Dsp - application of the fast fourier transform to digital audio electrical and acoustical measurement techniques

APPLICATION OF THE FAST FOURIER TRANSFORM TO DIGITAL AUDIO ELECTRICAL AND ACOUSTICAL MEASUREMENT TECHNIQUES

2254

(A-13)

Andre Perman Bruel and Kjaer Instruments, Inc. Foster City, California

Presented at the 78th Convention 1985 May 3-6 Anaheim This preprint has been reproduced from the author's advance manuscript without editing, corrections or consideration by the Review Board. The AES takes no responsibility for the contents. Additional preprints may be obtained by sending request and remittance to the Audio Engineering Society, 60 East 42nd Street New York, New York 10165 USA. .All rights reserved. Reproduction of this preprint or any portion thereof, is not permitted without direct permission from the Journal of the Audio Engineering Society.

_ I _

APPLICATION OF THE FAST FOURIER TRANSFORM TO DIGITAL AUDIO. ELECTRICAL AND ACOUSTICAL MEASUREMENT TECHNIQUES^ Andre Perman Bruel.and Kjaer Instruments, Inc. 1151 Triton Drive, Suite, B Foster City, CA 94404 0.

ABSTRACT This paper discusses applications of single and dual channel FFT analyzers to evaluate digital audio devices, such as Compact Disc players and PCM audio processors. With an appropriate choice of excitation and FFT instrument a very detailed analysis of the analog output signal is possible and surprisingly easy to perform. The information obtained in most cases allows the user to diagnose the function of both digital and analog circuitry involved with the least possible interaction with internal hardware,

1.

INTRODUCTION St Is not-immediately obvious that an off-the-shelf FFT-analyzer can be used in a straightforward way for very detailed analysis of Digital Audio products. Some initial points of doubt one has to consider are for example: a) Today's FFT analyzers employ 12 bit quantization. Isn f t that too crude for looking at 14-16 bits Digital Audio devices? b) Will looking at one sampling device with another sampling device cause problems due to different sampling rates? . c) Looking at anti-image and anti-alias filters found in Digital Audio products with a device that in itself incorporates a 20 or 25,6 kHz anti-alias filter (not necessarily any better than the one in the measured device) may seem somewhat dubious. As we shall discuss in the following, these objections are rather easily overcome. Numerous practical measurement examples, theoretical results regarding quantization .distortion and optimal FFT sampling rates and suggestions regarding improved test signals on Compact Discs and optimal FFT.-based instrumentation are presented.

- 2 2.1

TIME VS. AMPLITUDE RESOLUTION - DITHER CASE The essence of applying FFT-analyzers to Digital Audio lies in the fact that they operate on blocks of time-data rather than Individual samples. This possibility of exchanging listening (measuring) time for amplitude resolution is already well known in Digital Audio from the use of dither [ 1 ] , The function of dither can be summarized in the following way: "If the listener is given enough listening time and the relevant information is present - even at a level below 1 LSB - during all that time, he (i.e. the listener or a measuring device) will - by averaging be able to recover the information from (dither) noise." In other words by randomizing the quantization error, dither allows accurate recovery of the original analog waveform, if given sufficient time and consecutive signal repetitions. If we step aside for a moment from the main scope of this paper, there is a very important, " B U T . . . " to the dither's function that we would like to highlight here: Obviously time averaging and hence dither work only for signals that are repetitive (periodic) and sustain in time. These kinds of signals (typically sinusoids) are relevant and easy to produce as test signals. "Real life" program material (music) is, however, often of quite a different breed. A singer following, say, a guitar solo within a few milliseconds is quite a dramatic change in waveforms. The listener's possibility to "lock on to" and time average these signals is.only limited to a few hundred samples. Dither or no dither, quantization artifacts on transients remain a "staircase distortion" rather than average out as harmless noise. In particular, this may offer an explanation for the often expressed impression that digital recordings, even in mono, render a subjectively more "dead" room sound than an equivalent analog recording. If we concentrate on the mono case, so that we can disregard the possible differences in stereo channel separation and cross channel mixing, we may imply that the room information is then encoded in the reproduction of reverberation "tails" following each orchestra chord. These relatively short, low level "tails" will become highly distorted in a digital quantization process. In opposition to this an analog recording yields low level signals virtually distortion free. Although the broadband noise floor is typically higher than for a digital recording medium, the "analog medium" noise is uncorrelated to the signal hence presumably allowing the human brain to track the reverberation "tails" all the way down to insignificance. The incurable digital quantization distortion presumably inhibits this tracking process at an earlier point. Consequently it makes sense to assume that the brain interpretes this as a shorter reverberation time, i.e. the ambience is perceived more "dead", the acoustic surroundings appear smaller.

„ 3 -

2.2

TIME VS.. AMPLITUDE RESOLUTION - FFT-CASE A 16 b i t linear d i g i t a l encoding has a dynamic range between the highest and the smallest resolvable sine amplitudes of 90.3 d B . For a t y p i c a l 12-bit FFT analyzer, which in the following has been applied to the analog o u t p u t of various d i g i t a l audio devices, the theoretical dynamic range is c o r r e s p o n d i n g l y 66.2 d B . Too little? Not necessarily, because: The analyzer operates w i t h blocks containing from 1,024 to 10,240 time samples and the Fourier T r a n s f o r m a t i o n involves a v e r a g i n g (weighted i n t e g r a t i o n ) over all these values. T h i s on the whole p r o v i d e s a resolution much b e t t e r than i n h e r e n t l y present in each i n d i v i d u a l sample and is the reason f o r the fact that 12-bit FFT analyzers t y p i c a l l y display spectra over an 80 dB s p a n . Most of the limitation lies in the accuracy of the a v e r a g i n g routines and c o r r e c t l y applied d i t h e r (at the FFT i n p u t ) r a t h e r than the 12 b i t quantization i t s e l f . 90 dB is still more than 80 d B . B u t where the 16 b i t s of d i g i t a l audio are locked to cover a f i x e d analog voltage r a n g e , the 12 b i t s of the FFT analysis can via the i n p u t a t t e n u a t o r s be "zoomed" to p r o v i d e detailed analysis of fine s t r u c t u r e of the analog output v o l t a g e . When the dynamic span between a test tone and its d i s t o r t i o n p r o d u c t s becomes too wide the f u l l power of FFT analysis can be aimed at the d i s t o r t i o n p r o d u c t s b y employing an analog notch f i l t e r in f r o n t of the F F T . With a t y p i c a l 60 dBs attenuation of the test tone, FFT analysis of d i s t o r t i o n down to a comfortable -140 dB can be p e r f o r m e d . It must be noted that since a 12 b i t ADC of the FFT analyzer can be manufactured f a r more linear than 14 or 16 b i t s DACs £• ADCs of Digital Audio and since an analog notch f i l t e r , is not capable of c r e a t i n g n o n harmonically related d i s t o r t i o n , the obtained d i s t o r t i o n p i c t u r e s will be f u l l y d e s c r i p t i v e of the d i g i t a l device under test and NOT the measuring c h a i n . T h e following measurements on an oversampiing (Unit D) and a n o n oversampling ( U n i t A) compact disc players will i l l u s t r a t e t h i s .

2.

TEST TONE FREQUENCIES Sn o r d e r to get the optimum a m p l i t u d e / f r e q u e n c y resolution out of an FFT time record consisting of 1,024 samples, all of these samples should be equally employed in the F o u r i e r t r a n s f o r m a t i o n , i n other words t h i s means that a r e c t a n g u l a r time w e i g h t i n g ( a . k . a " F l a t " o r "No Weighting") should be employed r a t h e r than the generally used Manning w e i g h t i n g . To users of FFT analyzers it is well known t h a t use of the r e c t a n g u l a r time w e i g h t i n g is only permissible for a limited range of signals, namely those that only contain frequencies exactly coinciding w i t h f r e q u e n c y lines of FFT a n a l y s i s . Since f o r a g i v e n FFT analyzer 4£=-™r, whereof is the spacing between f r e q u e n c y lines and T is the time record length in seconds, the p r e f e r r e d test tone frequencies are ^C, 2^f f 3 ^ , . . . , n M . . e t c .

-

U -

Let us consider what t h i s means in terms of p r a c t i c a l l y available CD test discs and FFT a n a l y z e r s : T y p i c a l test frequencies f o u n d on test disc are 20 Hz, 1 kHz and 16 k H z . A 20 kHz 400 lines FFT analyzer, like f o r example B r u e l Bnd Kfaer T y p e 2033 that has been employed in the f o l l o w i n g , is hence w i t h a A ^ o f 50 Hz p e r f e c t l y applicable to all 3 of these f r e q u e n c i e s . On the other hand a 25.6 kHz 800 lines FFT a n a l y z e r , like for example B r u e l and KJaer t y p e 2032 that also has been employed in the f o l l o w i n g , however, for other measurement t y p e s , with i t s ^ f o f 32 Hz doesn't contain a 20 Hz nor a 1 kHz line in its baseband mode. By r e s o r t i n g to Hanning w e i g h t i n g when r e c t a n g u l a r w e i g h t i n g i s n ' t possible, the e f f e c t i v e f r e q u e n c y resolution is compromised from equaling the line s p a c i n g ! ^ t o a p p r o x . 1 . 5 A 4 - [ 2 ) . Using a t y p e 2032 one could hence p e r form analysis of 20 Hz or 1 kHz tones w i t h the same resolution of 1,5 x 32 = 50 Hz as w i t h t y p e 2033 and the 16 kHz tone can be analyzed withdj-= 32 Hz. T h e remarks above a p p l y to the " B a s e b a n d " mode from 0 Hz to Full Zoom mode f r e q u e n c y resolution in at the expense of longer measuring Fig.

analyzers o p e r a t i n g in the so called Scale F r e q u e n c y . When operated in the f r a c t i o n of a Hertz range is achievable, time.

1. A - 80 dB 1 kHz test tone replayed by a C D - p l a y e r and 0-20 kHz w i t h a Flat w e i g h t i n g .

analyzed

F i g . 2. " P h i l i p s Test Sample" test disc contains test tone at prime n u m ber - f r e q u e n c i e s , h e r e : 997 Hz. For analysis of such frequencies n o n - f l a t window types as f o r example K a i s e r - B e s s e l , F l a t - T o p or like h e r e , H a n n i n g , must be employed. Compare to F i g . 1. and note the loss of f r e q u e n c y r e s o l u t i o n . 3.

A N T I - A L I A S FILTER OF THE FFT If the anti-alias and the anti-image f i l t e r s of Digital A u d i o do t h e i r job of bandlimtttng well, it is advisable to bypass the i n p u t a n t i - a l i a s f i l t e r of the FFT analyzer when analyzing with a b a n d w i d t h of 20 kHz or more. T h i s will allow the greatest accuracy in analysis of amplitude and phase r e l a t i o n s h i p s . S h o u l d , however, the d i g i t a l test object leak out s p u r i o u s frequencies above 20 k H z , s w i t c h i n g of the FFT's anti-alias f i l t e r in and out can be used as a simple way of d e t e c t i n g such a leakage. See f i g s . 3-8. F i g . 3. T h e noise floor of an oversampling Compact Disc Player w i t h d e emphasis f i l t e r OFF. The anti-alias f i l t e r of the F F T - a n a l y z e r is I N ,

- 5 T h e noise floor level is NOT -125 dB as could be Implied b y u n c r i t i c a l number r e a d i n g . Remember that t h i s broadband noise is measured here w i t h a 50 Hz b a n d w i d t h ( T y p e 2033, 20 kHz s p a n , 400 lines, Flat w e i g h t i n g ) . The total 0-20 kHz noise rms value fs hence -125 dB + 20 i o g V ^ W = -125 dB + 26 dB - -99 dB F i g . 4. Same as F i g , 3. b u t with the anti-alias f i l t e r of the FFT analyzer OFF. The " 5 . 6 k H z " component is in fact the second harmonic of the oversampling f r e q u e n c y 176.4 k H z , i . e . 352.8 k H z . 5.6 kHz equals 352.8 kHz - 7X(51.2 k H z ) , where 51.2 kHz is the sampling rate of the a n a l y z e r . The amplitude measured is affected by the specifications of the analog f r o n t end c i r c u i t r y of the FFT above 20 k H z . Please note also t h a t only frequencies l y i n g w i t h i n ±20 kHz (or whatever other b a n d w i d t h of FFT analysis is employed) from FFT's sampling f r e q u e n c y or it f s harmonics will become " v i s i b l e " due to a l i a s i n g . F i g . 5. & 6. 1 kHz tone at 0 and -10 dB relative to f u l l scale r e s p e c t i v e l y , analyzed w i t h o u t FFT ! s anti-alias f i l t e r . All spurious components except the one at "10.25 k H z " " t r a c k " the -10 dB fall in level of the 1 kHz t o n e . T h i s indicates that they are intermodulation p r o d u c t s between 1 kHz and some spurious frequencies from the d i g i t a l signal treatment in the p l a y e r . T h e "10.25 k H z " component, which is also seen on F i g . 4 . , is however signal independent and must be created b y some other mechanism, presumably the t r a c k i n g s e r v o . F i g . 7. & 8. Same as F i g s . 3 and 4 however f o r the non-oversampling CD-pJayer* 14700 Hz is one t h i r d of the sampling f r e q u e n c y and is used for frame merging inside the player (1 frame = 6 left + 6 r i g h t samples). The o r i g i n of the other leaking f r e q u e n c y of 13240 Hz in F i g . 7 is more o b s c u r e . Most evident in- F i g . 8 is the aliasing of the sampling f r e q u e n c y of 44.1 kHz at "7100 Hz" ( = 51200 - 44100 H z ) . 4.

S I G N A L - T O - Q U A N T I Z A T I O N NOISE RATIO? The theoretical formula for Signal to Quantization Noise Ratio f o r an N - b i t linear PCM system is well known (see f. e x . [ 3 ] ) : SNR (dB) = 6.02 • N + 1.76 A practical measurement of the quantization noise level is however not an easy t a s k : If t h e r e is no signal t h e r e will be no quantization noise! [ O b v i o u s l y f i g u r e s 3 and 7 do not represent quantization noise. Such measurements are however commonly used for o b t a i n i n g a "SignaS-To-Noise Ratio" f o r CD-players and "compared" to the theoretical value a b o v e . ] The p r e f e r r e d test s i g n a l : a sine wave, is not applicable since it does not induce random quantization noise b u t a v e r y deterministic harmonic (quantization) . distortion.

- 6 A d d d i t h e r to the sine wave and b y notching down the sine wave at the o u t p u t it is now possible to measure the sum of d i t h e r - p l u s quantization noise rms v a l u e . T h i s is p r o b a b l y the most reasonable way, since d i t h e r should be an i n h e r ent p a r t of all d i g i t a l a u d i o . When it is t h e r e one often does the test w i t h o u t the sine wave, l e t t i n g the d i t h e r - n o i s e generate a o n e - b i t random oscillation of the D A C . T h i s test is however of questionable value also since In a practical DAC or ADC the quantization steps are not e q u i d i s t a n t , so looking at just a selection of few adjacent quantization i n t e r v a l s may y i e l d a r e s u l t , n o n r e p r e sentative of the f u l l range of the D A C / A D C . .5.

THEORETICAL DISTORTION IN AN IDEAL LINEAR . 16. B I T . PCM.SYSTEM The steady state d i s t o r t i o n in d i g i t a l audio results from one of the following f o u r mechanisms: T y p e 1 " H i g h Level Distortion' 1

Which is c l i p p i n g due to overload of either the d i g i t a l or the analog p a t h s .

T y p e 2 "Medium Level D i s t o r t i o n "

Which is d i s t o r t i o n due to nonequidistant q u a n tization steps m the t r a n s f e r c h a r a c t e r i s t i c s of ADC and DACs.

T y p e 3 "Small Level D i s t o r t i o n "

Which is the fact that small level analog signals only use few b i t s and hence come out as " s q u a r e w a v e s " . T h i s k i n d of d i s t o r t i o n is present at all levels of the i n p u t s i g n a l . I t - h o w e v e r becomes more s i g n i f i c a n t relative to smaller signal levels.

T y p e 4 "Slewing and timing d i s t o r t i o n "

Which is due to imperfections in exact timing of the sampling and deviations from the id€*al " i n f i n i t e " t r a n sition speed by the DACs.

Whereas the other t h r e e types can be minimized by employment of a p p r o p r i a t e l y well f u n c t i o n i n g h a r d w a r e , t y p e 3 d i s t o r t i o n represents an i n h e r e n t a r t i f a c t , of the technology itself. It is however to a c e r t a i n extent curable w i t h d i t h e r , as discussed In p a r a g r a p h 2 . 1 .

- 7 Assuming all other t h i n g s to be ideal, we have simulated how much T y p e 3 d i s t o r t i o n is to be expected on a 1 kHz tone sampled w i t h 16 b i t s at 44.1 k H z . The calculation was performed on an HP 15-C pocket calculator ( ! ) u s i n g the Discrete Fourier T r a n s f o r m f o r m u l a : 440 S ( k x 1 kHz) ^

_

^ y t t ^ e x p C - ^ ^ A . ) rs^O

With a O d B y ( t ) represented as 32767 x sin ( 2 l F x 1000 x t) the effects of ideal quantization are simulated simply using " I n t e g e r p a r t " f u n c t i o n of the HP 15-C. D i t h e r can be added u s i n g the "Random #" f u n c t i o n . 441 point Fourier T r a n s f o r m is employed in o r d e r to avoid windowing problems. The results are shown in table 1. Most s t r i k i n g in table 1 is the way in which the level of the fundamental a f t e r quantization becomes less and less f a i t h f u l to the o r i g i n a l level of the recorded sine wave f o r small signals. Since "room ambience" or r e v e r b e r t i o n , as discussed in p a r a g r a p h 2.1 is heard as a "small" signal in-between the h i g h l e v e l , d i r e c t sound s i g n a l , we see that quantization will increase the dynamic span between the two in f a v o r of the h i g h l e v e l , d i r e c t s o u n d . So, once a g a i n , we have an argument for the d i g i t a l r e c o r d i n g sounding less "ambient" than an analog r e c o r d i n g . Also at t h i s point we should note t h a t , - 6 0 , - 7 0 , --80 dB or -90 dB 1 kHz (level of the fundamental) signals found on many test CDs obviously must be encoded u s i n g a "master" sine wave of a successively h i g h e r amplitude than the nominal. T h i s means that while an 0 dB test t r a c k can only be obtained by d i g i t i z i n g an 0 dB sine wave, a -90 dB test t r a c k can be obtained from a range of sine waves a n y w h e r e between -85 and -90 dB b y d i f f e r e n t combinations of phase and DC-offset relationships (see p a r a g r a p h 6) r e s u l t i n g in q u i t e d i f f e r e n t d i s t o r t i o n s p e c t r a . F i g s . 9. and 10, i l l u s t r a t e this ambiguity in available -90 dB " t e s t " s i g n a l s . "Computer generated to an accuracy of 99.999...%, as it states on many test d i s c s , indeed doesn't mean identical! F i g . 9. Spectrum of the " - 9 0 dB test tone" of "Super A u d i o check C D " . Replayed b y an analog f i l t e r CD~player. Fig.

10.Same as F i g . 9.. except the test disc is now "SONY 3", A much b e t t e r d i s t o r t i o n c h a r a c t e r i s t i c s of t h i s disc over the p r e v i o u s one is e v i d e n t . I t is possible that some of the improvement is due to. use of d i t h e r on t h i s r e c o r d . However, no mentioning of such is f o u n d on the.sleeve of the r e c o r d .

We often see test r e p o r t s on C D - p l a y e r s in d i f f e r e n t publications d i s p l a y i n g the FFT spectrum of - 6 0 , - 8 0 , or -90 dB test tones. A t these test tone levels the tone itself and its d i s t o r t i o n components a r e easily accommodated and displayed by an FFT analyzer w i t h o u t the need of a notch f i l t e r at the f r o n t e n d . However, based on the discussion above, we feel it is advisable to go t h r o u g h the t r o u b l e of u s i n g a notch f i l t e r , so that d i s t o r t i o n measurement of tones at 0 dB is p e r f o r m e d . These tones are much more consistent on d i f f e r e n t makes of test d i s c s . A t lower levels r e s u l t s from d i f f e r e n t C D - p l a y e r are only comparable if performed with the same test disc make.

A n o t h e r valid point is that an 0 dB tone exercises alt 65,536 quantization levels available, whereas, a -60 dB tone only goes t h r o u g h 65 of them and a -90 dB tone only 3-4!

T A B L E 1. THEORETICAL_QUAN.TJZATION DISTORTION OF A 1 kHz SINE WAVE Sampled with 16 b i t s at 44.1 kHz All levels in dBs r e f . maximum amplitude sine wave of 32767 x LSB

Signal

2. harm

3. harm

4, harm

5. harm

6. harm

7. harm

THD

0.01

-119.08

-104.11

-130.56

-108.27

-133.45

-109.38

1.5

~60 d B

-60.18

-129.78

-104.95

-128.29

-106.41

-122.03

-113.06

1%

-80

dB

-81.58

-136.17

-100.34

-133.90

-123.40

-136.59

-105.94

19%

-84

dB

-87.15

-134,67

-101.80

-163.13

-118.62

-136.13

-

46%

-90

dB

-99.89

-143.14

-100.72

-136.39

-102.47

-137.18

-105.44

0

dB

Fundamental -

99.51

222%

10-5

- 9 FURTHER RESULTS FROM THE DISTORTION MODEL Table 2 lists some other results obtained from o u r HP 15-C simulation. Line A shows a test r u n w i t h o u t any q u a n t i z a t i o n , i . e . it is Just a check of t h e numerical accuracy of the F o u r i e r T r a n s f o r m employed. St is s a t i s f a c t o r y . In line B, u s i n g a d d i n g d i t h e r . As Many more samples distortion-reducing

the "Random #" f u n c t i o n , we have simulated the effect of a n t i c i p a t e d its effect on j u s t 441 samples is n e g l i g i b l e . ( i . e . a longer "time r e c o r d " ) would be needed to. see the effect of d i t h e r .

In line C , we removed d i t h e r and changed the a n a l o g - t o - d i g i t a l c o n v e r t e r to be symmetrical, i . e . analog i n t e r v a l [ 0 ; 1 [ into quantization value 0, it into 0. Somewhat s u r p r i s i n g l y t h i s increases the

f u n c t i o n of the simulated r a t h e r t h a n c o n v e r t i n g the is now c o n v e r t i n g [ - 1 / 2 ; 1/2 [ THD from 19% to 24%.

For the -90 dB tone in line D on the o t h e r h a n d , a symmetrical ADC improves the performance q u i t e d r a s t i c a l l y from 222% to 112%.

T A B L E 2.

Signal

Fundamental 2. harm

3. harm

4. harm

5. harm

6. harm

7. harm

THD

A

~80dB no quantization

-80.00

-265.40

-260.74

-267.64

-240.76

-258.97

-251.48

15 10~ 8

B

-80dB dither added

-81.82

-111.13

-106.14

-112.23

-113.86

-121.28

-113.76

19%

C

-80dB symmetrical ADC

-82.05

-110.66

-109,52

-111.07

-104.91

-111.78

-117.90

24%

D

-90dB symmetrical ADC

-95.38 .

-101.73

-127.64

-107.13

-110.06

-127.66

-111.75

112%

- 10 7,

PRACTICAL DISTORTION MEASUREMENTS ON TWO CD-PLAYERS Using a tunable notch filter (B&K Type 2120) followed by a 400 line, 20 kHz FFT analyzer (BSK Type 2033) we have investigated 0 dB distortion of two "first generation" CD players: Unit A and Unit D. The Unit A player employs a conventional analog anti-image filter, whereas the Unit D player utilizes oversampling and digital filtering. The anti-alias filter of 2033 was ON in all the following measurements. Fig. 12. This is the -60 dB, 1 kHz .test tone replayed by Unit A and analyzed without a notch filter. Fig, 13. 0 dB, 1 kHz test tone replayed by Unit A, The notch filter provides 68.8 dB attenuation of the 0 dB fundamental. Note that presence of the notch filter raises up the noise floor of the measuring situation by some 10 dB relative to f i g . 12. This filter is hence too noisy for measurement of quantization noise as discussed in paragraph 4, but provides us with sufficient resolution for distortion measurements. Fig. 14. Same test tone as in f i g . 13., but replayed by Unit D. Except for the 3rd harmonic there is noticeably more high order distortion than in f i g , 13. Fig. 15. This is the difference between f i g . 13. and 14., i.e. between an analog filter and a digital filter unit, displayed using the "Instant divided by Memory" - mode of the FFT analyzer. We can see that the player with the analog filter exhibited just second and third order distortion, whereas, the oversampling unit created a multitude of harmonic components with the second and eighth through tenth as the dominating plus a range of non-harmonically related discreet frequencies in the level range -94 to -108 dB. This distortion picture is indicative of "crosstalk" between the digital and the analog signal treatment circuits via the internal DC-power supply and possibly slew rate limiting in the very fast operating DAC. This diagnosis complies well with the fact that a British re-make of the here tested oversampling unit has addressed the insulation of the internal DC-power supply with a clearly improved sound as the result. Although a -90 dB distortion of a sine pure tone is definitely inaudible as such, the measurement results indicate imperfections in the player causing (possibly) audible distortion of more complex signals. Fig. 16. 0 dB, 20 Hz tone replayed by Unit A. Fig. 17. 0 dB, 20 Hz tone replayed by Unit D. unit performs better.

Again non-oversampling

- 11 20 Hz is typically the lowest test frequency found on test discs.

*) The two CD-players were compared both subjectively and objectively using a piece of recorded music containing subsonic frequencies. A measurable ^difference in the rendition existed only below 20 Hz and "should" hence be inaudible. However the player outputtsng more subsonic frequencies sounded "fuller" over a pair of headphones. This presumably indicates that the subsonic frequencies although not transmitted by the headphones created distortion components In the headphones in the lower frequency range hence increasing the perceived Sow frequency - "listening pleasure". Unfortunately no test CD containes a single subsonic test frequency allowing a more accurate estimation of the created distortion. Fig. 18. 0 dB, 16 kHz test tone causes in the analog filter player a -90.9 dB spurious at 12.1 kHz and a -102.5 dB spurious at 3.9 kHz. These are clearly second and third order harmonic distortion (32 and 48 -kHz respectively) created during the digital signal processing and undergone aliasing back into the audio range. They reflect the non-linearity (not equidistant transition steps) In the employed 16 bits DA-converter. "Fig. 19. 0 dB 16 kHz test tone. For the oversampling player, because of a 4 times higher sampling frequency (176.4 kHz) the first 9 distortion components of 16 kHz will not be aliased back into the audio frequency range. This measurement confirms this. 8.

MAGNITUDE AND PHASE OF THE FREQUENCY RESPONSE,

CD-PLAYERS.

By far the best method of measuring group delay distortion is by FFT analysis. Here one can artificially create a reference spectrum of any test signal and compare this reference to the spectrum of the signal actually replayed by the CD player under test. Any broadband signal can be used, but since the reference spectrum of a single impulse has zero phase and flat magnitude, catching a single one of them will directly give us what we want without further postprocessing!* The pha** of th« pomptm ln»t#ntn««guai spectrum will reflect the group delay distortion that the Impulse has undergone in the filter, and its magnitude will show ripples in the filter passband characteristics. But alas! No test disc with Just a single impulse is available. Many discs contain a series of fast repeated impulses, which is good for providing a stable display on a non-storage oscilloscope but is non-optimal for FFT analysis. Closely spaced impulses - even if one artificially manages to "window out" just one single of them, ring into one another and cause erroneous ripples in the calculated spectrum. In order to be able to see the roll-off behaviour of the filters above 20 kHz we have used the 25.6 kHz, 800 lines BSK type 2032 FFT Analyzer for the following measurements. Its ants-alias filter was switched off in order not to introduce extra phase and magnitude modification and off course no notch filter etc. was employed.

*) See Appendix on page 14.

- 12 Fig.. 20. Repeated 0 dB impulses as they come out of U n i t A . F i g . 2 1 . Using the TRANSIENT window of the FFT analyzer to isolate only one impulse, the instantaneous spectrum yields the magn i t u d e and phase of U n i t A ' s analog anti-image f i l t e r . F i g . 22. As f i g . 20. b u t f o r the oversampling U n i t D. F i g . 23. As f i g . 2 1 . b u t f o r U n i t D. Where the U n i t D is s u p e r i o r to U n i t A in terms of phase response, its magnitude response on the other hand is i n f e r i o r in r o l l - o f f steepness. At 23 kHz it p r o v i d e s about 12 dB less attenuation than the analog f i l t e r in Unit A . Also t h i s has been c o r r e c t e d in the B r i t i s h re-make o f the oversampling u n i t . T h e softly o f f - r o l l i n g Bessel f i l t e r at the o u t p u t stage has been replaced b y a s h a r p e r B u t t e r w o r t h f i l t e r w i t h improved sound q u a l i t y as a r e s u l t . Please note in the f i g u r e s above that the r i p p l e s on t h e shown f r e q u e n c y responses are due t o : .a) Limited time window w i d t h ( T = 2 . 9 msec gives a M % ^ 350 Hz) b) Previous impulses on t h e test r e c o r d keep on r i n g i n g and hence i n t e r f e r e w i t h the impulse which we pick out f o r spectrum c a l c u l a t i o n . T h e r e f o r e t h e y should not be i n t e r p r e t e d as r i p p l e s in the passband c h a r a c t e r i s t i c s of the tested anti-image f i l t e r s ( t h e y are f l a t w i t h i n ^ 0 . 5 d B ) . 9.

MAGNITUDE AND PHASE OF THE FREQUENCY RESPONSE. RECORD/REPLAYDEVICES. DUAL CHANNEL FFT MEASUREMENTS. Whenever we have access to both the analog i n p u t and the analog o u t p u t of a d i g i t a l tmamtti / i^^r&tluem dgviets, dual eh«stiri@l i^FT m©thoc|# can be employed. Using the dual channel B r u e l and Kjaer T y p e 2032 analyzer we have p e r formed such measurements in o r d e r to evaluate the f u n c t i o n of a U n i t F audio processor alone and in conjunction w i t h an analog "phase c o r r e c t i o n " filter. As p r e v i o u s l y d i s c u s s e d , the optimal signal f o r FFT-analysis is one that only contains frequencies coinciding with the lines of a n a l y s i s . If all those frequencies are present simultaneously and w i t h equal s t r e n g t h we have one special t y p e of test s i g n a l , often employed in dual channel a n a l y s i s : the so called PseudO' Random Noise. (For a more t h o r o u g h discussion of d i f f e r e n t excitation signals pros et cons see f o r example [ 4 J ) . For the 25.6 k H z , 800 lines analyzer employed, the pseudo random noise signal from the b u i l t - i n signal g e n e r a t o r , hence contains frequencies of 32, 64, 96 H z . . . e t c . all they way u p to 25.6 k H z . It is t h u s r e p e t i t i v e w i t h a p e r i o d length of 1/32 Hz = 31.25 ms. A n y analog test object to which we would a p p l y such a signal would a f t e r a while stablize and p r o d u c e an o u t p u t signal w i t h exactly the same r e p e t i t i o n rate of 31.25 ms. T h i s is however, not the case f o r a d i g i t a l test object, like f o r example U n i t F. Since 31.25 ms. does not equal an integer amount of 44.1 kHz sampling periods b u t exactly 1378 + 1/8 of them, the U n i t F will r e p e t i t i v e l y keep p r o d u c i n g 8 d i f f e r e n t responses to the same i n p u t signal!

- 13 T h i s effect of "time v a r i a n c e " is similar to the effect of wow and f l u t t e r in analog r e c o r d i n g and i t ' s effect upon dual channel FFT analysis of such a device is t h o r o u g h l y discussed in [ 5 ] . In s h o r t : a) T h e coherence will d r o p and the magnitude of the f r e q u e n c y will be underestimated w i t h h i g h e r frequencies

response

b) T h e averaged f r e q u e n c y response will not necessarily converge into a smooth c u r v e , b u t will keep looking " n o i s y " . F i g . 24, T h e measurement setup and the phase response f o r the d i g i t a l audio processor. F i g . 25. T h e c o r r e s p o n d i n g magnitude of the f r e q u e n c y response. T h e measurement object is NOT - 1 . 7 dB down at 20.000 Hz as indicated in the u p p e r r i g h t c o r n e r ! T h i s number is an underestimation due to "wow and f l u t t e r " - like effects caused b y non-optimal matching of the FFT analyzer's b a n d w i d t h a n d the sampling rate of the audio processor. F i g . 26. T h e coherence of the above measurement confirms problems at higher frequencies. F i g . 27. Phase response of the analog G r o u p Delay C o r r e c t o r . Note how smooth the c u r v e is compared to measurements above on a d i g i t a l device. F i g . 28. T h e combined response of the G r o u p Delay C o r r e c t o r and the d i g i t a l audio processor. T h e phase response is g r e a t l y improved t h r o u g h o u t most of the audible r a n g e . Pseudo Random n o i s e - t y p e signals p r o v i d e the best u t i l i z a t i o n of an FFT a n a l y z e r , due to the fact that a r e c t a n g u l a r time w e i g h t i n g can be used and hence optimum f r e q u e n c y resolution o b t a i n e d . In o r d e r to avoid discrepancies between the FFT analyzer and the test o b j e c t , such as we have encountered above, a v e r y simple requirement can be imposed on the sampling rate of the FFT a n a l y z e r : One FFT time r e c o r d must contain an integer amount of d i g i t a l audio sampling p e r i o d s . Based on t h i s simple requirement we can hence " d e s i g n " an optimal FFT analyzer aiming p a r t i c u l a r l y at analyzing both " 4 4 . 1 k H z " and "48 k H z " d i g i t a l audio: T h e necessary numbers a r e : 44100 = (2 x 3 x 5 x 7 ) 2 48000 = 2 7 x 3 x 5 3 and if we want an 800 lines f r e q u e n c y resolution we need 2048 samples ( = 2^ 1 ) in each FFT time r e c o r d . It is easily shown that only f o u r FFT sampling rates satisfy the compatibility c r i t e r i o n above. They a r e : f - j : 2 1 3 x 3 = 24,576 Hz f2:

2

13

x 5 = 40,960 Hz

(too low for practical application) (too low)

- 14 f 3 f

:

%:

2 1 3 x 3 x 5 = 122,800 Hz 2

13

x 5 x 5 =204,800

(better) (best)

I t is o u r suggestion that an optimal FFT based analysis of " 1 1 . 1 kHz 8 ' a n d "48 k H z " d i g i t a l audio should be based on an FFT sampling rate of 204800 Hz or 122800 H z . To the best of o u r knowledge, at present date none of the commercially available FFT analyzers u t i l i z e any of these sampling f r e q u e n c i e s . However, b y u s i n g an e x t e r n a l sampling f r e q u e n c y source many of them can be c o n t r o l l e d to do i t . 10. CONCLUSION We have demonstrated and discussed methods of a p p l y i n g discreet Fast F o u r i e r T r a n s f o r m to analyze d i g i t a l audio p r o d u c t s . Based on the many applications of todays FFT analyzers to d i g i t a l a u d i o , optimal f u t u r e i n s t r u m e n t a t i o n f o r t h i s purpose can be designed and an imp r o v e d understanding'of the audible effects of d i g i t i z i n g audio signals g a i n e d .

APPENDIX

T h e objective comparison of t h e two CD u n i t s u s i n g music was done in the following manner: We connected a p a i r of 600 ohms headphones in parallel w i t h a t y p e 2033 FFT analyzer (1 Megaohm i n p u t resistance) to t h e line o u t p u t s of the u n i t u n d e r t e s t . T h e analyzer was hence m o n i t o r i n g t h e electrical signal g o i n g to t h e headphones, u s i n g which we conducted a subjective comparison. T h e piece of music we used was " S a t i n D o l l " , t r a c k 16 of " S u p e r Audio Check C D " . F i g . 29. A v e r a g e of 20 consecutive music spectra as replayed by Unit A . 0 to 100 Hz. F i g . 30. A v e r a g e of 20 consecutive music spectra as replayed by U n i t D . ' 0 to 100 Hz. F i g . 3 1 . T h e d i f f e r e n c e between U n i t A and U n i t D ( f i g . 29 - f i g . 30) is most obvious in t h e subsonic r a n g e . T h e combination of 600 ohms load and 22 uF o u t p u t capacitance of u n i t D y i e l d a - 3 d B point at 12 Hz. F i g . '32. Connection'of a 600 ohms load ( i . e . the headphones) d i r e c t l y to the line o u t p u t s was convenient in o r d e r to obtain the same l i s t e n i n g level in t h e headphones, as well as when u s i n g t y p e 2033 in t r i g g e r e d mode to ascertain that the t r i g g e r i n g takes place at same i n s t a n t s in time in b o t h cases. 600 ohms i s , however, on t h e low side of allowable load. T h e removal of the headphones, as seen h e r e , y i e l d s the two u n i t s v i r t u a l l y i d e n t i c a l , -also in the subsonic range.

- 15 -

1)

"Resolution Below the Least S i g n i f i c a n t B i t in Digital Systems w i t h Dither 8 '. John Vanderkooy and Stanley P. L i p s h i t z JAES 32:3, p. 10$ (1984)

2)

"Zoom P P T " , N . T h r a n e , p. 9 B r u e l a n d Kjaer Technical Review no.. 2 - 1980

3)

" D i g i t i z a t i o n of A u d i o : A Comprehensive Examination of T h e o r y , a n d c u r r e n t P r a c t i c e " , B a r r y A . Blesser JAES. 26:10, p . 743, (1978)

4)

"Dual Channel FFT A n a l y s i s Parts I and I I " , H. H e r l u f s e n B r u e l and Kjaer Technical Review n o s , 1 and 2 - 1984

5)

"Dual Channel FPT A n p l y s l i for the Oevetopmnrtt and Evaluation of T i p t i R e c o r d e r s " , A n d r e Perman B r u e l and Kjaer A p p l i c a t i o n Note BO 0098-12

Implementation,

F i q . 2H

Fig. 25

Fig.

26

Fig.

27

Fig.

28