l £Q mus t
10. Minimu m redundanc y FI R transceiver s
316
come fro m grou p 1 . However , th e se t A$ , obtained b y deletin g th e ^ot h row , does no t hav e £o in grou p 1 . Therefore , th e intersectio n ( \ = o ^k contain s n o other element s t h a n th e thre e complet e rotations . W e hav e
A5(z) = (l-aiz-
6 3
).
T h e t e r m A^z) i s th e gc d o f al l possibl e combination s o f 4 term s o n th e diagonal o f T,(z). T o obtai n A^z), agai n w e ca n us e th e rotatio n table . W e construct subset s A^ b y deletin g tw o row s from th e rotatio n tabl e i n Tabl e 10. 2 and find thei r intersection , i.e .
kes 2 For example , whe n k = [ 0 l ] , row s # 0 an d # 1 ar e delete d fro m th e rota tion table . Grou p 3 lacks th e number s 3 and 4 to complet e a rotation. W e note t h a t thes e tw o number s ca n b e foun d i n grou p 2 , henc e A^ in thi s cas e contains a complete rotation . I n a similar way , w e ca n verif y t h a t eac h A^ fo r k G 6 2 contain s a complete rotation . T h a t is , th e intersectio n Pike s ^ k 1S { 0 , 1 , . . . , 5} . Thus , A^(z) = 1 — afz~ 6. Usin g a similar approach , w e ca n find = an Pikes ^ k 0 5 empty set , an d thu s A^(z) = 1. I t follow s t h a t A2(z) = 1 and Ai(z) = 1. Th e abov e discussio n demonstrate s t h a t w e ca n obtai n Ai(z) by examinin g th e rotatio n table . Althoug h workin g wit h th e rotatio n tabl e may see m roundabou t a t first, w e wil l se e it is actuall y muc h easie r whe n w e consider a more genera l cas e o f C(z) later . Denote th e i t h diagona l elemen t o f th e Smit h for m o f 5](z ) b y /3i(z). T h e n Pi(z) = Ai+1(z)/Ai(z) ar e give n b y f35(z) =
(1 - a 6lZ-6)\ f3
4(z)
=
(l-a6lZ-6)2, f3
/30(z)=/31(z)=/3= 2(z)
3(z)
=
1 - a\z~^
l.
We se e t h a t th e nonunit y term s ar e o f the for m ( 1 — a^z~NYi, an d th e exponents Hi happen t o b e th e sam e a s rai,ra2,and vn 3. I n particular , /3N-i(z) =
(1 - a?z~ N)m\ i
= 1,2,3 .
This t u r n s ou t t o b e tru e i n general , a s w e wil l se e next .
10.6.4 Smit
h for m o f S(z )
To sho w t h a t th e Smit h for m o f 5](z) i s of th e for m i n (10.23) , w e nee d to show t h a t /3N-i(z) = (1 - a? z- N)m>, i = l,2,...,q. This i s equivalen t t o showin g t h a t AN_i(z) = (l-a?z-N)^+im\ i
= 0,l,...,q-l. (10.24
)
As i n Subsection 10.6.3 , w e for m th e rotatio n tabl e b y listin g th e zero s o f C(zWl) i n a row , an d obtai n a tabl e o f N{m\ + rri2 + • • • + m q) entries . T h e se t A i s th e collectio n o f element s i n th e rotatio n tabl e an d i t contain s
10.6. Proo
f o f Theorem 10. 2
317
(777,1 + 777, 2 + • • • + m q) complet e rotations . W e wil l firs t loo k a t Ajy(z) an d then th e genera l t e r m A^-i(z). To conside r AN(Z), w e ca n examin e th e se t A, whic h consist s o f (777, 1 + 777,2 + • • • + m q) complet e rotations , i.e . th e se t { 0 , 1 , . . . , N — 1} repeatin g (777,1 + ^ 2 + • • • + m q) times . Eac h complet e rotatio n contribute s th e facto r (1 - a?z~ N) t o A N(z). Therefore ,
AN(z)=
( 1 - aTz-
N
)(mi+m2+---+m*\
Now le t u s conside r A J V - Z ( ^ ) - T o obtai n th e zero s o f Ajy-z(^) , w e ca n inspect th e intersectio n
where A^ i s a se t o f (N — i)(777,1 +777,2 + - • - + mq) element s obtaine d b y deletin g i row s fro m th e rotatio n table . T h e delete d row s correspon d t o th e i indice s in th e inde x vecto r k . W e divid e A^ int o q group s accordin g t o th e distinc t zeros a i , 0^2 , • • •, otq as in Sectio n 10.6.3 . T h e ^th grou p contain s th e entrie s o f A^ du e t o ai an d i t contain s (N — i)mg elements . W e will sho w t h a t AN-I{Z) is a s give n i n (10.24 ) b y provin g th e followin g tw o claims . C l a i m 1 . T h e se t Pike s ^ k contain s YH= i+i
m
t complet e rotations .
C l a i m 2 . I t contain s n o othe r elements . Proof o f Clai m 1 Eac h A^ i s obtaine d b y deletin g i row s fro m th e ro tation table . Non e o f th e group s o f A^ contain s a complet e rotatio n a s i numbers ar e missin g i n eac h column . T o for m a complet e rotation , number s in eac h grou p nee d t o b e combine d wit h entrie s fro m othe r groups . W e clai m t h a t fo r eac h o f the las t q — i group s o f A^ th e missin g number s ca n b e foun d in th e firs t i group s an d therefor e ther e wil l b e (TTT^+ I + 777^+ 2 + • • • + Tn q) complete rotations . Let u s conside r th e qth grou p o f A^ an d suppos e th e missin g i number s are ^ 1 , ^2, • • •, h- W e nee d t o sho w t h a t ^ 1 , ^2, • • •, h ca n b e foun d i n th e firs t i groups . W e wil l prov e i t b y contradiction . Suppos e thi s i s no t tru e an d th e number i\ canno t b e foun d i n an y o f th e firs t i groups . T h e n i\ i s missin g i n a tota l o f i + 1 groups (th e firs t i group s plu s th e qth group) . Not e t h a t eac h column o f th e rotatio n tabl e contain s a complet e rotation . T h a t a numbe r i s missing i n a grou p mean s t h a t i t i s i n th e row s t h a t ar e delete d whe n A^ i s constructed. W e ma y als o observ e t h a t , fo r ever y ro w i n th e rotatio n table , a number appear s i n a t mos t on e group . Thi s i s becaus e th e group s correspon d to distinc t zero s an d i n eac h ro w th e sam e numbe r canno t appea r i n mor e t h a n on e group . I n formin g A^ w e hav e i row s removed . I n thes e i rows , £1 ca n appea r i n a t mos t i row s o r appea r i n a t mos t i groups . I t canno t b e missing fro m i + 1 groups. Therefor e w e mus t b e abl e t o fin d t\ i n on e o f th e first i groups . Suppos e l\ appear s i n th e k^th grou p o f A^ on e o f th e firs t i groups. Becaus e m ^ > 777^ , we ca n fin d i\ a t leas t 777 ^ times. Similarly , w e can fin d £2 , • • •, (-% i n th e firs t i group s o f A^. Therefor e w e hav e 777 ^ complet e rotations du e t o th e zero s i n th e qth group . Using th e sam e approac h w e ca n alway s fin d th e missin g number s fo r groups # ( z + l ) , # ( z + 2 ) , . . . , # ( g — 1 ) . W e will hav e problem s usin g th e sam e
318
10. Minimu m redundanc y FI R transceiver s
approach onl y i f th e sam e numbe r i s missin g i n mor e t h a n on e o f th e las t (q — i) groups . Suppos e H\ i s missin g i n b o t h group s #q an d #(q — 1) . W e know i\ appear s m q time s i n on e o f th e firs t i group s a s show n earlier . Ca n we fin d th e sam e numbe r vn q-\ mor e time s fo r grou p #(q — 1) ? Th e answe r is i n th e affirmative . Th e reaso n i s a s follows . I f H\ is missin g i n b o t h group s #q an d #(q — 1) , thi s mean s t h a t H\ appears i n tw o delete d row s already . A s only i row s ar e deleted , H\ ca n appea r i n a t mos t (i — 2 ) othe r delete d rows ; we ca n fin d H\ i n a t leas t tw o o f th e firs t i group s o f A^. I n conclusion , w e have show n t h a t , i n eac h A^ ther e ar e (m q + m q-i + • • • + TT^+I ) complet e rotations, whic h prove s th e resul t i n Clai m 1 . Proof o f Clai m 2 Fro m Clai m 1 , w e kno w t h a t al l th e number s i n th e last (q — i) group s o f A^ ar e alread y i n th e intersectio n Pike s ^ k - ^ ^ n e intersection contain s anothe r number , sa y no , thi s numbe r mus t com e fro m the firs t i groups . Conside r onl y th e firs t i group s o f th e whol e rotatio n table . T h e numbe r n o wil l appea r i n exactl y i rows , sa y row s #/ci , # f e , . . . , #A^ . T h e n fo r k = [ki k^ • • • ki\ , A^ doe s no t hav e n o i n an y o f th e firs t i groups , whic h implie s t h a t f]ke s ^ k canno t contai n TIQ. Therefor e w e ca n conclude t h a t f]ke s ^ k contain s n o entrie s othe r t h a n (77^+1+77^+ 2 + ' • - + ^ q) complete rotations .
10.7. Furthe r readin g
10.7 Furthe
319
r readin g
T h e us e o f filter bank s t o introduc e redundanc y i n th e transmitte d signa l wa s proposed b y Xi a i n 199 6 [181 , 182]. T h e schem e wa s terme d filter ban k precod ing. Usin g filter ban k precoding , redundanc y wa s adde d an d IS I cancellatio n for F I R channel s achieve d wit h F I R transmitter s an d receivers . I t ha s bee n shown t h a t F I R transceiver s exis t unde r ver y genera l condition s eve n i f th e redundancy K i s a s smal l a s unit y [182] . T h e filters i n filter ban k transceiver s are LT I filters. I n [135] , time-varyin g system s wer e employe d fo r designin g F I R transceivers . Suppos e th e channe l i s F I R wit h distinc t root s an d th e number o f subchannel s M i s large r t h a n th e channe l order . I t wa s show n i n [135] t h a t , w e ca n alway s find a channel-independen t time-varyin g transmit ter suc h t h a t F I R time-varyin g receiver s exist . B y extendin g th e result s i n [135], necessar y an d sufficien t condition s t h a t guarante e th e existenc e o f F I R zero-forcing equalizer s wer e presente d i n [124] . A reductio n i n th e lengt h o f the F I R equalize r ca n thu s b e achieved . Precodin g wa s combine d wit h O F D M to reduc e redundanc y an d improv e robustnes s agains t channe l spectra l null s in [184] . Filter ban k precodin g ha s als o bee n extende d t o MIM O (multi-inpu t multi output) systems . Minimu m redundanc y fo r th e existenc e o f F I R transceiver s in th e cas e o f MIM O channel s wa s considere d i n [66] . Simila r t o th e cas e o f single-input single-outpu t channels , th e minimum-redundanc y conditio n ca n be give n i n term s o f th e congruou s zero s o f th e channe l matrix .
10.8 Problem
s
10.1 Suppos e w e hav e a scala r filter C(z) = block siz e i s N = 2 .
( 1 + z~ 2){\ —
£_ 1 ) ; assum e th e
(a) Determin e th e correspondin g pseudocirculan t matri x C (b) Fin d a Smit h for m decompositio n o f C
ps(z).
ps(z).
(c) W h a t i s th e minimu m redundancy ? 10.2 Repea t Proble m 10. 1 fo r N =
3.
10.3 Giv e on e exampl e o f C(z) suc h t h a t th e correspondin g 3 x 3 C ps(z) ha P3 = 3 .
s
10.4 Fin d th e genera l expressio n o f a n Lth-orde r channe l C(z) suc h t h a t 10.5 Le t C(z) = ( 1 + z~ 2)(\ — z~1), assum e th e bloc k siz e N = the correspondin g pseudocirculan t matri x b e denote d a s C (a) W h a t ar e th e zero s o f de t C (b) Fin d th e Smit h for m o f C
4 , an d le t ps(z).
ps(z)7 ps(z).
(c) Determin e th e diagona l matri x 5](z ) i n th e D F T decompositio n o f
10. Minimu m redundanc y FI R transceiver s
320
10.6 Fin d a n ISI-fre e F I R transceive r ( G ( z ) , S(z)) fo r Proble m 10.2 . Ca n w e find anothe r F I R transceive r solution ? 10.7 Suppos e th e channe l i s th e third-orde r F I R filter C(z) =
( 1 + z~ 2)(l —
(a) W h a t i s th e smalles t bloc k siz e N suc h t h a t PN = 1 ? (b) Fin d al l interpolatio n ratio s N fo r whic h F I R transceiver s exis t with redundanc y K = 1 . 10.8 Sho w t h a t N-l
1 J ] [l-aizW*)= ] l-a
N
z~N,
k=0
where W =
e~^l
N
'.
10.9 Suppos e a channe l C[z) i s F I R wit h distinc t zero s an d th e blocke d N x N channe l matri x i s C ps(z). I t i s know n t h a t detC p s (2;) ha s q distinct zero s /?i , / ? 2 , . . ., /3 q with multiplicitie s respectivel y t\, £2, ..., £ q, with £1 > £2 > • • • > £ q- Determin e th e Smit h for m C ps(z). 10.10 Sho w (10.9 ) usin g th e followin g tw o steps . (a) Sho w t h a t th e matri x 'D(z)C ps(zN)~D(z~1) i
s circulant .
(b) Us e (a ) t o prov e th e D F T decompositio n i n (10.8) . 10.11 Suppos e C{z) i s a n F I R channe l an d th e correspondin g N x N channe l matrix i s C ps(z). Th e Smit h for m o f C ps(z) ha s PN nonunit y diagona l terms. Le t redundanc y K > PN- Suppos e th e transmitte r doe s nothin g other t h a n zer o padding , i.e . G(z)
IM
0
Is i t alway s possibl e t o find a n F I R zero-forcin g receive r fo r th e zero padding transmittin g matri x give n above ? I f you r answe r i s yes, find a zero-forcing receiver . I f not , justif y you r answer . 10.12 Conside r th e channe l i n Exampl e 10.6 . Suppos e th e channe l nois e i s AWGN wit h varianc e J\f 0. (a) Fin d a Smit h for m decompositio n fo r C ps(z) whe n N = 3 . Us e th e decomposition t o find a n F I R transceive r solutio n wit h minimu m redundancy. (b) Determin e th e tota l nois e powe r fo r th e o u t p u t s o f th e receive r i n (a). 10.13 Conside r a n F I R channe l C{z) = En= o c(n)z_n- Sho w tha t F I R transceivers wit h n o redundanc y ca n achiev e th e ISI-fre e propert y onl y if C{z) i s a delay .
321
10.8. Problem s
10.14 Suppos e w e bloc k a scala r channe l b y usin g a dela y chai n o f z~ p rathe r t h a n z~ x a s i n Fig . P5. 3 o f Proble m 5. 4 (wher e p an d N ar e coprime) . T h e resultin g blocke d channe l matri x wil l b e differen t fro m th e pseudo circulant give n i n (10.2) . (a) I n th e discussio n o f minima l transceiver s i n Sectio n 10.3 , w e sa w t h a t solution s o f causa l stabl e zero-forcin g transceiver s exis t i f th e scalar channe l ha s minimu m phase . Doe s th e ne w blockin g schem e affect th e existenc e o f causa l stabl e zero-forcin g transceivers ? (b) Fo r a give n interpolatio n rati o AT , i s th e minimu m redundanc y affected b y th e ne w blockin g scheme ? 10.15 Suppos e th e diagona l matri x 5](z ) i n th e D F T decompositio n o f C ps(z) in (10.8 ) i s real . W h a t doe s t h a t tel l u s abou t th e scala r filte r C(z)7 10.16 I n Sectio n 10.6. 1 w e considere d tw o matrice s relate d b y B(z) =
D(z)E(z)D(z- 1),
where E ( z ) i s a polynomia l matri x i n z~ x suc h t h a t B ( z ) i s als o a polynomial matri x i n z - 1 , an d D ( z ) i s a s give n i n (10.8) . Le t Ai(z) b e the gc d o f al l th e i x i minor s o f E ( z ) an d A'^z) th e gc d o f al l i x i minors o f B ( z ) . Sho w t h a t Ai(z) = A[(z).
322 10
. Minimu m redundanc y FI R transceiver s
Appendix A Mathematical tool s In this appendix , w e summarize som e mathematical inequalitie s an d theorem s that ar e use d i n th e book . Fo r mor e detaile d description s o f thes e materials , please se e [52] .
Arithmetic-mean an d geometric-mea n ( A M - G M ) inequalit y Given a se t o f non-negativ e number s a>i fo r 0 < i < M — 1, th e arithmeti c mean (AM ) an d geometri c mea n (GM ) ar e respectivel y define d a s M-l
M^ ' i=0
/M-l\
GM=[\[al\
V M
The AM-G M inequalit y say s tha t AM > GM y with equalit y i f an d onl y i f a® = a\ = • • • = O M - I -
Matrix inversio n lemm a Let P an d R be , respectively , M x M an d N x N invertibl e matrices . The n (P + QRS)" 1 =
P1-
P ^ Q (SP^ Q + R"1)"1 S P 1 .
• Whe n M = N an d Q = S = I , th e lemm a reduce s t o (P + R ) " 1 = P - 1 - P -
1
(P- 1 + R-1)"1 P- 1.
• Whe n P _ 1 ha s a simpl e for m an d th e ran k o f R i s muc h smalle r tha n M, the n th e lemm a ca n b e employe d fo r a n efficien t computatio n o f (P + R ) - . Fo r example , whe n R ha s ran k 1 , R = qs T fo r som e M x l vectors q an d s . Th e invers e become s
(p+R)-=p--
l
+ s^T-i q
323
324
A. Mathematica l tool s
Singular valu e decompositio n Given a n M x i V matri x A o f ran k r , it s singula r valu e decompositio n (SVD ) is give n b y A=
UDV t,
where U an d V are , respectively , M x M an d N x N unitar y matrices . T h e matri x D i s a diagona l matri x an d it s r nonzer o diagona l entrie s satisf y doo > dn > • • • > d r-iiT-i > 0. • Th e quantitie s da ar e th e singula r value s o f A . I f w e le t A ^ b e th e nonzero eigenvalue s o f A ^ A o r A A ^ arrange d i n a nonincreasin g order , then da = y/Xi fo r 0 < i < r — 1 . • Th e colum n vector s o f U an d V ar e know n a s th e lef t an d righ t singula r vectors o f A du e t o thei r location s i n th e decomposition . Thes e vector s are th e eigenvector s o f A A ^ an d A"! " A, respectively . • Th e singula r value s o f a matri x ar e unique , wherea s th e singula r vector s are not .
Hadamard inequalit y For an y M x M positiv e semidefinit e matri x A , th e H a d a m a r d inequalit y say s that M-l
n [A]* * ^ det t A ]-
k=0
Furthermore, i f A i s positiv e definite , the n equalit y hold s i f an d onl y i f A i s diagonal.
Fischer inequalit y Let A b e a positiv e definit e matrix . Partitio n A a s BC Ct D where B an d D ar e square . The n th e Fische r inequalit y say s t h a t det[A] < det[B]det[D] . • Th e inequalit y become s a n equalit y i f an d onl y i f C = 0 . • Th e H a d a m a r d inequalit y ca n b e prove n b y repeatedl y applyin g th e Fischer inequality . Thu s th e latte r ca n b e viewe d a s a generalizatio n o f the former .
325
Rayleigh-Ritz theore m Let Xmin an d A m a x be , respectively , th e larges t an d smalles t eigenvalue s o f a Hermitian matri x A . T h e n vtAv
mm ■
v # 0 V ' Tv V
-Xrn
vUv
HiaX i
—
v#0 v t v
Xraax-
T h e minimu m an d maximu m value s o f A m ^ n an d A m a x ar e achieve d whe n v are th e eigenvector s o f A associate d wit h A m ^ n an d A m a x , respectively .
Convex and concave functions A functio n f(t) i s sai d t o b e conve x ove r a n interva l (T 0 , T\) if , fo r ever y to, h e (T 0 , Ti ) an d fo r al l 0 < A < 1 , A / ( t 0 ) + ( 1 - A ) / ( t i ) > / ( A t 0 + ( 1 - A)ti) . T h e functio n f(t) i s sai d t o b e strictl y conve x whe n equalit y hold s i f onl y i f to = t\. I f th e abov e inequalit y i s reversed , f(t) i s sai d t o b e concav e (o r strictly concav e whe n equalit y hold s i f an d onl y i f t o = t i ) . • A functio n f(t) i s (strictly ) conve x i f an d onl y i f —f(t) i s (strictly ) concave. • I f th e secon d derivativ e o f f(t) exist s an d i t satisfie s f"(t) > 0 fo r al l t E (T 0 , Ti) , the n /(£ ) i s conve x (i t i s strictl y conve x i f /"(£ ) > 0) . O n the othe r hand , i f /"(£ ) < 0 fo r al l t G (T 0 , T i ) , the n f(t) i s concav e (i t is strictl y concav e i f f"(t) < 0) . • Give n a se t o f M numbers , to , t i , . . . , £ M - I £ (To , T i ) , th e convexit y of f(t) implie s t h a t
^ \f(U) i=0
> f
where 0 < A ^ < 1 , an d J2i= o A ^ = 1 . I f f(t) i s strictl y convex , the n equality hold s i f an d onl y i f t o = t\ = • • • = £ M - I - O n th e othe r hand , the concavit y o f f(t) woul d impl y M-l (M-\
\
i=0 V
i= 0 J
W h e n f(t) i s strictl y concave , equalit y hold s i f an d onl y i f t o = t\ =
• • • = tM-i'
326 A
. Mathematica l tool s
Appendix B Review o f rando m processe s This appendi x give s a brie f overvie w o f rando m processes . A mor e detaile d treatment o f this topi c ca n b e foun d i n [111 , 179].
B.l Rando
m variable s
A real random variabl e x(rj) i s a real value that i s assigned t o ever y outcome77 of a rando m experiment . Th e argumen t7 7 i s usually droppe d fo r convenience . The probabilit y o f x < a can b e describe d b y th e cumulative density function (cdf) Fx(a) =
P(x < a).
We ca n als o compute i t usin g th e probability density function (pdf ) f P(x
I
f
x(x),
x(x)dx,
J — OO
where th e pd f f x{x) i s th e derivativ e o f F x(x). I n communications , a fre quently use d rando m variabl e i s th e Gaussian . Fo r a rea l Gaussia n rando m variable, th e pd f i s fx[x) =
J
-
e
\/2lXGx
- ( * - ^ ) 2 A ^ -0
0 < x < 00 ,
where m x an d a 2 are , respectively , th e mea n an d varianc e o f x. Expected value s Le t g(x) b e a functio n o f a rando m variabl e x. Th e expected valu e o f g(x) i s define d a s OO
/
g(x)fx(x)dx.
-00
Some commonly use d expecte d value s ar e the mean , mea n square d value , an d variance, give n respectivel y b y mean m
x
=
E[x],
mean square d valu e = ^[x 2 ], variance o
x
= E[{x — m x) ] .
327
B. Revie w o f rando m processe s
328
When m x = 0 , w e sa y th e rando m variabl e i s zero-mean . I n thi s book , al l random variable s hav e zero-mea n unles s otherwis e mentioned . Fo r a zero mean rando m variable , th e varianc e i s equa l t o it s mea n square d value . A s the transmitte d signal s an d nois e ar e ofte n modele d a s rando m variable s i n digital communicatio n systems , we will call the mea n square d valu e the power of the rando m variable . Two o r mor e rando m variable s Tw o rando m variable s x an d y ca n b e described b y th e joint probability density function f xy(xyy). Le t g(x yy) b e a function o f x an d y. Th e expecte d valu e o f g(x, y) i s define d a s oo />o
/
o
/ g(x,y)f(x,y)dxdy.
-oo J — oo
Two rando m variable s x an d y ar e independent i f fxy{x,y) = fx(x)f
y(y).
A useful quantit y tha t ofte n come s into the discussio n of two random variable s is the cross correlation, give n by R xy = E[xy]. W e say x an d y are uncorrected if Rxy = E[x]E[y], and w e sa y x an d y ar e orthogonal i f R xy = 0 . Not e tha t th e independenc e implies uncorrelatednes s bu t th e convers e i s usuall y no t true . However , fo r Gaussian rando m variables , th e propert y tha t tw o rando m variable s ar e un corrected als o implie s tha t the y ar e independent . Fo r tw o rando m variable s x an d y tha t ar e zero-mean an d uncorrelated , th e su m w = x + y ha s varianc e g i v e n b y cr 2w = cr 2x + (j 2y.
Similarly, a se t o f M rando m variable s xo , x\,..., % _ i , ca n b e describe d by a joint pdf . Th e expecte d value , independence , an d uncorrelatednes s ca n be define d i n a simila r manner . Complex rando m variable s A comple x rando m variabl e x = a + jb i s a complex quantit y whos e rea l par t a an d imaginar y par t b are bot h rando m variables. Fo r a comple x rando m variabl e x, th e mean , mea n square d value , and varianc e ar e define d respectivel y a s mx =
E[x], E[\x\%
E[\x-m
2
x\
}.
For tw o comple x rando m variable s x an d y, th e cross correlation i s Rxy =
E[xy*}.
Similar t o th e rea l case , w e sa y x an d y ar e uncorrelated i f R xy = E[xy*] = i£[#]i£[2/*], an d the y ar e orthogonal i f R xy = 0 . W e sa y a rando m variabl e is circularly symmetric complex Gaussian i f it s rea l an d imaginar y part s ar e independent Gaussia n variable s wit h th e sam e variance . I n passban d com munications, th e channe l nois e i s ofte n modele d a s a zero-mea n circularl y symmetric comple x Gaussia n rando m variable . It s pd f i s give n i n (2.5 ) an d it i s plotted i n Fig . 2.7 . I n th e followin g discussion , th e rando m variable s ar e assumed t o b e comple x unles s otherwis e mentioned .
B.2. Rando m processe s
329
Random vector s A rando m vecto r x o f dimension s M x 1 is a collectio n of M rando m variables , x = [xo x\ • • • XM-I] • It s mea n E[x]=[E[x0] E[x{\
••
T
• E[x
M-i]]
is als o a vector . T h e cross correlation matrix o f two rando m vector s x an d y i s defined a s E ^ x y t ] . W e cal l th e MxM matri x R x = ^ [ x x ^ ] th e autocorrelation matrix o f x . Fo r example , le t M = 3 ; the n w e hav e
E[\xo\2] R.T
rL
x0x1
rL
x0x2
ttxoX!
E[\xi\2} rL
x1x2
ftx0x2 %iX
2
E[\x2\\
where R XiXj denote s th e cros s correlatio n betwee n Xi an d Xj. T h e diagona l of R x consist s o f th e mea n square d value s ^[|x^| 2 ] whil e th e off-diagona l terms ar e th e cros s correlation s amon g th e rando m variables . I n th e zero mean cas e R x i s als o sai d t o b e th e covariance matrix. Observ e t h a t a n autocorrelation matri x i s alway s Hermitian , i.e . Rj , = R x . Moreover , i t can b e show n t h a t R x i s positiv e semidefinite . I n th e specia l cas e t h a t th e elements o f x ar e orthogona l t o on e another , th e autocorrelatio n matri x i s diagonal. If , i n addition , th e rando m variable s hav e th e sam e varianc e
B.2 Rando
m processe s
A rando m proces s x(n, rj) i s a functio n t h a t i s assigne d t o ever y outcom e 7 7 of a n experiment . W e usuall y writ e i t a s x(n) an d dro p th e argumen t 7 7 for convenience. Fo r a particula r outcome , x(n) i s a fixed tim e function , whic h we cal l a realization o r a sample function. W e ca n vie w x(n) fo r ever y n a s a r a n d o m variable . Ofte n times , th e first- an d second-orde r statistic s ar e ver y useful i n application s involvin g rando m processes , i n particula r th e mea n an d autocorrelation functions . I n genera l th e mea n functio n E[x(n)\ depend s o n the tim e inde x n. T h e autocorrelation function R x(rn,n) i s define d a s th e cross correlatio n betwee n x(m) an d x(n), Rx(rn,n)=
E[x(jn)x* (nj\.
This i s generall y a functio n o f b o t h tim e indice s m an d n. T h e rando m processes t h a t aris e i n th e applicatio n o f communicatio n system s ca n ofte n b e assumed t o hav e certai n stationar y properties , e.g . wid e sens e stationar y an d cyclo wid e sens e stationar y properties .
Wide sens e stationar y processe s A rando m proces s x(n) i s sai d t o b e wid e sens e stationar y (WSS ) i f x(n) satisfies th e followin g tw o conditions : (1) E[x(n)] = (2) E[x(n)x*(n
m x ( a constan t independen t o f n) ; , k)] = R x(k) ( a functio n o f k). ^
, >
R
T h e mea n functio n E[x(n)] i s independen t o f n an d th e autocorrelatio n func tion E[x(n)x* (n — k)] depend s onl y o n th e time lag /c , bu t no t n.
B. Revie w o f rando m processe s
330
Power spectru m Fo r a WS S process , th e powe r spectru m (o r power spectral density) i s define d a s th e Fourie r transfor m o f R x(k), CO
/e= —o o
Conversely R x{k) i s th e invers e Fourie r transfor m o f S x{e^). I n particular , the powe r o f x(n) ca n b e obtaine d fro m it s powe r spectru m usin g Rx(0) =
/ S Jo
x(e^)duj/27r.
It ca n b e show n t h a t S x{e^) > 0 fo r al l ou; the powe r spectru m i s a non negative function . Th e are a unde r S x(e^UJ) fro m CJQ t o uo\ ca n b e viewe d a s the powe r o f x(n) i n th e frequenc y rang e (c^o^i) Jointly W S S processe s Tw o rando m processe s x{n) an d y(n) ar e sai d t o be jointly WSS if eac h o f t h em i s WSS an d thei r cros s correlatio n E[x(n)y* (n— k)] i s a functio n o f tim e la g k only , independen t o f n. I n othe r words , thei r cross-correlation functio n i s Rxy(k)=
E[x(n)y*(n-k)].
We sa y t h a t x(n) an d y{n) ar e uncorrelated i f R xy(k)= m xm*, fo r al l k. I f one o f th e processe s ha s zero-mean , the n uncorrelatednes s implie s R xy(k)= 0 for al l k. W h e n w e hav e tw o jointl y WS S rando m processe s x(n) an d 2/(n) , zero-mean an d uncorrelate d wit h eac h other , thei r su m w(n) = x(n) -\-y(n) i s also a WS S rando m proces s wit h zero-mean , an d cr^ = o^. + cfy. Computation o f experimenta l statistic s A WS S proces s i s sai d t o b e ergodic i f th e statistica l averages , e.g . E[x(n)], R x(k), ar e equa l t o th e corre sponding tim e average s fo r an y singl e realization . Th e ergodicit y assumptio n allows u s t o estimat e th e expecte d value s b y usin g tim e average s obtaine d from a singl e realization . I n practice , th e numbe r o f availabl e sample s o f a realization xo(n) i s ofte n finite. Fo r example , give n N sample s o f a realizatio n xo(n), w e ca n estimat e th e mea n b y N-l 1
m
1 7V_ N
n=0
T h e autocorrelatio n functio n ca n b e estimate d b y Rx(k)=
1 7V_ JJ—J^ Y,
x
1
o(n)x*0(n -
k), k
= 0,1,... ,N - 1 .
T h e accurac y o f th e estimat e improve s a s th e numbe r o f availabl e sample s N increases.
B.2. Rando m processe s
331
W h i t e processe s A rando m proces s i s sai d t o b e whit e i f x(m) an d x(n) ar e uncorrelate d fo r any m ^ n , i.e . E[x(m)x* (n)]
= E[x(m)]E[x*(n)], m
^ n.
W h e n a whit e proces s i s als o WSS , w e hav e m
l + ali fc =
r> n\ i
[m^, otherwise
0,
.
If, i n addition , x(n) ha s zero-mean , the n Rx(k)=
5{k)a
2 x.
In thi s cas e th e powe r spectru m i s a constan t give n b y Sx(e>") =
o* x,Vw.
Such a proces s i s usuall y referre d t o a s white noise. A WS S proces s i s sai d t o be colored i f i t i s no t white . T h e channe l nois e q(n) i n wireles s environment s can ofte n b e assume d t o b e a n additiv e whit e Gaussia n nois e (AWGN ) wit h zero-mean. Fo r wire d applications , th e nois e i s often colore d additiv e Gaussia n noise (AGN) . Fo r AWG N an d AGN , wid e sens e stationarit y an d zero-mea n are usuall y implicitl y assumed .
Vector rando m processe s An M x l vecto r rando m proces s x ( n ) = [xo(n) x\(n) • •• XM-IW ]i s a vecto r o f M rando m processes , wher e eac h entr y Xi(n) i s a rando m process . For eac h n , x ( n ) i s a rando m vector . W e sa y x ( n ) i s WS S i f b o t h E[x(n)] an d E{x(n)x)(n — k)] ar e independen t o f n. T h e n th e autocorrelatio n i s R(fc) = £ [ x ( n ) x t ( n - / c ) ] , which i s a n M x M matri x fo r ever y tim e la g k. W h e n x ( n ) i s WSS , an y pai r of component s x m ( n ) an d Xi(n) ar e jointl y WSS .
Cyclo wid e sens e stationar y processe s A rando m proces s x(n) i s sai d t o b e cycl o wid e sens e stationar y wit h perio d M (abbreviate d a s C W S S ( M ) ) i f i t satisfie s th e followin g tw o conditions : (1) E[x(n + M)]= E[x(n)], . (2) E[x(n)x*(n k)] = E[x(n + M)x*(n +
M - k)\.
K
.
}
These tw o propertie s impl y t h a t th e mea n functio n E[x(n)\ an d th e mea n squared functio n ^ [ | x ( n ) | 2 ] ar e b o t h periodi c wit h perio d M . Als o th e aut o correlation E[x(n)x* (n — k)] i s a periodi c functio n o f n. W e defin e th e averag e autocorrelation functio n a s M-l
332
B. Revie w o f rando m processe s
which i s a functio n independen t o f n. Th e averag e powe r o f x(n) o r th e average mea n square d valu e i s M-l
n=0
Similar t o th e WS S case , w e defin e th e averag e powe r spectru m o f a CWS S process a s th e Fourie r transfor m o f th e averag e autocorrelatio n function , CO
We wil l omi t th e wor d "average " an d cal l R x{k) th e autocorrelatio n functio n and S x{e^) th e powe r spectru m o f x(n) whe n i t i s clea r fro m th e contex t t h a t x(n) i s a CWS S process .
B.3 Processin g o f rando m variable s an d rando m processes Passage throug h LT I system s W h e n w e pas s a WS S proces s x(n) throug h a stabl e LT I syste m H(z), th e output y{n) (Fig . B . l ) i s als o WSS . I n particular , y{n) ha s mea n an d auto correlation function s give n respectivel y b y h n E[y(n)] = m y= mx J2 n ( ), m Ry(k) = Rx(k) * h(k) * h*(-k). {
}
ox
T h e powe r spectru m o f y{n) i s relate d t o t h a t o f x(n) b y Sy(e?») =
S
2
x(e?»)\H(e?»)\
.
W h e n x(n) i s AWG N (WS S wit h zero-mean) , S y(ej") = power spectru m o f y{n) assume s th e shap e o f \H (eja;)|2. I compute th e outpu t powe r a 2 usin g
(T 2x\H{e^)\2] th e n thi s case , w e ca n
T h e inpu t powe r i s amplified b y a facto r o f J 0 \H (eja;) \ 2duj/27r1 whic h i s equa l to th e energ y o f th e filter. W h e n w e pas s a C W S S ( M ) proces s x(n) throug h a stabl e LT I filter H(z), as show n i n Fig . B . l , th e outpu t i s als o a C W S S ( M ) process . Th e (average ) autocorrelation functio n o f th e outpu t y{n) i s give n b y Ry(k) =
R x(k) *
fe(jfe)*h*(-k). (B.4
)
We ca n obtai n th e (average ) powe r spectru m o f th e outpu t b y takin g th e 2 Fourier transfor m o f th e abov e expression , S y{e^) = S x{e^)\H{e^)\ .
B.3. Processin g o f rando m variable s an d rando m processe s
x(n) »
] H(z) | ►
y
333
(n)
Figure B . l . Passag e o f a WSS proces s throug h a n LT I system .
Passage throug h multirat e block s Suppose w e pass a WSS proces s x{n) throug h a decimator (Fig . B.2(a)) , the n the outpu t i s give n b y u(n) = x(Mn). I t can b e show n tha t u(n) i s als o a WSS proces s wit h mea n an d autocorrelatio n functio n given , respectively , b y mu =
m x, R
u{k)
=
R
x(Mk).
On th e othe r hand , whe n w e pas s a WSS proces s x(n) throug h a n expande r (Fig. B.2(b)) , th e outpu t v(n) ha s mea n functio n give n b y . , X 1 I m x. n is a multiple o f M , E[v[n)\ = < 10, otherwise . Furthermore, _ r , N . ,7 Rx(k/M), n N1 I E[v(n)v* (n - k)\ = { v ' J 10, otherwise
and k are intege r multiple s o f M , .
The expande r outpu t v(n) i s n o longe r WSS . I t follows fro m th e definitio n of CWS S rando m processe s tha t v(n) i s CWSS(M) . I t turns ou t tha t th e (average) autocorrelatio n functio n o f v(n) i s simpl y a n expande d an d scale d version o f R x(k); tha t is , 1, , _ \±R x(k/M), k ^ W - M ^ W V " j o , otherwise
is a n intege r multipl e o f M , , (B.5) where (R x(k)) denote s th e M-fol d expande d versio n o f R x{k). W e ca n verify thi s b y first examinin g th e expressio n M-l
^(^-^KnKln-l)]. n=0
Only th e ter m E[v(fS)v* (—k)\ ca n b e nonzero , an d i t i s nonzer o onl y whe n k is a n intege r multipl e o f M . I n thi s case , th e ter m E[v(0)v* (—k)] i s equa l t o Rx(k/M). Whe n k is no t a multiple o f M , Rv(k) =
j
= iE[v(0)v*(-k)]
0.
From (B.5) , w e can obtai n th e powe r spectru m o f v(n):
It i s a scaled an d "squeezed " versio n o f the inpu t powe r spectrum .
334
B. Revie w o f rando m processe s
|M
x(n)
\M
x(n)
-► u(n)
(a)
-> v(n)
(b)
Figure B.2 . Passag e o f a WSS proces s throug h multirat e buildin g blocks .
Passage throug h matrice s In Fig . B.3 , we pass a n M x l rando m vecto r x throug h a n A ^ x M matri x P . The outpu t y = P x ha s autocorrelatio n matri x R ^ = i£[yy"l" ] give n b y R^ = P ^ x x ^ P 1 " = P R . P 1 . In man y case s th e DF T matri x ha s bee n foun d t o b e ver y usefu l i n th e pro cessing of random vectors . Conside r th e cas e that P i s the M x M normalize d DFT matri x W . Whe n th e inpu t vecto r x ha s uncorrelate d elements , it s au tocorrelation matri x i s diagonal , an d i n thi s cas e R ^ i s a n M x M matri x sandwiched betwee n W an d W" L Fro m Theore m 5.2 , w e kno w matrice s o f such a for m ar e circulant . Furthermore , th e variance s a^. ar e o f th e sam e value an d i t i s equal t o th e averag e o f the variance s a^ .
—n >
—A > •I
xM-\
I
p
•
•i
i
•
•i
i
•
> yn-i
—w ► Figure B.3 . Passag e o f a rando m vecto r throug h a matrix .
Blocking In man y applications , w e bloc k a scala r rando m proces s t o obtai n a vecto r random process , a s show n i n Fig . B.4 . Th e vecto r rando m proces s i s x(n) = [x(Mn) x(Mn
+ 1)•
• • x(Mn
+ M - 1)]
T
.
When x{n) i s WSS , s o i s th e blocke d versio n x(n) . I n som e applications , w e are only concerned with R(0 ) bu t no t R(/c ) fo r k ^ 0 , although the underlyin g
B.3. Processin
335
g o f rando m variable s an d rando m processe s
x(n) x(n)
w
zw
w
\M
w W
^wx{Mn+l)
|M Z
x{Mn)
W
T
x(Mn+M-l)
| M
Figure B.4 . Blockin g o f a scala r rando m proces s x{n)
process i s a vecto r rando m process . Fo r example , whe n w e conside r one-sho t block transmissio n i n Chapter s 6 an d 7 , onl y R ( 0 ) i s relevant . T h e auto correlation matri x R ( 0 ) o f x ( n ) i s a Hermitia n positiv e semidefmit e Toeplit z matrix. Suppos e th e autocorrelatio n functio n o f x(n) i s r^ , then , fo r M = 3 ,
R(0)
ro
ri
T2
We sa y R ( 0 ) i s th e M x M autocorrelatio n matri x associate d wit h th e scala r process x(n). I n th e specia l cas e t h a t th e scala r WS S proces s x(n) i s white , R ( 0 ) reduce s t o th e simpl e matri x cr^I. Fo r th e blockin g syste m i n Fig . B.4 , we ca n verif y t h a t th e outpu t vecto r rando m proces s x ( n ) i s WS S whe n th e input x(n) i s C W S S ( M ) .
Unblocking Consider th e interleavin g (unblocking ) syste m o f expanders followe d b y a dela y chain, a s show n i n Fig . B.5 . T h e syste m interleave s th e M input s t o produc e a singl e outpu t x(n). I f th e input s ykiji) ar e jointly WSS , the n th e interleave d o u t p u t x{n) i s C W S S ( M ) . T o se e this , w e firs t verif y t h a t th e mea n functio n of x(n) i s periodic . Le t u s writ e n = UQ -\- n i M, wher e n o = ((TI))M a n d the notatio n ( ( - ) ) M denote s modul o M operation . W e se e t h a t E[x(n)\= E[yno(ni)]. Als o not e t h a t E[x{n + M)\ = E[y no(l + n i ) ] , whic h i s equa l t o E[yno(ni)] becaus e y no(^) i s WSS . T o se e i f i t satisfie s th e secon d propert y i n (B.2), le t n - k = £ 0 + < i M , wher e £ 0 = ((n - k)) M. T h e n E[x(n)x*(n-k)] =
^ [ 2 / n 0 K ) 2 / | 0 ( ^ i ) ] = Ry nQviQ(ni ~
£
^'
Similarly, w e ca n verif y t h a t E[x(n + M)x*(n +
M - k)} = E[y no(ni +
1 ) ^ ( 4 + 1) ] = Ry
noVto
(n,
- 4 )
B. Revie w o f rando m processe s
336
00
0V '
An)
w
W
w
W
\M
fM A
•
^M-lO) J
W
-1 ^Z
| M
-1
^z
i
Figure B.5 . Interleavin g (unblocking ) o f M rando m processes .
and th e secon d propert y i n (B.2 ) holds . Conversely , i t ca n b e show n t h a t an y C W S S ( M ) rando m proces s ca n b e viewe d a s th e outpu t o f th e unblockin g system i n Fig . B. 5 wit h jointl y WS S inputs .
B.4 Continuous-tim
e rando m processe s
In thi s book , w e mostly dea l with discrete-tim e rando m processes . Continuous time rando m processe s als o aris e in som e o f the discussions . A continuous-tim e random proces s x a(t, rj) is a continuous-tim e functio n t h a t i s assigne d t o ever y outcome r] of a n experiment . Lik e th e discrete-tim e case , w e usuall y writ e i t as x a(t) an d dro p th e argumen t rj for convenience . Fo r a particula r outcome , xa(t) i s a fixed continuous-tim e function . W S S processe s Fo r continuous-tim e rando m processes , th e WS S prop erty ca n b e define d i n a simila r manner . W e sa y t h a t x a(t) i s WS S i f b o t h E[xa(t)\ an d E[x a(t)x^(t — r)] ar e independen t o f th e variabl e t. Th e autocor relation o f x a(t) i s th e continuous-tim e functio n R Xa(r)= E[x a(t)x^(t — r)]. T h e powe r spectru m (o r power spectral density) i s defined a s the Fourie r trans form o f R Xa(r),
sXaUty
oo
/
RXa(r)e~jnT dr.
-OO
Conversely, R Xa{r) i s th e invers e Fourie r transfor m o f S Xa(jQ,). W h e n x a(t) is white , WSS , an d zero-mean , it s autocorrelatio n functio n i s a n impuls e RXa(r) = a^. 5(T) an d th e powe r spectru m i s a constan t S Xa(jft) = cr^ , for al l Q . W h e n w e pas s a continuous-tim e WS S proces s x a(t) throug h a stabl e LT I filter H(jQ,), a s show n i n Fig . B.6 , th e outpu t i s als o a WS S process . T h e autocorrelation functio n o f th e outpu t y a(t) i s give n b y Rya(T)=RXa(T)*h(T)*h*(-T).
(B.6)
B.4. Continuous-tim e rando m processe s
xa(t) »|#(/Q)
337
y a(t)
|►
Figure B.6 . Passag e o f a WSS proces s throug h a continuous-tim e LT I system .
The powe r spectru m o f y a(t) i s relate d t o tha t o f x a(t) b y
sya(jn) =
s
2
Xa(jn)\H(jn)\
.
For input x a(t) tha t i s white, WSS, and zero mean, the output powe r spectru m Sya(jn)= aljH(jn)\ 2 assume s th e shap e o f \H(jn)\ 2. CWSS processe s W e sa y tha t a continuous-tim e rando m proces s x a(t) is cycl o wid e sens e stationar y wit h perio d T (denote d b y CWSS(T) ) i f (1) E[x (2) E[x
a(t
+ T)] = E[x a(t)], (t)x* (t-T)] = E[xa(t + a a
T)x* a(t + T-T)]. ^ '
j
Both th e mea n functio n E[x a(t)\ an d th e mea n square d functio n ^[|x a (t)| 2 ] are periodi c wit h perio d T. Th e (average ) autocorrelatio n functio n i s define d as RXa(r)
^J E[x
a(t)x*a(t-r)]dt,
which i s a function independen t o f t. W e define th e (average ) powe r spectru m as th e Fourie r transfor m o f the (average ) autocorrelatio n function , oo
/
-oo
RXa(r)e~jilT dr.
Like th e discrete-tim e case , th e wor d "average " wil l b e omitte d whe n i t i s clear fro m th e contex t tha t x a(t) i s a CWS S process . When w e pass a continuous-time CWSS(T ) proces s x a(t) throug h a stabl e LTI filter H(jQ), a s shown in Fig. B.6, the outpu t i s also a CWSS(T) process . The (average ) autocorrelatio n functio n o f th e outpu t y a(t) i s give n b y fiyjr)=
R Xa(r) * h(r) * ft*(-r).
(B.8
)
Taking th e Fourie r transfor m o f R ya(r), w e ca n obtai n th e (average ) powe r 2 spectrum o f th e outpu t a s S Va(jQ) = S Xa(jQ)\H(jQ)\ .
Passage throug h C / D an d D/ C converter s When w e pass a continuous-tim e rando m proces s r a (t) throug h th e syste m o f P2(t) followe d b y a C/D converter , a s shown in Fig. B.7, we get a discrete-tim e random proces s r(n). Th e proces s r(n) i s give n b y r(n) =
w a(nT) =
(r a *p 2 )(*)
t=nT
B. Revie w o f rando m processe s
338
W h e n r a(t) i s WSS , th e discrete-tim e proces s r(n) i s als o WSS . Th e autocor relation functio n i s R r{k) = E[r(n)r*(n - k)] = E[w a(nT)w*a(nT kT)}. W e can als o writ e i t a s Rr(k)=
R Wa(kT)=
R ra(r) * p
2(r)
*p*
2(-r)
=kT
which i s a sample d versio n o f R Wa ( r ) . Not e t h a t R r(0)= R Wa (0) , t h a t i s r(n) and w a(t) hav e th e sam e power . Conside r th e cas e wher e r a(t) i s white , WSS , and zer o mean . Suppos e th e receivin g puls e p 2(t) i s a n idea l lowpas s filte r with unit y gai n i n th e passban d an d cutof f frequenc y TT/T. The n w a(t) ha s a lowpass powe r spectru m an d S Wa (j£l) = of a i n th e passband . I t follow s t h a t RWa(0)= a 2 /T an d r(n) i s a discrete-tim e whit e nois e (WS S an d zero-mean ) 2 with S r(e^)= (T r /T. P2(t)
ra{i)
C/D
-+. r(n)
T T
Figure B.7 . Passag e o f a continuous-tim e rando m proces s throug h p2(t) followe d b y a C/ D converter .
x(n)
D/C
T
Px(t)
-► ^a( 0
T
Figure B.8 . Passag followed b y pi(t).
e o f a discrete-tim e rando m proces s throug h a D/ C converte r
Now conside r th e D / C converte r followe d b y a transmittin g puls e pi(t) shown i n Fig . B.8 . Assum e th e discrete-tim e inpu t x{n) i s WSS . W h e n th e transmitting puls e pi(t) i s a n idea l lowpas s filter , th e outpu t x a{t) i s WS S [131]. W h e n p\{t) i s no t ideal , th e outpu t x a{t) i s i n genera l a continuous time C W S S ( T ) process . Furthermore , whe n th e discrete-tim e inpu t x(n) i s C W S S ( M ) , th e outpu t x a(t) i s C W S S ( M T ) . I n th e followin g w e wil l conside r the cas e t h a t x{n) i s C W S S ( M ) , a s a WS S x{n) ca n b e viewe d a s a C W S S ( M ) process wit h M = 1 . For th e syste m show n i n Fig . B.8 , w e kno w th e outpu t ca n b e expresse d as xa(t)
E x(n)p (t-nT). 1
339
B.4. Continuous-tim e rando m processe s
We can verif y x a(t) i s CWSS (MT) b y directl y substitutin g th e abov e expres sion int o (B.7) . I t turn s ou t tha t th e (average ) autocorrelatio n functio n o f xa(t) i s related t o tha t o f x(n) i n a simpl e manner . T o derive R Xa(r), w e firs t note tha t E[xa(t)x*a(t-T)}=E
yy
x(n)x*
nm
(m)pi (t — nT)p\ (t — r — mT)
J
Y Y E [ * w** MI PI (* - nT )pi (t-T-
mT
nm
E
YY
nk
x
I
(nK (n -
k
M-l
= Y^2Y1 E[x(Me =i 0
k£
)
)\ Pi (* - nT )Pl (t-r-(n- k)T)
+ i)x*(M£ + i-k)]
x Pl (t - (Ml + i)T)p\(t -T- (Ml
+ i - k)T),
where we have used a change of variables k = n—m t o obtain the third equalit y and a chang e o f variable s n = Ml + i t o obtai n th e fourt h equality . A s x(n) is CWSS(M) , w e hav e E [x(M£ + i)x*(Ml + i - k)} = E [x(i)x*(i - k)}. I t follows tha t M-l
E[xa(t)x*a(t-r)] =
E
E[x{i)x*(i-k)]
EE k£
=i 0
x Pl (t - (Ml + i)T)p\(t -T- (Ml Therefore, w e hav e 1 /.M T ^ x 0 ( r ) = — j E[x
+ z - fc)T).
a(t)x*a(t-r)]dt,
M-l
^ ^ [ x ( t ) i * ( t - k ) ] MT A; z= 0 E/
/•MT
pi(
t - (M £ + i)T)p\(t -T- (Ml
+ z - fc)T)dt
M-l
MT /
A; z= oo
0
Pl(t
- iT)p\(t
- r - ( i - k)T)dt. (B.9
)
-oo
Let u s defin e
KT) = Pi( T) * P i ( then R Xa (r) ca n b e writte n a s 11
M_
r
)^
1
i=0
= ^ ^ ( f c ) M r - f c T ).
(B.10)
340 B
. Revie w o f rando m processe s
Taking the Fourier transfor m o f the abov e expression, w e obtain the (average ) power spectru m o f xa(t) a s SXa(jtt) =
^ ( e ^ l J M J f t ) ! 2 , (B.ll
where Pi(jft) i s the Fourier transfor m o f the transmitting puls e pi(t).
)
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Index additive whit e Gaussia n noise , see AWGN AGN, 14 , 33 1 A M - G M inequality , 32 3 analysis filte r bank , 9 5 autocorrelation CWSS, 33 1 estimate, 33 0 function, 32 9 matrix, 32 9 WSS, 32 9 AWGN, 14 , 18 , 33 1 bandwidth efficiency , 11 9 bandwidth expansio n ratio , 99 , 119 baseband communication , 12 , 1 8 BER, 1 9 PAM, 1 9 QAM, 2 4 B E R minimization , 19 3 bias removal , 4 3 biased estimate , 41 , 15 0 biased SNR , 41 , 20 6 SC-CP, 15 8 Z P - O F D M , 15 0 binary phas e shif t keyin g (BPSK) , 21 bit allocation , see bi t loadin g bit loading , 3 , 2 9 gain, 23 1 O F D M , 23 2 SC-CP, 23 4 integer, 30 , 174 , 23 5 optimal, 22 9 bit stream , 1 bits-to-symbol mapping , 1 block transceiver , 10 5 blocked version , 7 6 blocking, 7 6 r a n d o m process , 33 4
C / D converter , 1 1 random process , 33 7 carrier frequenc y offse t (CFO) , 179 carrier modulation , 1 2 C F O , 17 9 channel capacity , 2 0 channel dependence , 4 , 13 5 O F D M , 13 9 precoded O F D M , 20 2 SC-CP, 15 3 channel diagonalization , 13 7 channel equalization , 1 6 channel model , 9 AGN, 1 4 AWGN, 1 4 continuous-time, 1 0 discrete-time, 1 0 FIR, 1 4 frequency-nonselective, 1 5 iid channel , 1 6 random channel , 1 5 SIMO, 6 8 channel noise , 1 3 baseband transmission , 1 3 passband transmission , 1 3 channel shortening , 59 , 12 0 channel spectra l null , 144 , 156 , 203, 20 6 channel stat e information , 3 , 13 5 circulant, 108-111 , 137 , 16 0 definition, 10 9 diagonalization, 11 1 circular convolution , 11 1 circularly symmetri c comple x Gaus sian, 13 , 136 , 32 8 complexity O F D M , 13 8 SC-CP, 15 4 SC-ZP, 16 1
356 zero-padded O F D M , 14 9 concave function , 32 5 congruous zero , 29 7 almost congruous , 30 5 definition, 29 8 minimum-redundancy, 30 3 convex function , 32 5 crosstalk, 169 , 17 2 CWSS continuous-time, 33 7 discrete-time, 33 1 cyclic prefix , 2 , 13 6 O F D M , 13 9 SC-CP, 15 2 cyclic-prefixed systems , 115-11 9 cyclo wid e sens e stationary , see CWSS D / C converter , 1 0 random process , 33 7 decimation filter , 7 9 decimator, 7 1 D F T bank , 16 4 implementation, 266 , 27 7 receiver, 27 7 transmitter, 26 6 D F T decomposition , 29 5 D F T filter , 164 , 27 2 DFT-based transceiver , 3 , 13 5 digital modulation , 1 8 discrete multitone , see D M T D M T , 3 , 135 , 168 , 26 0 bit allocation , 17 4 filter ban k representation , 16 9 subfilter, 25 9 time domai n equalizer , 16 9 tone, 16 9 tone SNR , 17 4 downsampler, 7 1 DSL, 3 , 135 , 17 2 egress, 25 9 equalization, 1 6 equalizer, 1 6 linear equalizer , 1 6 equivalent discrete-tim e channel , 12 passband, 1 3 expander, 7 1
INDEX exponential powe r dela y profile , 16 F E Q , 137 , 17 4 F E X T crosstalk , 17 2 filter ban k representation , 26 3 filter ban k transceiver , 5 , 95, 29 1 distortion, 10 0 D M T , 16 9 no redundancy , 30 1 O F D M , 16 3 polyphase representation , 9 0 redundant, 9 7 Z P - O F D M , 16 8 F I R channel , 1 4 least-squares equalizer , 3 5 MMSE equalizer , 4 5 zero-forcing equalizer , 3 4 F I R equalizer , 17 , 3 3 MMSE, 4 5 zero-forcing, 3 4 F I R transceiver , 29 1 minimal, 30 1 minimum-redundancy, 30 3 solution, 30 3 Fischer inequality , 32 4 flatness, 23 3 fractionally space d equalize r (FSE) , 122, 12 8 frequency divisio n multiplexin g (FDM), 172 , 25 9 frequency domai n equalize r ( F E Q ) , 3, 13 7 frequency-division multiplexin g (FDM) , 97 frequency-selective channel , 1 5 frequency-shifting property , 164 , 263, 266 , 272 , 276 , 27 7 Gaussian, 1 3 AGN, 1 4 AWGN, 1 4 circularly symmetri c complex , 13 real, 1 3 Gaussian noise , 13 , 32 8 guard interval , 2 , 14 7 subfilter, 266 , 27 6 H a d a m a r d inequality , 32 4
INDEX IBI, 2 , 136 , 14 7 IIR minima l transceiver , 30 1 ingress, 25 9 integer bi t allocation , 30 , 17 5 inter subchanne l interference , 10 0 interblock interferenc e (IBI) , 2 , 102, 13 6 interference, 1 2 interpolator, 8 0 intersymbol interference , see IS I intra subchanne l interference , 10 0 intrablock interference , 2 ISI, 2 , 12 , 16 , 3 3 ISI-free property , 16 , 102 , 291 , 293 least-squares problem , 3 5 linear estimate , 3 9 MMSE, 4 4 multiple rando m variables , 44 linear estimator , 3 9 lower triangula r Toeplit z matrix , 35 minimal transceiver , 10 0 minimum mea n square d error , see MMSE minimum redundancy , 29 1 minimum-redundancy congruous zero , 30 3 MMSE, 3 9 precoded O F D M , 20 3 SC-ZP, 16 1 zero-padded O F D M , 15 0 modulation symbol , 1 , 1 8 Monte Carl o simulation , 2 0 nearest neighbo r decisio n rule , see NNDR N E X T crosstalk , 17 2 NNDR, 1 9 MMSE equalizer , 5 6 noble identity , 7 5 Nyquist frequency , 26 1 N y q u i s t ( M ) filter , 8 2 O F D M , 3 , 135 , 19 3 bandwidth expansion , 13 9 bit erro r rate , 14 2
357 channel diagonalization , 13 9 cyclic prefix , 13 9 MMSE, 14 2 oversampling, 18 8 P A P R , 13 9 one-shot transmission , 136 , 147 , 224 optimal precoder , 20 7 orthogonal frequenc y divisio n mul tiplexing, see O F D M orthogonality principle , 3 9 PAM, 1 8 BER, 1 9 SER, 1 9 SNR gap , 2 0 PAPR definition, 13 9 O F D M , 13 9 SC-CP, 15 4 parallel subchannels , 28 , 13 7 parallel t o seria l conversio n ( P / S ( M ) ) , 77 passband communication , 12 , 1 8 peak t o averag e powe r ratio , see PAPR phase shif t keyin g (PSK) , 2 6 pilot, 17 8 pilot tone , 17 8 polyphase matrix, 13 5 polyphase component , 8 4 polyphase decomposition , 84 , 8 7 polyphase identity , 7 6 polyphase implementation , 8 7 polyphase matrix , 9 0 analysis filter bank , 9 0 synthesis filter bank , 9 0 polyphase representation , 90 , 29 2 analysis bank , 9 0 synthesis bank , 9 0 power dela y profile , 1 6 power loading , 3 power spectrum , 33 0 CWSS, 33 2 D M T , 17 0 O F D M , 16 6 precoded O F D M , 19 3 MMSE, 20 3
358 zero-forcing, 19 4 precoder, 144 , 19 3 pseudocirculant, 106-108 , 29 3 definition, 10 6 determinant, 29 6 properties, 293-29 5 rank, 29 7 Smith form , 30 8 zeros, 29 6 PSK, 2 6 pulse amplitud e modulation , see PAM Q function , 1 9 QAM, 2 2 BER, 2 4 SER, 2 4 SNR gap , 2 5 QPSK, 23 , 2 5 q u a d r a t u r e amplitud e modulation , see QA M q u a d r a t u r e phas e shif t keying , see QPSK radio frequenc y interference , see RFI random channel , 1 5 Rayleigh-Ritz principle , 62 , 32 4 receiving filter , 26 0 receiving pulse , 1 1 redundant sample , 2 redundant transceiver , 97 , 10 0 RFI, 173 , 25 9 SC-CP, 5 , 135 , 152 , 19 3 MMSE, 15 7 zero-forcing, 15 5 SC-ZP, 153 , 16 0 MMSE, 16 1 zero-forcing, 16 0 SER, 1 9 PAM, 1 9 QAM, 2 4 serial to paralle l conversio n ( S / P ( M ) ) , 77 sidelobe, 259 , 26 0 signal constellations , 1 8 signal to interferenc e rati o (SIR) , 61
INDEX signal t o nois e interferenc e ratio , see SIN R signal t o nois e rati o (SNR) , 1 7 single-carrier syste m wit h cycli c prefix, see SC-CP , 15 2 single-carrier syste m wit h zer o padding, see SC-Z P single-carrier syste m wit h zer o padding (SC-ZP) , 16 0 single-input multi-outpu t chan nel, 6 8 singular valu e decompositio n (SVD) , 323 SINR, 17 , 6 2 SIR, 6 1 Smith form , 29 4 Smith for m decomposition , 29 4 SNR, 1 7 SNR gap , 2 0 PAM, 2 0 QAM, 2 5 spectral leakage , 6 , 261 , 27 2 spectral rolloff , 6 spectrum mask , 17 2 subchannel gain , 13 7 subfilter, 25 9 B E R performance , 27 4 D F T bank , 266 , 27 7 F E Q coefficients , 27 6 ISI-free condition , 26 3 receiver, 27 6 receiver complexity , 27 8 receiver design , 28 0 transmitter, 26 5 transmitter complexity , 27 1 transmitter design , 27 2 SVD, 226 , 32 3 symbol detection , 1 , 1 8 MMSE receiver , 5 6 symbol erro r rate , see SE R symbol space d equalize r (SSE) , 122 symbol-to-bits mapping , 1 8 synchronization, 17 9 synthesis filter bank , 9 5 T E Q , 60-62 , 169 , 17 4 time domai n equalizer , see T E Q
INDEX time domai n N y q u i s t ( M ) prop erty, 27 9 time multiplexing , 9 7 time-division multiplexin g (TDM) , 97 tone, 16 9 transmitting filter , 26 0 transmitting pulse , 26 1 transmultiplexer, 9 5 unbiased estimate , 41 , 4 3 unbiased SNR , 41 , 20 5 SC-CP, 158 , 18 3 Z P - O F D M , 15 0 unblocking, 7 7 r a n d o m process , 33 5 unitary precoder , 19 4 upsampler, 7 1 wide sens e stationar y property , see WS S windowing, 26 0 WSS, 13 , 3 3 continuous-time, 33 6 discrete-time, 32 9 zero padding , 2 , 112 , 13 6 O F D M , 14 7 single carrier , 16 0 zero-forcing, 2 , 1 6 precoded O F D M , 19 4 O F D M , 13 7 SC-CP, 153 , 15 6 SC-ZP, 16 0 zero-jamming system , 22 5 zero-padded O F D M , 14 7 efficient receiver , 14 9 MMSE, 15 0 pseudo-inverse receiver , 14 8 zero forcing , 14 7 zero-padded system , 112-115 , 22 5 ZJ system , see zero-jammin g sys tem ZP system , see zero-padde d sys tem
