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= Y, UnT2n4> = Y, «»*»» • n = v{T)(t). m Equation (7.50b) gives an interpretation of the down-sampling operator: S^v(T)(p is obtained by deleting all the odd samples in v(T) . {2rnuj)\2. To see how the different scales interact in the frequency domain, we now add the third (and most fundamental) property of t h e scaling function, namely the dilation equation D
( 7 ' 44 )
n
S^ "doubles" the number of samples by separating them to A t = 2 and inserting a zero between each neighboring pair (which restores At = 1). Thus Stu(T)
or
H* = h(T)S^
on V0.
(7.45)
Equation (7.45) has a simple and elegant interpretation: H* acts on u(T)(/> G VQ by first up-sampling with *S\ , then averaging with h(T). This averaging process fills in the missing "pickets," and the result, according to (7.42), is precisely the stretched version Du(T)(/) of the original signal! For this reason, H* is called an interpolation operator. It zooms in on the sampled signal, just as H zooms out.
7. Multiresolution Analysis
149
The advantage of the form (7.45) of H* is that it immediately gives the adjoint operator H : VQ —> Vo as H = S;h(T)* = SMT)*
onV 0 ,
(7.46)
where S^ = S* is called the down-sampling operator. To find H, we must find S$ explicitly. Note that every v(T) G V can be written uniquely as a sum of even and odd polynomials: v(T) = v+(T2) + Tv_(T2), where
v+(T2) = l[v(T)+v(-T)}
^v2nT2"
= n
2
v_(T ) = IT-^T)
- v(-T)]
(7.47)
^
= n
^v2n+1T2«.
(7-48)
Equation (7.3) can be used to give v+(T) and v_(T) directly in terms of v(T) and v(—T): v+(T) = Y,V2nTn
= \D-l\v{T)
+
v(-T)]D
"
(7.49)
n
P r o p o s i t i o n 7.3. The up-sampling and down-sampling operators have the fol lowing properties: (a)
S^St = IyQ
(b)
S^v(T)(f) = V+(T)
(c) (d)
S ^ ( r ) S * = r; + (T), 5 ^ +T5^r-
1
(7.50)
S.T^v^S^ = v_(T) = /Vo.
Proof: Since S^ = S* , we have < u(T)0, S»S«v(T)
= {u(T2)4,, v(T2)4>)
= (u(T)4>,v{T)4>), where the last equality follows from the orthonormality condition, since (4>2n, 4>2k ) = &2k ~ ^k ~ (0nj 4>k) • That proves (a). To prove (b), note that for any u(T) G V, we have (u(T)0, S\v(T)
' '
150
A Friendly Guide to Wavelets
The orthogonality of the
= (u(T2)
(7.53)
Since this holds for all u(T)(f) G Vb, it proves (b). To show (c), note that S^u{T)v{T)4) = u(T2)v(T2)(p = w(T 2 )^v(T)(/); hence we have an operator com mutation relation similar to (7.3): S^T)
= w(r2)5t,
u(T) G V.
(7.54)
Thus
= S,[v+(T2) + Tv.(T2)]St = S,S,v+(T) + S,TS,v_(T). 2 But S^TS^T)^ = S^Tu{T )(f> = 0 by (b), since Tu(T2) is odd; hence S^TS* = 0. Furthermore, S^S^ = Iy0 by (a), thus (7.55) reduces to S^v(T)St = v+(T). The second equality in (c) follows from the first since the even part of T~1v(T) is v_(T2). (d) follows from S,v(T)St
StS*v(T)
= Stv+(T) + TS^v_{T)
= v+{T )
Hu(T)$ = SMTYu(T)
[
'
j
Hence Hu(T) = 0 if and only if h(T)*u(T) is odd. This can be arranged by choosing u(T) = h(-T)*v(T) with v(T) odd, (7.57) since h(T)*h(—T)* is necessarily even. In fact, this gives a characterization of W\ that complements that of V\ given by (7.43), as shown next. Proposition 7.4. Fix an arbitrary odd integer A and let g(T) = -Txh(-T)\
(7.58)
Then every u(T) G V can be written uniquely as u(T) = h(T)v{T2)
+ g{T)w{T2)
for some v{T),w{T)
G V,
(7.59)
and Wx = closure of {g{T)w{T2)(j): w(T) G V}.
(7.60)
7. Multiresolution Analysis
151
Proof: Equation (7.59) will be proved below. Assuming it is true, note that h{T)v{T2)(t) e Vi and g{T)w{T2)(f) e Wx (by (7.57), since -Txw(T2) is odd). Hence h{T)v(T2)(t> = Piw(T)0, g{T)w{T2)(t) = QlU(T)
SMT)*u(T)St
= SMT)*h(T)Stv(T) = v(T).
SMTyg(T)w(T2)St
+ SMT)*9(T)SMT)
(7.62)
Similarly, SMT)'u(T)St = w{T). (7.63) Thus v(T) and w(T) are uniquely determined, i/ (7.59) holds. It remains to show that given any u(T) £ V, the substitution of (7.62) and (7.63) into the right-hand side of (7.59) indeed gives back u(T). By (7.49) and (7.50)(c), v(T2) = Dv(T)D-x
=
DS^h{T)*u{T)StD-1
= \ [h(T)*u(T) + 2
W(T
l
) = Dw(T)D~ =
h(-T)*u(-T)}
=
DS^giTyu^S^D-1
l[g(Tyu(T)+9(-Tyu(-T)}.
Hence h(T)v(T2)
+ g(T)w(T2)
= 1 [h{T)h{T)* + g{T)g{T)'\ + I [h(T)h(-Ty
u(T)
+ g{T)g(-T)*]
u(-T).
But by (7.66) of the Lemma, h(T)h(Ty h(T)h(-T)*
+ g(T)g(Ty + g(T)g(-Ty
Thus (7.59) holds as claimed.
= h(T)h(T)* + h(-T)*h(-T)
= 2IVo
= h(T)h(-T)*
= 0.
- h{-T)*h{T)
^ '
■
L e m m a 7.5. h(T) and g(T) satisfy the following "orthogonality" relations: SMT)'h(T)S,
= S,g(TT9(T)St
= IVo
S>g(X)'h(T)St
= SMT)*g{T)St
= 0.
Equivalently, h(T)'h(T)
+ h(-Tyh(-T)
= g(T)*g(T) + g(-T)'g(-T)
g(T)'h(T)
+ g(-T)'h{-T)
= h(T)*g(T) +
= 2IVo
hi-T)'g(-T)=0.
}
152
A Friendly Guide to Wavelets
Proof: Recall from (7.39) that HH* = P0 on L 2 (R). For the restrictions to Vb, this means that HH* = IVo. Thus by (7.45) and (7.46), HH* = SMT)*HT)St Furthermore, since g(T)*g(T) = h(-T)h(-T)* SMTrg(T)S, = IVo. Finally,
= IVo.
(7.67)
= h(-T)*h(-T),
= -Sih(T)*h(-T)*TxSi
SMTYgfflS*
since h(T)*h(-T)* Tx is odd, and similarly S^Tyh^S^ (7.65). Equation (7.66) follows from (7.65) and (7.49).
we also have
=0
(7.68)
= 0. That proves ■
The intuitive meaning of g(T) is that it is a differencing operator on Vb, just as h{T) is an averaging operator. Equations (7.64) can be interpreted as stating that these two operators complement each other. The significance of the inte ger A will be discussed later, once the wavelets have been defined. Clearly, a change in A merely amounts to a change in the polynomial w(T) representing We are at last able to define the wavelets associated with the multiresolution analysis. The mother wavelet belongs to Wo, just as the scaling function belongs to Vb- Since Wo = D~1W±1 a dense set of vectors in Wo is given by I T ^ ( I X r 2 ) ^ = D-'wiT^giT)^
= w^D-'giT)^
(7.69)
with w(T) G V. We therefore define the mother wavelet as ^ = D-lg{T)ct>^W0,
i.e.,
i>{t) = V ^ ^ g ^ t
- n),
(7.70)
n
so that W0 = closure of {w(T)ip : w{T) G V}.
(7.71)
The shifted wavelets ^ n = Tn*l> = T n D - ^ ( T ) 0 = D-lT2ng{T)(j)
(7.72)
therefore span Wo. It follows that for any fixed m G Z , the vectors V>m,n = D™TniP,
i.e.,
m,n (t) = 2-m^(2-mt
- n),
(7.73)
span W m . Proposition 7.6. The wavlets {iprn,n '■ rn,n G Z} form an orthonormal basis forL2(K).
7. Multiresolution Analysis
153
Proof: We show that the wavelets ipn at scale m — 0 are orthonormal, and hence they form an orthonormal basis for their span Wo- It then follows from the unitarity of D that for any fixed ra, {ipm,n : n E Z} is an orthonormal basis for W m . Since £ 2 (R) is the orthogonal sum of the W m 's, the proposition follows. =
{^i>k)
=
(T2n-2k
But 4> = St(f) and T2n~2k(t) = StTn~k(f), so by Lemma 7.5 and the orthonormality of the 0 n 's, {fn, i,k) = ( 5 t T » - V , 9(T)*g(T)St
(7.75)
We can now express the orthogonal projections Qm to W m in terms of the wavelet basis, just as the projections Pm to Vm were given in (7.12):
Qm=£>m,»C»-
(7-76)
n
Thus (7.28) and (7.30) become / - E f e n & , n / + neZ
M E Z>*,n^,„/ k=-oo n<EZ
(7.77)
and
/=EEWi,n/. fcez nez
(7-78)
respectively. The operator H* = DP0j originally defined on L 2 (R), was restricted to H* : VQ —> Vo. Together with its adjoint, this operator was used to define the wavelets. It is useful to restrict G* = DQQ similarly, since the four restrictions H,H*M,G* collectively give rise to "subband filtering," a scheme for wavelet decompositions that is completely recursive and therefore useful in computa tions. According to (7.41), the image of Wo under G* is W± C Vo . Hence G* restricts to an operator from Wo to Vo, which we denote by the same symbol to avoid introducing more notation. This restriction G* : Wo —► Vo acts by = Dw(T)*P = w(T2)DiP
G*w{T)tl> = DQ0w(T)^
= w(T2)g(T)d> =
g(T)SMT)4>-
(7.79)
It can be shown that G : VQ —»• WQ is given by Gu{T)cf> = SMT)*u{T)tl>,
(7.80)
where S^ and St act on Wo exactly as they do on Vo: S>(T)V> = w(T2)i/>t
=»
S^w{T)i) = w+{T)tl>.
(7.81)
A Friendly Guide to Wavelets
154 Together with H*u(T)
Hu{T)4> = S^h{Tfu{T)(t>,
(7.82)
(7.79) a n d (7.80) give a complete description of t h e Q M F s H, G, a n d their adjoints. I n actual computations, these operators act on t h e sequences {un} corresponding t o u(T)4> G VQ or u(T)tp G Wo- We have given t h e above "ab stract" description because it seems t o b e easier t o visualize. (Compare t h e actions of H* given by (7.82) a n d (7.88) below, for example.) We now find t h e corresponding actions on sequences. Note, first of all, t h a t g(T) = -Txh{-T)*
= -]T(-l)n/inTA-n n
_
(7.83)
n
Hence gn = (-l)nhx-n-
(7.84)
{gn} is the set of filter coefficients for rp, or the high-pass filter sequence. Daubechies wavelets will have 27V nonzero coefficients ho, hi,..., /i2Ar-i, with N = 1, 2, 3 , . . . . (N — 1 gives t h e Haar wavelet.) Then t h e nonvanishing coefficients of g(T) are in the same range (i.e., gn ^ 0 only for n = 0 , 1 , . . . , 2N — 1) if and only if we choose A = 27V — 1. This, in turn, results in ip having t h e same support as (p. A change of A t o A' = A + 2\x gives g'(T) = T2^g(T), resulting in a new wavelet i/>' = D-lT2»g(T)
= T*V,
(7.85)
which is merely a translated version of ip. This, then, is t h e significance of A. T h e action of H* on sequences is obtained by first considering its action on the basis elements (j)k\
H*(/>k = h(T)cf>2k = J2 hnTn+2k(l> = ] £ K-2k
(7.86)
n
Hence H*^2uk(pk k
= ^2^2UkK-2k
(7.87)
which gives H*{uk}kez
= I J2 hn-2k uk \ • I k ) n<EZ
(7.88)
Similarly, G*1pk = ^ # n - 2 f c 0 n , H(j)n = ^
hn-2k
(7.89)
7. Multiresolution Analysis
155
These result in = < 22gn-2kUk I k
3*{uk}kez
| ) neZ (7.90)
I n
) kez
G{un}
= S Yl9n-2kUn ? In J fcGZ In the case of compactly supported <j> and ^ , all the above sums are finite; hence these equations are easy to implement numerically. H and G are then said to be finite impulse response (FIR) filters. The above gives a recursive scheme for the wavelet decomposition and reconstruction of signals, as follows: Note that for the restrictions H : Vo —► Vo and G : VQ —> WQ and their adjoints, the relations (7.39) imply HH* = IVo,
GG* = IWo,
HG*=GH*=0
H*H + G*G = IVo.
(7.91)
This can be applied to a sequence u° = {u^} € £2(Z) « Vo to give the decom position u° = if V + G*d\ where w1 = Hu°, d1 = Gw°. (7.92) 1 2 1 Since u G ^ (Z) « Vb, we may similarly decompose w . Repeating this, we obtain the recursive formulas for synthesis and analysis, um =
H*um+1 + G*dm+\
where u m + l =_ Hum,
dm+1 = Gum,
(7.93)
where every step involves just a finite number of computations. In the electrical engineering literature, this scheme is known as subband filtering (Smith and Barnwell [1986], Vetterli [1986]). (It is also called subband coding; the cofficients {um1dm} are interpreted as a "coded" version of the signal, from which the original can be reconstructed by "decoding.") In view of the expressions (7.79), (7.80), and (7.82) for the filters, (7.93) can be written in a diagrammatic form as in Figure 7.1.
h(T)'
T w
I
s»
m+l.
>u
[S7Udm+1-
i
eT
Figure 7.1. Subband filtering scheme.
u"
A Friendly Guide to Wavelets
156
Repeated decomposition and reconstruction are represented by the pair of "dual" diagrams in Figures 7.2 and 7.3.
H
o
u
H
I
> u
► u
G
G
G
H
2
i
► u
d2
dl
Figure 7.2. Wavelet decomposition with QMFs.
u° *-
H*
H* U
G*
G*
U2
<-
-e-
H* W
G*
G*
d3 Figure 7.3. Wavelet composition with QMFs. Let us now review the various requirement t h a t have been imposed on the scaling function so far, in order to translate them into constraints on h(T) and g(T) or, equivalently, on the filter coefficients hn and gn. The requirements will be reviewed in an order t h a t makes it possible to build on earlier ones to extract the constraints on h(T) and g(T). (a) The averaging property 0(0) = J dt (j)(t) = 1 imposes no constraints on the filter coefficients. (b) The dilation obtain
equation D(j> = h(T)4>: Integrating over t and using (a), we
(f>(t) = V2 Y^ hn(/)(2t -n) (c) The orthogonality
=> ^2hn
= V2.
(7.94)
condition:
$ = (faA) = (D4>k,D) = (h{T)
(7.95)
= 2_^hnhn+2k ■ Note t h a t both constraints (7.94) and (7.95) come from the interaction between t h e dilations and translations. These are all the requirements imposed so far;
7. Multiresolution Analysis
157
more will be added later to obtain scaling functions and wavelets with some reg ularity properties. Since we have not set a limit on the number of nonvanishing coefficients hn , any finite number of conditions can be imposed as long as they are mutually consistent. The relation between the linear condition (7.94) and the quadratic one (7.95) must be examined to ensure they do not conflict. An easy way to do this is to note that by (7.66), 2IVo = h{T)*h{T) +
h(-T)*h(-T)
= £ [ l + (-!)'] hnhn+lT\
(™6)
[l + ( - l ) i ] ^ f t n f e n + ; = 26(°.
(7.97)
n,l
and hence n
For odd I, (7.97) is an identity, whereas for I = 2/c, (7.97) reduces to (7.95). The advantage of the form (7.96) of the orthogonality condition over (7.95) is that its relation to (7.94) is completely transparent. According to (7.18), taking the Fourier transform results in the substitution T —* e~2ntu} = z. Hence h(T) becomes h(z), an ordinary Laurent polynomial in the complex variable z with |*| = 1. Similarly, h(T)* = £ n KT~n becomes £ n hnz~n = ^ n hnzn = h(z). Thus in the frequency domain (7.96) becomes
|fc(*)|2 + |/>(-*)|2 = 2.
(7.98)
Similarly, (7.94) translates to h(l) = y/2.
(7.99)
Now the relation between the two constraints is clear: Together they imply that / i ( - l ) = 0,
i.e.,
^ ( - l ) n / i n = 0.
(7.100)
n
Hence the sum of the even coefficients /i2fc must equal the sum of the odd ones /i2fc+i- Since the total sum equals \/2, it follows that
Recall that g(T) = -Txh(-T)*.
Hence \g(z)\ = \h(-z)\
| ^ ) | 2 + |<7(-z)|2 = 2,
(/(1) = 0,
g(-l)
for \z\ = 1 and = >/2.
(7.102)
Thus Z ) 92k = - Yl 92k+i = ^= •
(7.103)
A Friendly Guide to Wavelets
158
Equation (7.103) actually implies that the mother wavelet ip satisfies the admissibility condition: CO
/
-l
dt^(t) = -7=y2gn = 0.
(7.104)
The filter coefficients hn and gn thus represent all the properties oi
(a)
/(f) = ^fc,n(()&,J+ E E^*.»WK»/neZ
k=-oo
neZ
(7.105)
(b) /(t) = E E w w i , / . fcez nez Since J dt
/
A«00
dt <j>M,n(t) = 2 M / 2 > -oo
/ J —oo
dt lPm,n (t) = 0,
(7.106)
a formal term-by-term integration of (7.105) gives oo
/
n<EZ
(7.107) oo
/
dt f(t) = 0.
-oo
(7.107b) cannot hold in general, since the left-hand side need not vanish for / e L 2 (R). This "paradox" is a result of the term-by-term integration performed on (7.105b). The point is that the expansions (7.105b) converge in the sense of
7. Multiresolution Analysis
159
L 2 (R) but not necessarily in the sense of L1. That means that the partial sums N
M
N
(a) Mt)= E
y
N
(b) M * ) = E E
k=-N n=-N
J
^(m.nf
approach / in the L2 norm || • H2 (which we have been denoting simply by || • || for convenience), CO
/
dt\f(t)-fN(t)\2^0
as N -»«>,
(7.109)
as TV ->oo,
(7.110)
-OO
but not necessarily in the L1 norm || • ||i : OO
/
dt\f(t)-fN(t)\-f>0 -00
in general. For example, let
'"«={r" 4 *♦!"** T h e n as A7" —► 00, ||SN||| = 2 A T - 1 / 2 ^ 0
I, U
(7 ni)
-
otherwise.
and HsrjvMi = 2N1'* - oo.
(7.112)
The function /(£) — /jv(t) in (7.108b) behaves much like gw- It becomes long and flat for large TV, since more and more of the detail has been removed. As N —*■ oo, its L2 norm vanishes, but its L1 norm does not vanish unless Jdtf(t) = 0, causing the "paradox" in (7.107b). As seen from (7.108a), no such paradox arises from (7.105) (a) because it contains the "remainder" term PMIHence the expansion (a) is preferable to (b). (In fact, (a) holds in a great variety of function and distribution spaces, including L p , Sobolev, etc.; see Meyer (1993a).) Example 7.7: The Haar Wavelet. As a simple illustration, we apply the above construction to the Haar MRA of Example 7.2, where cf> = X[o,i)- Equa tions (7.36) and (7.58) give g{T)
= -Tx h{-T)* = 4= (T A_1 - ^ A ).
(7.113)
V2
Hence
«, = D-ig(T)* = -J= D-> (Xlx.lt A) - XlK A+1))
^ n4)
A Friendly Guide to Wavelets
160
Note that the support of 4> is [0,1], while that of ip is [^^, ^^]- The two supports therefore coincide only when A = 1 is chosen in g(T). This illustrates the point made earlier. For A = 1, (7.114) is the Haar wavelet. 7.3 The View in the Frequency Domain Although it is sometimes claimed that wavelet analysis replaces Fourier anal ysis (which may or may not be the case in practice), there can be no doubt that Fourier analysis plays an important role in the construction of wavelets. Furthermore, it gives wavelet decompositions like (7.28) and (7.30) a vital inter pretation by explaining how the frequency spectrum of a signal / G Vm is divided up between the spaces Vm+i and Wm+i, which enhances our understanding of what is meant by "average" and "detail." Thus, although the natural domains for wavelet analysis are time and the time-scale plane, it is also useful to study it in the frequency domain. In Section 7.2, we have referred to the inner products ( 0 n , / ) as "samples" in VQ (i.e., with At = 1) of / G L 2 (R). However, these samples are not exactly the values / ( n ) , since <j> is not the delta function. In fact, oo
/
dw e3**nufa)f(u>)
= F(n),
(7.115)
-OO
where the function F(t) is defined by its Fourier transform: F(u)=J(uj)f(uj).
(7.116)
Note that fdv\F(uj)\ < \\4>\\\\f\\ = |MIH/|| by the Schwarz inequality and Plancherel's theorem; hence F(LJ) is integrable. F(t) is therefore continuous, and its values F(n) are well defined. Thus 0* / are actually the samples of the function F(t) defined by (7.116) rather than samples of f(t). In fact, a general function / G L2(H) cannot be sampled as is, since its value at any given point may be undefined (Section 1.3). In Section 5.1, we sampled band-limited func tions, which are necessarily smooth and therefore have well-defined values. A general / G £ 2 ( R ) must be smoothed (e.g., by being passed through a low-pass filter) before it can be sampled. Equation (7.115) shows that 0 performs just such a smoothing operation, the result being the function F(t). The operator / — i >• F is a filter in the standard engineering sense, since it is performed through a multiplication (by ) in the frequency domain. Note that fl
OO
/
du
e27rinUJ
F{ij)
-oo
= f JO
k
due^^J^Fiu
duJ
= Y, JO
+ k)^ u
e 2T
p^j
+
I dw e2lTi™ A{u). Jo
fc)
(7.117)
7. Multiresolution Analysis
161
The samples F(n) are therefore the Fourier coefficients of the 1-periodic function A(u) — J2k F(oj + k). The copies F(UJ + k) with k ^ 0 are aliases of F(CJ), and if the support of F(LJ) is wider than 1, these aliases may intefere, making it impossible to recover F(LJ) from the samples. In other words, knowing F(t) only at t = n G Z gives information only about the periodized version of F obtained by aliasing. So much for the sampling of / G £ 2 (R) in VJ). Now let us look at the partial reconstruction of / by Pof = Y^n un(pn — u(T)(f), where un = 0 £ / are the above samples. Recall that the time translation operator T acts in the frequency domain by (7.18): {Tg)A{ij) = zg(u),
where z = z{u) = e~27Tiu},
(7.118)
for any g G L 2 (R). It is convenient to use the notation g(w) = e^g, even though eu(t) = e27Xtujt does not belong to L2(TV) (Section 1.4), for then we can write (7.118) as e*uTg = ze*ug, or e*uT = ze*u(7.119) 27r The last equation is just the adjoint of T^e^t) = e M*+i) = zew{t). Equa tion (7.119) states that e^ is a "left eigenvector" of T with eigenvalue z{u). (This can be made rigorous in terms of distribution theory.) Thus ( P o / ) » = e^Pof = e>(T) = u ( z ) e > = u(z)0(w).
(7.120)
But un = F(n), hence by (7.117), u(z) = ^ n
e~
27rinu;
u n = A(u;) = ^
H" + *0>
(7-121)
k
the "aliased" version of F(u>). We therefore recognize our u(T) notation of Section 7.2 as a symbolic form of Fourier series, to which it reverts under the Fourier transform. In fact, (7.120) characterizes VQ. That is, VQ is the closure (in L 2 (R)) of the set of all functions whose Fourier transform can be written as the product of an arbitrary trigonometric polynomial u{z) with the fixed function (LJ). (Taking the closure enlarges the set by allowing u(z) to be any 1-periodic function square-integrable on 0 < u < 1, as in (7.19).) Let us digress a little in order to get an overview that ties together (7.115) through (7.121). The sampling process / »-> {un}nez defines an operator So : L 2 (R) —*• ^ 2 (Z), i.e., Sof = {un}n€zWe call SQ the Vo-sampling operator. (The V^-sampling operator Sm can be defined similarly, and everything below can be extended to arbitrary ra G Z. For simplicity we consider only m — 0.) If {un} G ^ 2 (Z), then the corresponding Fourier series u(z) = ^2nUnZn *s a square-integrable function on the unit circle T = {z G C : \z\ = 1}. Let us denote the space of all such functions by L 2 (T). The map {un}nez »-»• u(z)
A Friendly Guide to Wavelets
162 then defines an operator Fourier transform.*
A
: £2(Z) -► L 2 ( T ) , which is a discrete analogue of the
We write {un}A(z) = u(z). We define the operator S0 : L2(R) -+ L 2 ( T ) as t h e "Fourier dual" of So, i-e., as the operator corresponding t o So in the frequency domain: (S0f)(z) = (S0f)A(z) = J2 znrnf-
(7.122)
n
T h a t is, So is denned by the "commutative diagram" in Figure 7.4.
/(«)ei 2 (R)
-^-*
/ H e i 2 ( R ) —^-»
{
u(z)el?{T)
F i g u r e 7.4. The Vo-sampling operator So and its Fourier dual, the aliasing operator So. Equations (7.121) and (7.116) give an explicit form for SQ: (S0f)(z) = $ ^ ( w + k)f(u + k).
(7.123)
k
(The right-hand side is 1-periodic; hence it depends only on z = e~27Ttu}.) Thus So can be called the aliasing operator associated with Vo- Figure 7.4 gives the frequency-domain precise relation between sampling and aliasing: Aliasing is the equivalent of sampling in the time domain. From the definition of S0 it easily follows t h a t S£ : £2(Z) -+ L 2 ( R ) is given by S'o{w n } = u(T)4>. SQ is a n interpolation operator, since it intepolates the samples {un} t o give the function ^nun<j)(t — n). The relation between the adjoint operators SQ and SQ of So and SQ is summarized by the diagram in Figure 7.5, where (7.120) has been used. t The proper context for this is the theory of Fourier analysis on locally compact Abelian groups (Katznelson [1976]), where T (which is a group under multiplication) is seen to be the dual group of Z, just as the "frequency domain" R n is the dual group of the "time domain" R n ; cf. Section 1.4. (The case n = 1 is deceptive since R is self-dual.)
7. Multiresolution
Analysis
163
{un} e £2(Z)
u{T)4> G L 2 ( R )
U{Z)(J>{UJ) G
u(z) G L 2 ( T )
L2(R)
Figure 7.5. The Vb-interpolation operator SQ and its Fourier dual, the "de-aliasing" operator SQ. T h e operator SQ a t t e m p t s to undo the effect of aliasing by producing the nonperiodic function u(z)<j>(u) from u(z). Of course, for a general / G L 2 ( R ) , the aliasing cannot be undone. In fact, by (7.123), (7.124)
(SSS0f){u) = $ > ( * + *)/(" + *)&")»
which does not coincide with f(u>) in general. When is S^Sof = / ? Note t h a t #oSo = Po, hence 5 5 5 0 / = / if and only if / G V0. If / = u{T)(j) G V&, then / ( C J ) = u(z(<J))
U(Z(UJ
+
£;))(CJ
+ k) = u{z{u))<j){uj +
fc).
(7.125)
Hence (7.124) becomes (S5S0f)(u>)
= \£\$(u
+ k)\2]u(z)4>(u>).
(7.126)
k
T h e next proposition confirms t h a t the right-hand side is, in fact, U(Z)4>(UJ) =
just
f((j).
P r o p o s i t i o n 7.8. The orthonormality
property (<j)ni
K(Lj) = Y^\4>(u + k)\2 = 1
a.e.
to
(7.127)
Proof: Note t h a t K(LJ) is 1-periodic and JQ dwK(u) = ||||2 = 1; hence K is integrable and therefore possesses a Fourier series. Equations (7.115) and (7.117), with / = 0, give = /
dw e2ninu> K(cu).
(7.128)
JO
Hence the orthonormality condition states t h a t the Fourier coefficients of K(ui) are cn = <S°, which means t h a t K(LJ) = 1 a.e. ■
A Friendly Guide to Wavelets
164
Note: (a) W h e n <j>(t) has compact support, then (LJ) is analytic and therefore continuous, and (7.127) holds for all UJ, not just "almost everywhere." (b) W h e n ±27Vlujt , then (7.127) becomes e±iwt a r e u s e d m ^he Fourier transform instead of e Y^ \4>{u +
2TT/C)|2
= —
a.e.
(if e±iujt
are used).
(7.129)
k
This is the convention used in Chui (1992a), Daubechies (1992), and Meyer (1993a). Proposition 7.8 gives the following intuitively appealing intepretation of
(a)
4>(k) = 8l
keZ oo
/ (c)
du f(u)
{
a.e. for all integrable f € V0
of unity, i.e., y^(f)n(t)
= 1
.
.
a.e.
n
Proof: Since <j> is integrable, cf) is continuous; hence its values at the individual points UJ = k are well defined. Together with (7.127), <^(0) = 1 implies t h a t
7. Multiresolution Analysis
165
<j>{k) = 0 for all integers k ^ 0, which proves (a). Now let / = u{T)(j> G Vo and consider the time-aliased signal P(t) = ^2nf{t — n), which is 1-periodic and integrable (since u(T) is a polynomial) and hence possesses a Fourier series. Its Fourier coefficients are [ dte-2niktJ2f(t-n) n °
ck=
dte-2vik^-n^f(t-n)
= ^2[ n
°
(7.131) v
OO
/ -oo
7
rft e- 2 "**/(t) = /(fe).
But (a) implies that /(fc) = w(e~ 27rifc )0(fc) = 6g, hence OO
* /(*),
/
(7.132)
-OO
proving (b). Assertion (c) now follows by choosing f = (f>. ■ Note: The term "partition of unity" refers to the fact that an arbitrary (but sufficiently reasonable) function can be written as a sum "localized" functions: /(*) = E „ / n ( < ) , where fn(t) = <*„(«)/(*). Example 7.10: The Ideal Low-Pass Filter. Let
^ 33 >
*">={; !:!>! In the time domain, <j> is given by m
= ^
-
(7.134)
Since (7.127) holds, so does the orthonormality condition, which can also be easily verified directly. <j> cannot be integrable, since <j> is discontinuous. Never theless, (LJ) is bounded for all u and continuous in a neighborhood of a; = 0, and it satisfies <j>{k) = 6® for all k £ Z. Hence <j> generates a multiresolution analysis (see the comments below Proposition 7.1). The function defined by (7.116) is F(t)=
due2*™* / H ,
/
(7.135)
which is the band-limited function obtained by passing / through the low-pass filter <j>. The projection of / to Vo is (using 0 * / = F(n)) (P0f)(t)
Sini
?~nn)F(n)
= $>(*)*;/ = £ n
= F(t),
(7.136)
n
by Shannon's sampling theorem of Section 5.1. (The simple relation F = Pof does not hold in general; it is special to MR As generated by ideal filters.)
A Friendly Guide to Wavelets
166
Up to now, we have only considered sampling and reconstruction with (pn, hence in Vo- Only the orthogonality and averaging properties of
(7.137)
where = -7= Kz{u))) = m0(uj + 1). (7.138) v2 Equation (7.98), which was a consequence of orthonormality and the dilation equations, stated t h a t |fo(z)| 2 + \h(—z)\2 = 2. For ra0(u;), this reads raoH
\m0(uj)\2
+ \m0 (LJ + .5) | 2 = 1,
(7.139)
since z(u + .5) = —z{ui). The averaging property implied t h a t h(l) = y/2 and h(—l) = 0, or equivalently m 0 (0) = 1,
mo(.5) = 0.
(7.140)
(When the Fourier transform is defined with the complex exponentials e±lu}t, mo(w) is 27r-periodic instead of 1-periodic. Then (7.139) and (7.140) read \mo(u)\A + I mo (CJ + ir) \z = 1 and mo(0) = l,mo(7r) = 0, respectively.) T h e mother wavelet was defined in (7.70) by Dtp = g{T)(j). This implies IP(2UJ)
If g(T) = -Txh{-T)*,
= —g(z(u>))4(uj)
= mi(w)^).
(7.141)
with A an odd integer, then mi(w) = - e ~ 2 7 r i X u J m0(u> + .5).
(7.142)
Thus (7.137), (7.139), and (7.141) imply | ^ ) |
2
+ |^(2a;)|2^|0(a;)|2.
(7.143)
Equation (7.143) can be viewed, roughly, as describing how the "spectral energy" of / £ Vo is split up between V\ and W\. For example, if 0 is an ideal low-pass filter, then tp is an ideal band-pass filter, as shown in the next example, and the spectral energy is cut up sharply between V\ and W\. In general, the splitting is not "sharp" in the frequency domain (see Example 7.12). In any case, (7.143) suggests t h a t the energy tends to be split evenly between V\ and W i , since /dhj\(p(2uj)\2 — Jdcj\ip(2uj)|2 = \. (The actual amount going to V\ and W\ depends, of course, on / . )
7. Multiresolution Analysis
167
Example 7.11: Shannon Wavelets. We have noted that the ideal lowpass filter of Example 7.10 defines a multiresolution analysis. To construct the corresponding wavelets, we must find m0(uj) so that (7.137) holds. Since (J>(2UJ) is the characteristic function of [—.25, .25], (7.137) implies that mo(w) = 4>{2LO) on [—.5, .5]. By periodicity, m0(uj) = Y^ 0(2^ + 2k).
(7.144)
k
Since <j> is real-valued, (7.142) gives m1(u)=
27TiX
e-
"
^2J>(2u; + 2k + l).
(7.145)
k
The support of mi(u>) intersects with that of <j>{u) only in [—.5, —.25] (for k = 0) and in [.25, .5] (for k = - 1 ) . Thus (7.141) gives 4>(2u) = - e~27riXuJ ——e
-2ir iXoo
U{2LJ
+ 1) + 4>{2u - 1)) 0(<j)
(7.146)
(X[-.5,-.25]H +X[.25,.5]H) •
Thus $(2LJ) is an ideal band-pass filter for the frequency bands [—.5, —.25] and [.25, .5]. Similarly, 4>(UJ) is an ideal band-pass filter (up to the normalization factor y/2) for the bands [—1, —.5] and [.5,1]. ip is sometimes called the Shannon wavelet. In the time domain it is given (taking A = 1) by
This wavelet is not very useful in practice, since it has very slow decay in the time domain due to the discontinuities in the frequency domain. In a sense, the Shannon wavelet and the Haar wavelet are at the opposite extremes of multiresolution analysis: The former gives maximal localization in time, the latter, in frequency. Each gives poor resolution in the opposite domain by a discrete version of the uncertainty principle. (They are analogous to the delta function and f(t) = 1, which are at the extremes of continuous Fourier analysis.) All other examples we will encounter are intermediate between these two. Example 7.12: Meyer Wavelets. The slow decay of the ideal low-pass filter in the time domain makes it far from "ideal" from the viewpoint of multireso lution analysis, since it gives rise to poor time localization. The Meyer scaling function solves this problem by smoothing out ((j). Choose a smooth function 9(u) with 0(u)=O
forw
and
6{u) = -
for u > §,
(7.148)
A Friendly Guide to Wavelets
168 with the additional property e{u) + e(l-u)
= ^
for all u.
(7.149)
"Smooth" means, e.g., t h a t 9 is d times continuously differentiate, for some integer d > 1. Such functions are easily constructed, as shown in Exercise 7.1. Given 9(u), define <j>(t) through its Fourier transform by (7.150)
4>(LJ) = C O S 0 ( M ) .
Then 4>(UJ) is also d > 1 times continuously differentiate, with (f>(uj) — 1 for \UJ\ < 1/3 and <j>(u) = 0 for |CJ| > 2 / 3 . Hence the support of <j> is [ - 2 / 3 , 2 / 3 ] , which exceeds a single period of mo- It is easily seen t h a t 4> satisfies (7.127). Only neighboring terms \4>(UJ + k)\2 in the sum (i.e., with Ak — 1) have overlapping supports. For simplicity we concentrate on the two terms |(t<;)|2 + \(f)(uj — 1)| 2 . In the interval [—1/3,1/3], the supports do not overlap and the sum equals 1. For 1/3 < UJ < 2 / 3 , 0(\u - 1|) - 0(1 - UJ) = TT/2 - 6(LJ) by (7.149). Hence <j>{w - 1) = cos (^ - 6{u))) = sin<9(cj)
(7.151)
and \(J)(LU)\2
+
\4>(LJ
- 1)| 2 = cos 2
9{LJ)
+ sin 2 9(u) = 1,
This proves (7.127) for —1/3 < UJ < 2 / 3 and hence Clearly, (f)(k) = <5° holds as well. Since
± <
UJ
< f.
(7.152)
for all UJ by periodicity. (j>(t) has reasonable decay. the more rapid the decay Just as in Example 7.11,
m0{uj) = ^T (j)(2uj + 2k).
(7.153)
k
(This is a consequence of j)(2uj-\-2k)4>(uj) — b\ (/)(2UJ), which is not true in general but holds in both examples.) Equation (7.145) also holds in this case, as does the top expression in (7.146) for the associated mother wavelet: IP(2UJ)
= - e~27TiXuJ
(J>(2u - 1) +
4>{2UJ
+ 1))
4>(UJ).
T h e arguments about overlapping supports are similar (exercise).
(7.154)
7. Multiresolution
Analysis
169
7.4 Frequency Inversion Symmetry and Complex Structure* The MRA formalism described in Section 7.2 exhibits an as yet unexplained symmetry between low-frequency and high-frequency phenomena. Recall that we can start with the filter coefficients hn, subject to certain constraints, or equivalently with h(T) £ V. h(T) determines g(T) by g(T) = -Txh(-T)\
AGZodd,
(7.155)
and the scaling function and mother wavelet are determined (up to normaliza tion) by D = h(T), D4> = g(T)
'
}
Gu(T) = Sig{T)*u{T)il>. The symmetry of these relations with respect to h(T) <-► g{T),
(7.158)
where A is the odd integer used in the definition of g(T). The next theorem shows that (a) J behaves much like multiplication by i, so it is a kind of "90° rotation" in Vb; (b) J "rotates" D(j) e V\ to Dip G W\\ and (c) it "rotates" the whole subspace V\ onto W\ and W\ onto V\. The frequency inversion symmetry This section is optional.
A Friendly Guide to Wavelets
170
is therefore explained as an abstract rotation in Vo- A similar operator Jm can be defined on Vm for every m G Z by Jm = Z)m«7X)_m. Theorem 7.13. (a) (b)
J2 = -IVo JD(f) = Dip,
(c)
JVi = Wu
JDtP = -D
(7.159)
JWi = Vi.
Proof: J2u(T)
= T A [(-T) A ]* u(t)
-u(T)^
(7.160)
since A is odd and (T A )* = T~x. This proves (a), (b) follows from JDcj) = Jh{T)(P = -Txh(-T)*<j)
= g{T)(j) = D^
(7.161)
and JDip = J2D(j) = —D(p, by (a). To prove (c), recall that a typical vector in V\ has the form Du(T)(f> = h(T)u(T2)(f), while a typical vector in W\ has the form Du(T)ip = g(T)u(T2)<j). Hence JDu(T)
0
2
= 2(TKT )*e^i.
(7.162)
Since {Du{T)<j) : u{T) G P } is dense in Vi and {#(rXr 2 )* : u(T) G P } is dense in W\, this proves that JVi = { < / / : / £ Vi} = W\ as claimed. It now follows from (a) that JWl
= J2Vx = { J 2 / : / e Vi} = { - / : /
G Vi} = Vi. m
(7.163)
The relation between the filters is also implicit in (7.162): JH*u{T)
(7.164)
Replacing u(T) by u(T)* and applying J to both sides then gives JG*u(T)^
= J2H*u(T)*(t) = -H*u(T)*
(7.165)
J is real-linear, i.e., J[cu(T) + v(T)0] = cJu(T)
Jc = cJ.
VQ, note that for (7.167)
7. Multiresolution Analysis
171
That means that J is antilinear with respect to complex scalar multiplication. This results in an interesting extension of the complex structure defined by J . Define K : V0 -> V0 by K = iJ,
i.e.,
Ku{T)(j) = iJu{T)(t)=-iTxu{-TY(t),
(7.168)
where i = \/^T. According to (7.167), we have K2 = UiJ = -i2J2 = -IVo
(7.169)
and J K = JU = -iJ2 = i, X i = iJz = - z 2 J = J. (7.170) Hence {?', J, K} defines a quatemionic structure on Vo! This means that we can extend the "rotations" generated by J to ones very similar to general rotations in R 3 . I do not know at present what use can be made of these, but it seems to be an interesting fact. The above implementation of frequency-inversion symmetry by the com plex structure seems rather abstract, since no explanation was given as to why J should interchange low-frequency and high-frequency phenomena. To get a better understanding, we now view the action of J in L2 (T), using the diagram of Figure 7.5. The operator corresponding to J is j : L 2 (T) —► L 2 (T), defined by (ju)(z) = -zxu(-z). (7.171) Since z = e~ with period 0 < \LJ\ < .5, low frequencies are represented by 0 < \u>\ < .25 and high frequencies, by .25 < \u\ < .5. (Recall Example 7.10.) Clearly, j exchanges the low- and the high-frequency parts of u(z). The corresponding operator j : £2(Z) —► £2(Z) is given by 2nluJ
j{un}n€Z
= {(~l)n
UX-n}neZ
(7.172)
(see Figure 7.6).
£2{Z)
£2(Z)
L 2 (T)
L 2 (T)
Figure 7.6. The complex structures in £2(Z) and L 2 (T). Note that u(-T)* = E n ^ ( - T _ 1 ) n = w ( - r _ 1 ) , where u(T) is the polynomial defined by u(T) = J^ n unTn. In particular, if the w n 's are real, then u(—T)* = u(—T~l). Hence (ju)(z) = —zxu{—z) and the mapping j amounts to a reflection about the y-axis in the z-plane, followed by multiplication by —zx.
A Friendly Guide to Wavelets
172
As an extreme example, consider sampling f(t) = 1. ( / ^ L 2 ( R ) , hence this example does not fit into t h e L2 theory, but it does make sense in terms of distributions!) Then un = 4>^f = 1 for all n, and = {(-l)n}n€Z-
j{Un}neZ
(7.173)
T h a t is, j transforms the least oscillating sequence { l } n G z into the sequence with t h e most oscillations. T h e quaternionic structure can likewise be transfered to £2(Z) and L 2 ( T ) by defining k = ij on £2(Z) and k = ij on L 2 ( T ) .
7.5 W a v e l e t Packets* One of t h e main reasons t h a t wavelet analysis is so efficient is t h a t the frequency domain is divided logarithmically, by octaves, instead of linearly by bands of equal width. This is accomplished by using scalings rather t h a n modulations to sample different frequency bands. Thus wavelets sampling high frequencies have no more oscillations t h a n ones sampling low frequencies - very unlike the situation in t h e windowed Fourier transform. However, it is sometimes desirable to further divide t h e octaves. T h a t amounts t o splitting the wavelet subspaces W m , and it can actually be accomplished quite easily within the wavelet frame work. T h e resulting formalism is a kind of hybrid of wavelet theory and the windowed Fourier transform, since it uses repeating wavelets to achieve higher frequency resolution, much as t h e W F T uses modulated windows. Wavelet pack ets were originally constructed by Coifman and Meyer; see Coifman, Meyer and Wickerhauser (1992). To construct wavelet packets, coefficients are allowed in u(T) G implies t h a t the complex structure \s skeiu-Hermitmu, \.e., 3"" — — 3 and F\ :Vo —* Vo by
let us assume that Vo is real, i.e., only real V. (Then Vm = DmVo is also real.) This defined in Section 7.4 (which is now linear) ^ x s i c \ s € ) . TJetme xtoe, opeYaXois F$ ; V$ —* VQ
F0 = H,
F i = -HJ.
(7.174)
Essentially, F\ performs like t h e high-pass filter G, extracting detail. In fact, by (7.40) and (7.41), FXVX = -HJVi F i W i = -HJWi
= -HWi = -HVi
= {0} = V0
FfV0 = JH*V0 = JVi = Wi. This section is optional.
(7.175)
7. Multiresolution Analysis
173
The actions of F\ and Fj* on sequences are Fi{uk}keZ (7.176) Fi{un}T (exercise) which are almost identical with the actions of G* and G, as given by (7.90). (Keep in mind that we now assume everything to be real.) The main difference is that, unlike G : Vo —» Wo, F\ maps Vo to itself. That makes it possible to compose any combination of Fo, Fi, F0* , and Fx*, which leads directly to wavelet packets. The next proposition shows that Fo and Fi can be used exactly like H, G to implement the orthogonal decomposition of Vo = V\ 0 W\. Proposition 7.14. (a)
FaFb* = 8abIVo (a, b = 0,1)
(b)
FSFo +
F^F^Ivo.
Proof: We already know from (7.91) that F0F^ = HH* = Iy 0 . Since J J * = - J 2 = IVo, it follows that FiFf = IVo. By (7.41), (7.159)(c), and (7.40), FiFSVo = -HJH*V0
= -HJVi
= -HW1
= {0}.
(7.178)
Hence FiF0* = 0, which also implies FoFj* = 0. That proves (a). For the proof of (b) and the exact relation between F\ and G, see Kaiser (1992a). ■ To consider arbitrary compositions of the new filters, it is convenient to introduce the following notation: Let A = {ai, a 2 , . . . , a m } and B = {bi, 62,..., bm} be multi-indices, i.e., ordered sets of indices with ak,bk = 0,1. We also write the length of A as \A\ = m. Define %=%:S£-S£,
\A\ = \B\ = m.
(7.179)
Then we have the following generalization of Proposition 7.14. Proposition 7.15. (a)
FAFZ = 6$IVo
(b)
^
F:FA = IVo
if
|A| = | B | > 1 . for any m> 1,
|i4|=m
where the sum in (b) is over all 2m multi-indices with \A\ = m.
(7-180)
174
A Friendly Guide to Wavelets
Proof: Follows immediately from (7.177) (exercise).
■
Proposition 7.15 states that the operators UA = F*AFA,
(7.181)
with \A\ = m any fixed positive integer, are a complete set of orthogonal pro jections on VQ. That is,
n : - n A = ni, nAnfl = ^n A ,
]T uA = iVo.
(7.182)
\A\=m
It follows that any / G VQ can be decomposed as
(7-183)
/ = £ i w = Y, f*> |A|=m
|A|=m
with (fA,fB) = 0 whenever B ^ A. The subband filtering scheme in (7.92) and (7.93) can now be generalized. In the first recursion, an arbitrary u G Vo is decomposed by
u= Y^ F>A>
where
\A\=m
uA = FAu.
(7.184)
Repeating this gives wAxA2...A7
where
= E
\Ar+i\=m = uAiA2-Ar+1
nr+yiA2-Ar+\ FAr+1u
AlA2
Ar
(7.185)
-~ .
The essential difference between the wavelet decomposition (7.92)-(7.93) and the above wavelet packet decomposition is that in the former, only the "averages" get decomposed further, whereas in the latter, both averages and detail are subjected to further decomposition. Exercises 7.1. In this exercise it is shown how to construct functions 9{u) such as those used in the definition of the Meyer MRA and wavelet basis. Let fcbea positive integer, and let Pk(x) be an odd polynomial of degree 2k + 1: Pk(x) = a0x + aix 3 + • • • + akx2k+1.
(7.186)
We impose the conditions Pk(l) = 1 and Pkm){l) = 0 for m = 1, 2 , . . . , k. This gives k + 1 linear equations in the k + 1 coefficients, with a unique solution. For example, P1{x) = \ x - \ x 3 ,
P2{x) = f x - | x3 + | x5.
(7.187)
7. Multiresolution Analysis
175
Now define 0 7T
0(U) = ~ X
l +
Pk(6u-3)
ifu<§ if \ < u < | ifu>§.
(7.188)
(a) Prove that 6(u) is k times continuously differentiable. (b) Verify that 0{u) satisfies (7.149). (c) Find P3{x). 7.2. Verify (7.153). 7.3. Prove that if Vo is real, then the complex structure of Section 7.4 is skewadjoint, i.e., J* = —J. (Use the orthonormahty condition.) 7.4. Prove (7.176). 7.5. Prove (7.180).
Chapter 8
Daubechies' Orthonormal Wavelet Bases Summary: In this chapter we construct finite filter sequences leading to a family of scaling functions (f)N and wavelets tjjN, where TV = 1, 2 , . . . . N = 1 gives the Haar system, and iV = 2, 3 , . . . give multiresolution analyses with orthonormal bases of continuous, compactly supported wavelets of increasing support width and increasing regularity. Each of these systems, first obtained by Daubechies, is optimal in a certain sense. We then examine some ways in which the dilation equations with the above filter sequences determine the scaling functions, both in the frequency domain and in the time domain. In the last section we propose a new algorithm for computing the scaling function in the time domain, based on the statistical concept of cumulants. We show that this method also provides an alternative scheme for the construction of filter sequences. Prerequisites: Chapters 1 and 7.
8.1 C o n s t r u c t i o n of F I R F i l t e r s In Chapter 7, we saw t h a t a multiresolution analysis (MRA) is completely de termined by a scaling function satisfying the orthonormality and averaging con ditions and the dilation equation Dcj) = /i(T), with h(T) given. , in turn, is determined up to normalization by h(T) or, equivalently, by the filter coefficients hn. So it makes sense to begin with {hn} in order to construct the MRA and the associated wavelet basis. When all but a finite number of the /i n 's vanish, the same is true of the gn's. Then h(T) and g(T) are called finite impulse response (FIR) filters. T h e reason for this terminology is as follows: Since h(T) and g(T) operate by multiplication in the frequency domain, they are filters in the usual engineering sense, i.e., convolution operators. When applied to 0, which corresponds (under the sampling operator So '• L2(R) —> £2(2*)) to the "im pulse" sequence {<S°}n<EZ, these operators give the sequences {hn} = Soh(T)(f) and {gn} = S$g(T)(f), respectively. Hence {hn} and {gn} are called the impulse responses of h(T) and g(T). T h e scaling function and wavelet corresponding to a FIR filter are com pactly supported: hence they give very good localization in the time domain. The simplest example of a multiresolution analysis with FIR filters is t h a t as sociated with the Haar system (Examples 7.2 and 7.7). Although (or because) it is extremely well localized in time, the Haar system is far from ideal. Its discontinuities give rise to very slow decay in frequency, consequently to poor
G. Kaiser, A Friendly Guide to Wavelets, Modern Birkhäuser Classics, DOI 10.1007/978-0-8176-8111-1_8, © Gerald Kaiser 2011
176
8. Daubechies' Orthonormal Wavelet Bases
177
frequency resolution. We have already seen examples of MRAs with good fre quency resolution (the Shannon and Meyer systems), but they were at the other extreme of the uncertainty principle since they had very poor time localization. In particular, they have infinite impulse responses. Until recently, the Haar MR A was the only example of an orthonormal FIR system. In this chapter, fol lowing Daubechies (1988), we construct a whole sequence of low-pass FIR filters hN(T), N = 1,2,3,..., and their associated high-pass FIR filters gN(T). The associated scaling functions and wavelets, obtained by solving D
\ = -j=g(T) =
-TXP{-T)*
(8.1)
(see (7.58)). The Haar filters are now P(T) = (I + T)/2, Q(T) = (I - T)/2. For N > 2, the strategy is to find polynomials P(T) satisfying the averaging property (7.99) and orthonormality condition (7.98), which now take the form P ( l ) = l,
\P(z)\2 + \P(-z)\2
= l,
z=e~27viu}.
(8.2)
By translating if necessary, we may assume that the first nonzero coefficient is Co, so M
P(z) = ^ c
n
z",
(8.3)
with c 0 , CM 7^ 0. We assume that the c n 's are real, which implies that
M-2k
Y^ cn = 2,
] T cncn+2k
n=0
n=0
= 2$ ,
k = 0 , 1 , . . . , [M/2],
(8.4)
where [M/2] denotes the largest integer < M/2. If M is even, then the second equality with k = M/2 gives COCM = 0, which contradicts the assumption that Co , CM 7^ 0. Hence M must be odd, and we take M = 2N — 1 with N > 1, so that [M/2] = N - 1. Then (8.4) gives N + 1 equations in the 2N unknowns
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Co, c i , . . . , C2Ar_i. When N = 1 (so N + 1 = 2iV), the unique solution is CQ = c\ = 1, which gives the Haar system. When JV > 2, we can add TV — 1 more constraints to (8.4). This freedom can be used to obtain wavelets with better regularity in the time domain, which then give better frequency resolution than the Haar system. (This is in accordance with the uncertainty principle: The support widths of <j> and ip will be seen to be 2TV — 1, hence the larger we take Ar, the more decay ij> and ty can have in the frequency domain. In turn, this means more smoothness for <j> and ip.) To find an appropriate set of additional constrains, note that the averaging condition P ( l ) = 1, together with orthonormality, implies that P(—1) = 0. By (7.137) and (7.138), m0(u>) = P(e-27riuj) and the dilation equation implies 4>{k) = P(e-i7rk)(j>(k/2) = 0,
k <E Z odd.
(8.5)
When k is even but not zero, then k = 2ek' for some integers £, k' with k' odd, and (8.5) (applied repeatedly) implies <j>{k) =
,
/-,
\N
P{z) = ( ^ )
\N
N-l
W(z) = (i±i)
Y, w^-
(8-6)
We have chosen the normalization so that the averaging condition reads W(l) = 1. An argument similar to the above now shows that the derivatives of ij> satisfy 0(m)(fc) = O,
m = 0,l,...,JV-l, 0//cGZ.
(8.7)
That means that <j> is very flat at integers k ^ 0, further improving the separation of low frequencies from high frequencies. The larger N is, the better the expected frequency resolution. (Note that the ideal low-pass filter of Example 7.10 satisfies (8.7) with TV = oo.) Equation (8.6) has several equivalent and important consequences, the most well known of which is that the wavelet tj) associated with <j> has N zero moments, i.e., oo
/
dt •tmip{t) = 0,
m = 0,1,...,N-l.
(8.8)
-oo
For m = 0 this is, of course, just the admissibility condition, which can be interpreted as stating that I/J has at least some oscillation, unlike a Gaussian (or 0, for that matter). The vanishing of the first N moments of ip implies more and more oscillation with increasing N. See Daubechies (1992); see also
8. Daubechies' Orthonormal Wavelet Bases
179
Beylkin, Coifman, and Rokhlin (1991) for applications to signal compression. To see what (8.6) means for the filter coefficients, note t h a t {zdz)mP{z)
Y^^rncnzn.
= n
Hence by (8.6), 2JV-1 m
= ] T ( - l ) n n m c n = 0,
(zdz) P(-l)
m = 0 , 1 . . . , JV - 1.
(8.9)
71=0
This is a simple set of equations generalizing the admissibility condition, which is the special case m = 0. It can be restated in terms of the high-pass filter coefficients dn = (—l) n C2jv-i-n as ] T nmdn
= 0
m = 0 , 1 . . . , iV - 1.
n
In this form, it is a discrete version of (8.8). Our task now is to find polynomials of the form (8.6) satisfying the orthonormality condition. T h e set of all such polynomials can be found using some rather sophisticated theorems of algebra. We follow a shortcut suggested in Strichartz (1993), showing t h a t the solution for N = 1, which is the Haar sys tem, actually generates solutions for all N > 2. For N = 1, the unique solution is P(z) = (1 + z)/2 = e~iiru} COS(TTU) (8.10) P(-z) = (1 - z)/2 = ie~i7VUJ sin(Trcj). To find a solution for any N > 2, introduce the abbreviations ^1/2
C(z) =
COS(TTUJ)
=
+
^1/2
^1/2 _ ^ l / 2
, S(z) = sin(7Rj) =
—
,
(8.11)
Zi i
ZJ
and raise the identity C2 + S2 = 1 to the power 2N - 1: 1 = (C2 + S2)™-1
= ^ (2N fc=0 ^
"
^ ^ - 2 - 2 ^ 2 ^ ^
(8.12)
where ( ^ ) = M\/k\(M - k)\ are the binomial coefficients. T h e sum on the right contains 2N terms. Let VN{C) be the sum of the first N terms (using S2 = 1 - C 2 ) , a polynomial of degree 4 N - 2:
VN(C) = J2 (2N~ fc=o ^
X
/
W - 2 - 2 f c ( 1 - C2)k > 0.
(8.13)
If we can find a polynomial P(z) such t h a t \P(z)\2 = VN(C), then \P(-z)\2 is automatically the sum of the last N terms in (8.12) and the orthonormality
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condition in (8.2) is satisfied. (This follows from the fact that C(—z) — —S(z) and S(—z) = C(z).) Note that (8.13) can be factored as = C2NWN(S)
VN(C)
11 4- z
= |-±- |
\2N
WN(S),
(8.14)
where
WN(S) = £ ( 2 J Y *) (1 - S2)N-1~kS2k > 0
(8.15)
and WTV(0) = 1. In order to obtain a filter P(z) of the form (8.6), it only remains to find a "square root" W(z) of WN{S), i.e., N-l
W{z) = J2
w
nzn
such that
\W{z)\2 = WN(S),
(8.16)
n=0
with W{1) = 1 (which is consistent with WJV(O) = 1) and real coefficients wn (since c n must be real). The existence of such W(z)'s is guaranteed by a Lemma due to F. Riesz, but the idea will become clear by working out the construction of W(z) in the cases N = 2 and N — 3, where it will also emerge that W(z) is not unique if N > 1. Example 8.1: Daubechies filters with N = 2. For iV = 2, (8.15) gives W2(S) = 1 + 2S2 = 2-
Z
-^-.
(8.17)
W(z) must be a first-order polynomial W(z) — a + bz, with a, 6 G R. Thus | ^ W | 2 = W 2 (5) implies a2 + b2 = 2, 2a6 = - 1 , => (a + &)2 = 1, (a - 6)2 = 3.
(8.18)
Two independent choices of sign can be made, giving four possibilities, but only two of these satisfy W(l) = 1: W±(z) = i [ ( l ± V 3 ) + ( l T V3 )*] •
(8.19)
The Daubechies filters with N = 2 are therefore P±(^) = ^ (
1
Y
£
)
[ ( 1 ± V 3 ) + (1=FV3)*].
(8.20)
Aside from the zero of order 2 at z = — 1, -P±(^) has a simple zero at 2 ± = 2 ± \ / 3 . 2_ is inside the unit circle, while z+ is outside. This "square-root process" |W(2;)|2 —»• W(z) is called spectral factorization in the electrical engineering literature, and the multiplicity in the choice of so lutions is typical. The filter whose zeros (not counting those at z = — 1) are all chosen to be outside the unit circle is called the minimum-phase filter, and the
8. Daubechies' Orthonormal Wavelet Bases
181
filter with all the zeros inside the unit circle is called the maximum-phase These two filters are related by inversion in the complex 2-plane:
filter.
P.(z) = *>»-lP+(z-1) = ( i l f y V - 1 ^ * - 1 ) ,
(8.21)
and the filter coefficients are related by C
n = 4v-l-n >
W
n = WN-l-n
(8-22)
•
This means t h a t the corresponding scaling functions are related by reflection about the midpoint t = N — 1/2 of their support: (j)-(t) = (l>+{2N-l-t).
(8.23)
Because of this simple relation, there is no essential difference between the max imum and minimum phase filters. Both are therefore commonly called extremal phase filters. Thus we see that, up to inversion, there is just one Daubechies filter with N = 2. Suppose t h a t <j>{t) is symmetric about t = to, i.e., (f>(2to — t) = (t). In the frequency domain this means t h a t e-*™t°4>(-Lj)
= j>((j).
Since (— UJ) = (/>(UJ) because <j>{t) is real, it follows t h a t the phase of
(8.24) (/>(LJ)
<£(u,) = e-2niut0\4>(Lj)\.
is
(8.25)
For this reason, filters with a symmetric impulse response are called linear phase filters. In some applications (image coding, for example), linear phase filters a r e prefered. The Haar scaling function is a linear phase filter with to = 1/2, as can be seen directly from 4>(t) = X[o,i]W o r from
- 1 + 3 5 2 + 6S4 = l [38 - lS{z + z) + 3(z2 + z2)] . = a + bz + cz2 with a, 6, c € R and solving |V^(^)| 2 = Ws(S)
P±(z) = ( ^ )
W±(*)»
W±(*) =a± + bz + c±z2
(8.26) gives (8.27)
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182
(exercise), where a± = \(l + Vld±
y / 5-h2v / 10 j
b=l(l-VT6>j
(8.28)
c± = \ f l + VlO=F\/5 + 2 \ / l o Y Only these two solutions (out of eight solutions) survive the requirement that W(z) be real with W(l) = 1. Again, P+(z) has minimum phase and P-(z) has maximum phase, so there is effectively only one Daubechies filter with N = 3. For N > 4, however, the spectral factorization also gives "in-between filters" that have neither minimum nor maximum phase. It is then possible to choose the zeros of W(z) so that <j>(t) is "least asymmetric" about t = N—1/2, although complete symmetry is impossible, as already mentioned. (See Daubechies (1992), Figure 6.4 and Section 8.1.1.) The reader will have guessed by now that numerical methods must be used to obtain the Daubechies filters with large N. The values of the coefficients for the minimum-phase filters are listed in Daubechies' book on page 195. Actually, the above procedure does not give the most general solution of (8.2) and (8.6). If WN{S) is replaced by W(S) = V M S ) + S2NR (2 - AS2) = WN(S) + S2NR (z + z),
(8.29)
with R any odd polynomial such that the positivity condition W(S) > 0 is retained for \S\ < 1, then solving W(S) = |VF(2;)|2 for W(z) by spectral factor ization still leads to a solution P(z), since \P(z)f
+ \P(-z)\2
= C2NWN(S) 2N
+ S WN(C)
+ C2NS2NR(z 2N
2N
+ S C R(-z
+ z) - z),
(
'
and the second and fourth terms on the right cancel. Indeed, this gives the most general solution. If the degree of R is M, then that of W(S) is 2Af -f 2M and that of W(z) is N + M. Hence the degree of P{z) is 2N + M, which is odd as necessary since M is odd by assumption. The support width of both <j> and ip is then also 2N + M. On the other hand, we could have chosen a filter with 2N = 2N + M + 1 coefficients and a zero of order N at z = — 1, in which case any of the resulting scaling functions and wavelets 0 and ijj have support width 2N — 1 = 2N + M, the same as 4> and ip. That is, scaling functions and wavelets with R ^ 0 do not have the minimal support width for the given order of their zero at z = — 1. The additional freedom gained by requiring fewer than N factors of (1 + z)/2 for a filter with 2N coefficients can be used to make
8. Daubechies' Orthonormal Wavelet Bases
183
a more detailed discussion of these ideas, we refer the reader to Sections 6.1, 7.3, and 7.4 in Daubechies' book. A completely different and beautifully simple method of arriving at |P(^)| 2 has been developed recently by Strang (1992). Yet another new way is the main topic in Section 8.4, 8.2 Recursive Construction of
(8.31)
n
In this section we describe a recursive scheme in the frequency domain. This uses the polynomial P( e~ 27riw ) = mo(uj) to build up <j>(u>) scale by scale. When the desired accuracy is obtained in the frequency domain, the inverse Fourier transform must still be applied to obtain the corresponding approximation to 4>(t). By Plancherel's theorem, the global error in the time domain (measured by the norm) is the same as that in the frequency domain. A repeated application of the dilation equation (7.137) gives M 4>{LJ)
= m0{u;/2)4>(uj/2) = Yl m0{Lu/2m)4)(Lj/2M).
(8.32)
m—\
A s M - > oo, product:
4>{<JU/2M)
—» (0) = 1,* and we obtain (&) formally as an infinite oo
oo
0 ( « ) = n ™ o ( W 2 ) = n p ( e_2 " iu,/2m ) • rn—l
m
m=l
(g-33)
This shows that a given set of filter coefficients can determine at most one scaling function (given the normalization (f)(0) = 1). The sense in which the infinite product converges must now be examined to determine the nature of <j>. Recall that <J>(UJ) is an entire-analytic function since <> / is compactly supported.
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It must also be shown that 0(u;), now defined by (8.33), is actually the Fourier transform of a function in the time domain, and the nature of this function must be investigated. Obviously, rao(O) = P ( l ) = l i s a necessary condition for any kind of convergence, since otherwise the factors do not approach 1. If rao(O) = 1, then m o M = 1 + ±J2
c
-( e _ 2 7 F i U U } - !) =
x
- * ] £ c « e _ 7 r i n W sin(7mu;).
n
(8.34)
n
Hence \m0(uj)\ < l + ^ | c n | | s i n ( 7 r W ) | < 1 + A\u\ < e A|u;| ,
(8.35)
n
\ncn\ (since |sinx| < |rc|), and
where A = ^Yln
OO
/
OO
\
JJ |m0(o;/2m)|<expL4|a;|^2-m ra=l
\
n=l
= eAK
(8.36)
/
This shows that the infinite product converges absolutely and uniformly on compact subsets. In fact, it was shown by Deslauriers and Dubuc (1987) to converge to an entire function of exponential type that (by the Payley-Wiener theorem for distributions) is the Fourier transform of a distribution supported in [0,27V — 1]. However, this in itself is not very reassuring from a practical standpoint, since 0 could still be very singular in the time domain. (Consider the case P(z) = 1, which satisfies P ( l ) = 1 but not the orthogonality condition (8.2). It gives 4>(LJ) = 1, hence 0(f) = 6(t).) Mallat (1989) proved that if P(z) furthermore satisfies the orthogonality condition (8.2), then <j> belongs to L2(H) and has norm \\<j)\\ < 1. Hence the averaging and orthonormahty properties for P(z) imply that (j) G L2(H) (by Plancherel's theorem). For ip to generate an orthonormal basis in L 2 (R) in the context of multiresolution analysis, we only need to know that <j> satisfies the orthonormahty condition 0*0 = 6° (Chapter 7), although some additional regularity would be welcome. However, this is not the case in general! The reason is that although the condition (8.2) on the filter P(z) was necessary for orthonormahty, it may not be sufficient. (This is somewhat reminiscent of the situation in differential equations, where a local solution need not extend to a global one; roughly speaking, P(z) is a "local" version of 4>(uo)\) As a counterexample (borrowed from Daubechies' book), consider P(*) = l
1
^ )
(! - * + *2) = ^
= e~i7"W coe(3*w).
(8.37)
The averaging and orthonormahty conditions (8.2) are both satisfied, and the infinite product is easy to compute in closed form. (This fact alone is enough to raise the suspicion that something is wrong!) Using
8. Daubechies' Orthonormal Wavelet Bases
185
we have M
M
i _
. 1 2(1 — e
m=l
Hence =
0(CJ) 0(w)
r vV
and
m
e-12nitj/2
^
y
'
2 _
_6?riu; 2m
/
)
e-6iriu>
M
2 (1 — e~67ria;/2M)'
— Qiriuj I1 _ e„-67riu;
37riuJ _, —37riu; ._, lim -—— _ . / o M x = e~ sine MM 67rtcJ 2 M-*oo2 ( M-,oo 2 ( 1 - e ~ / )
(3TTCC;)
(8.40)
v
J
1/3 0 ^ < 3 t 0 otherwise. This is just a stretched version of the Haar scaling function. Obviously, the translates
=
] T \
(8.42)
k
which vanishes for u — 1/3. This shows t h a t {n} is not even a "Riesz basis" for its linear span Vo, which means that, although the 0 n ' s are linearly independent, instabilities can occur: It is possible to have linear combinations J ^ n un<j>n with small norm for which J ^ n \un\2 is arbitrarily large (exercise). (Furthermore, the coefficients un t h a t produce a given / G Vo cannot be obtained simply by taking the inner products with 0 n , due to the lack of orthonormality; we must use a reciprocal basis instead.) Consequently, 0 does not generate a reasonable multiresolution analysis. If the recipe for the MRA construction of the wavelet basis is applied, the result is not a basis but still a tight frame, with frame constant 3. This is just the redundancy equal to the stretch factor in (8.40). T h e above example shows t h a t additional conditions are required on P(z) (aside from (8.2)) to ensure t h a t the 0 n ' s are orthonormal and, consequently, t h a t {ipm,n } is an orthonormal basis. Although conditions have been found by Cohen (1990) and Lawton (1991) t h a t are both necessary and sufficient, they are somewhat technical. A useful sufficient condition, also due to Cohen, is the following: T h e o r e m 8 . 3 . Let P(z) satisfy the averaging and orthonormality conditions (8.2), and let <j) be the corresponding scaling function defined by (8.32). Then the translates n(i) = (£ — n) are orthonormal if mo{uj) = P(e~2lxlu)) has no zeros in the interval \UJ\ < 1/6. See Daubechies' book (Corollary 6.3.2) for the proof. Note t h a t in the above counterexample, rao(±l/6) = 0. This proves t h a t the bound 1/6 on |u| is optimal, i.e., no lower value exists t h a t guarantees orthonormality. The above considerations are unnecessarily general for our present purpose, since they do not use the special form (8.6) of P(z). We now examine the
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186
regularity of the Daubechies scaling functions (fiN, N > 2, constructed from the FIR filters P(z) given by (8.6), with W(z) obtained from (8.15) and (8.16) or (8.29) by spectral factorization. How does the factor A(z)N = ((l + z)/2)N affect the regularity of 0? Before going into technicalities, note that the dilation equation in the time domain is now D(j> = y/2 P(TU = y/2 A(T)NW(TU (8-43)
M
rN
= y/2W(T)A{T)
0M
e
-2™/2"
\_rn=l
jjjW(e-2*iu,/2™y
(8>44)
TO=1
The first factor has already been computed in (8.38) through (8.40), with z replaced by z3. Letting u —► u/3 in (8.40), we have II (
i
^
) = e-™sincf^) =
™t™>.
(8.45)
Let CO
F(w) = J ] W{e-^iu^m
).
(8.46)
Since W(l) = 1, the estimates (8.34) through (8.36) show that the infinite prod uct (8.46) converges absolutely (and uniformly on compact subsets). Since W(z) is a polynomial of degree N — l, the theorem of Deslauriers and Dubuc (1987) invoked earlier for P(z) shows that F(LJ) is an entire function of exponential type that is the Fourier transform of a distribution F(t) with support in [0,7V — 1], We have N N „, x e-™ "sm (7ru;)
HLJ) =
—*w
F{u)
'
(8 47)
'
8. Daubechies' Orthonormal Wavelet Bases
187
Now everything boils down to the behavior of F(LJ). Since both factors in (8.47) are analytic, only the behavior of F(cu) as \LO\ —> oo concerns us, since this will determine the regularity of the inverse Fourier transform of (/>. Suppose that | F ( C J ) | grows like \u>\@ as \u>\ —* oo, for some / 3 E R , which we denote by |F(CJ)| ~ \UJ\P. Then |<^(o;)| ~ \UJ\(3'~N. If <j> is to be continuous in the time domain, it suffices for <j> to be absolutely integrable, and this will be the case if 0-N < - 1 , i.e., ,3 = N-l-efor some e > 0. Similarly, ti 0 = N - 1 - k - e k for some integer k > 1, then u 4>(uj) is absolutely integrable. Since the A:-th derivative of <j>(t) is related to <J>(UJ) by OO
/
do; {2
(8.48)
-oo
it follows that <j>^ is continuous, denoted as / G Cfc. What \i (3 = N — 1 — a — £ for some positive noninteger a? Let a = fc + 7 with k being a nonnegative integer and 0 < 7 < 1. Then ujk(f) is absolutely integrable, and hence <j)^k\t) is continuous. Although (fc+1)(£) is not continuous, it is clear that 0 is more regular than the mere statement <j) G Ck would allow. This suggests a notion something like a "fractional derivative," which can be made precise as follows: A function f(t) is said to be Holder continuous (also called Lipschitz continuous) with exponent 7 if \f(t + h)-
f(t)\ < C\hp
for all t, h G R.
(8.49)
Obviously / is then also uniformly continuous. We denote the (vector) space of all such functions by C1'. This definition applies only for 0 < 7 < 1. The choice 7 = 1 (which is not allowed in (8.49)) would make / almost differentiate, but not quite. (Take f(t) = |t|, for example.) To see the relation between / G C 7 (defined in the frequency domain) and differentiability (defined in the time domain), note that
It is not difficult to show that if f(u) is absolutely integrable and if | / ( C J ) | ~ j^j-i-7-e^ then the integral on the right-hand side is absolutely convergent and bounded by a constant C independent of h; hence f £ C7. That gives a close relation between the rate of decay of / in frequency and its smoothness in time. The definition (8.49) can be extended naturally to arbitrary a > 0 as follows. If a = k + 7 with k G N and 0 < 7 < 1, then / is said to be Holder continuous with exponent a if / G Ck and f^ G C 7 . (When 7 = 0, this reduces to / G Cfe, since C° is the space of continuous functions.) We then write / G Ca. The relevance of all this to the regularity of (j) is as follows: Equation (8.47) shows that if the growth of F(LJ) can be controlled, then 4>{uS) will decay ac cordingly. This can then be used to give information about the regularity of 4>
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188
in the time domain, in the form of Holder continuity. In order to prove t h a t <j> belongs to Ca, it is sufficient to show t h a t |.F(u;)| ~ \u;\N~l~a~£ for some e > 0. Thus, one looks for conditions on W(z) t h a t guarantee such growth estimates for |F(c^)|. The following result is taken from Daubechies' book (Section 7.1.1). T h e o r e m 8.4. Suppose
that max\W(z)\ \z\=i
for some a > 0. Then <j) €
<2N-1~a
(8.51)
Ca.
For example, let N = 1 and W(z) = 1 (Haar case). Then m a x | W ( 2 ) | = 1, and a would have to be negative in order for (8.51) to hold. Thus Theorem 8.4 does not even imply continuity for 0 1 (which is, in fact, discontinuous). More interesting is the case of 0 2 , where N = 2 and W(z) With z =
COS(2TTLJ)
\W{z)\2
= a + bz,
a=i^,
b=l-±^.
(8.52)
+ zsin(27rcj), we obtain = 2-
COS(2TTU;),
hence
To find a from (8.51), first solve y/3 = 22'1~^
m&x\W{z)\
= y/Z.
(8.53)
= exp[(l - /i) In 2], which gives
\x = 1 - -^— « .2075. ^ 21n2
(8.54) '
v
Thus (8.51) holds with any a < /i, showing t h a t (f)2 is (at least) continuous. This does not prove t h a t 4>2 is not differentiable, since (8.51) is only a sufficient condition for e C a . Actually, much better estimates can be obtained by more sophisticated means, but these are beyond the scope of this book. (See Daubechies' book, Section 7.2.) It is even possible to find the optimal value of a, which turns out to be ~ .550 for <j)2. Furthermore, it can be shown t h a t 0 2 possesses a whole hierarchy of local Holder exponents. A local exponent 7(2) for / at t € R satisfies (8.49) with t fixed; clearly the best local exponent is greater t h a n or equal to any global exponent, and it can vary from point to point, since / may be smoother at some points t h a n others. It turns out t h a t the best local Holder exponent a(t) for
8. Daubechies' Orthonormal Wavelet Bases
189
By a variety of techniques involving refinements of Theorem 8.4 and detailed analyses of the polynomials WAT in (8.15), it is possible to show that all the Daubechies minimal-support scaling functions
(Again, this does not mean that a is optimal!) The factor (i was already en countered in (8.54) in connection with 0 2 . Thus, for large N, we gain about //, « 0.2 "derivatives" every time N is increased by 1. 8.3 Recursion in Time: Diadic Interpolation The infinite product obtained in the last section gives 0(CJ), and the inverse Fourier transform must then be applied to obtain 4>(t). In particular, that construction is highly nonlocal, since each recursion changes <j)(t) at all values of t. If we are mainly intersted in finding 0 near a particular value to, for example, we must nevertheless compute 0 to the same precision at all t. In this section we show how 0 can be constructed locally in time. This technique, like that in the frequency domain, is recursive. However, rather than giving a better and better approximation at every recursion, it gives the exact values of {t). Where is the catch? At every recursion, we have 0(£) only at a finite set of points: first at the integers, then at the half-integers, then at the quarter-integers, etc. To obtain 0 at the integers, note that for t = n the dilation equation reads 2JV-1
2N-1
0(n) = J2 ck(f)(2n-k)= fc=0
J^
c2n-k4>(k)1
(8.56)
k=0
with the convention that c m = 0 unless 0 < m < 2N — 1. Both sides involve the values of (f> only at the integers; hence (8.56) is a linear equation for the unknowns <j>(n). If (f> is supported in [0, 2N — 1] and is continuous, then only 0(1), 0 ( 2 ) , . . . , (p(2N - 2) are (possibly) nonzero. Thus we have M = 2N - 2 unknowns, and (8.56) becomes an M x M matrix equation, stating that the column vector u = [0(1) • • • <j>(M)\* is an eigenvector of the matrix An^ = Qn-ki n,k — 1 , . . . , Af. The normalization of u is determined by the partition of unity (Proposition 7.9). This gives the exact values of 0 at the integers. From here on, the computation is completely straightforward. At t = n + .5, the dilation equation gives 2JV-1
0(n + .5) -
] T ck
fc),
(8.57)
A Friendly Guide to Wavelets
190
which determines the exact values of <j> at the half-integers, and so on. Any solution of Au = u thus determines <j> at all the diadic rationals t = k/2m (k,m G Z, m > 0), which form a dense set. If <j> is continuous, that determines 0 everywhere. Conversely, any solution of the dilation equation determines a solution of Au — u. Since we already know that the dilation equation has a unique normalized solution, it follows that u is unique as well, i.e., the eigenvalue 1 of A is nondegenerate. Example 8.5: The Scaling Function with N = 2. Taking the minimumphase filter of Example 8.1, we have 9
S
(8.58) where a = (1 + V3)/2 and b = (1 - y/S)/2. This gives l + >/3
co = —
c
i =f
3 + >/3
-, c 2 =
3-V^
-, c 3 =
l-\/3
(8.59)
Au = u becomes C\
[c3
C0 0(1) c2\ L0(2)J
Together with the normalization J2k 0 W
«D = ^ ,
0(1)
(8.60)
L0(2)J* =
1' t m s g i y e s
tne
unique solution
«2) = i ^ .
(8.61)
Hence 0(.5) = ^ c f c ^ ( l - f c ) = c o 0(l) =
2 + N/3
fc=0 3
0(1.5) = J ] cfc0(3 - A;) = ci^(2) + c 2 0(l) = 0
(8.62)
fe=0 3
2-x/3 0(2.5) = ] T ck
The scaling functions 4>N with minimal support (R = 0) and minimal phase are plotted at the end of the chapter. Figures 8.1 through 8.4 show <j>N for N = 2, 3, 4, and 5, along with their wavelets ipN. Note that for N = 2 and 3, <j> and ip are quite rough at the diadic rationals, for example, the integers. As N increases and the support widens, these functions become smoother.
8. Daubechies' Orthonormal Wavelet Bases
191
8.4 The Method of Cumulants Since f dt
/
dt-t(f>(t)
(8.63)
-oo
is the mean time with respect to cf>. That is, the "probability" picture suggests Condition E'\ faf = f{n + r ) , n € Obviously, (8.64) cannot hold for all / G L 2 (R), since / defined values (Section 1.3). For / = , it implies Condition E: 4>[r + n) = fat = <5° , Conversely, (8.65) implies (8.64) for every / £ VQ. For if /
Z. (8.64) does not have welln € Z. (8.65) = u(T)(f), then
f(r + n) = ] T uk
(8.66)
A-
Definition 8.6. A scaling function is exact if rf satisfies Condition E. Note that the Haar scaling function X[o,i)M is exact, with r = .5. There is no obvious reason why (8.65) should hold for any N > 1, and we will see that in general it is only an approximation, although an unexpectedly good one. But for N = 2 and N = 3, there is a surprise: Condition E appears to hold exactly! At the time of this writing, I have only a partial proof of this (see below). Nevertheless, (8.65) is useful for the following reasons: (a) Whether exact or not, Condition E gives a new algorithm for constructing and plotting . This algorithm is a refinement of diadic interpolation, and is somewhat simpler than the latter because no eigen-equation needs to be solved at the beginning. (b) To the extent that it is exact, Condition E gives a simple and direct interpre tation to the filter coefficients cn of (/>: They are samples of at t= (r + n)/2. (c) Research into the validity of Condition E leads to a new method for con structing filter coefficients for orthonormal wavelets, and possibly other multiresolution analyses, based on the statistical concept of cumulants. This method is independent of the validity of Condition E.
A Friendly Guide to Wavelets
192
This section gives a brief account of the cumulant method. The new algorithm for constructing <j> from the filter coefficients begins by interpreting Condition E as defining the initial values of (j) in a recursion scheme: 4>{0)(r + n) = S0n.
(8.67)
This is analogous to finding
4,w(Z±^=J2c^(0)(T
+ n-k) = cn,
(8.68)
and each iteration is obtained from the previous one by
^+1)(^)=E^(m)(^^)-
(^9)
If Condition E holds, then (p^ = (j) for all m > 0 and (8.68) shows that the filter coefficients cn are values of (f> at t = (r + n)/2 . (This gives the c n 's a nice intuitive interpretation!) In order to use this scheme we need to find r in terms of the filter coefficients. We claim that n
which is just a discrete analogue of (8.63) using {c n /2} as a discrete probability distribution on Z. (Recall that ^2n cn = 2.) Equation (8.70) will be seen to be a special case of a relation that leads, as it turns out, to the heart of the matter. Since <j> is integrable with compact support, we can define oo
/
dt est (t) = 4>
-OO
is \
2i)
(8.71)
(Remember that (f)(u>) is an entire function.) Then (8.33) becomes oo
*(«) = r j P ( e ^ ) -
(8-72)
771=1
and the dilation equation reads $(2s) = P(e s )$(s).
(8.73)
Since $(s) is an entire function with $(0) = 1, there is a neighborhood D$ of s = 0 where a branch of log $(s) can be defined. A similar argument shows that
8. Daubechies' Orthonormal Wavelet Bases
193
logP(e s ) is also defined in some neighborhood Dp of s = 0; hence logP (e s / 2 m ) is defined in Dp for all m > 0. Equation (8.72) implies that oo
log $oo = x>g p ( e S / 2 m )
(8.74)
in D$ n Dp. Define the moments of 0 about £ = 0 by oo
/
dttk
k = 0,1,2,....
(8.75)
-OO
In particular, M 0 = 1 and Mi = r. 3>(s) is a generating function for these moments, since #*(«)
s=0
= M fc .
(8.76)
The dilation equation (8.73) gives a recursion relation for the moments that allows their computation from the filter coefficients. But a more direct relation exists in terms of logarithms because of the multiplicative nature of (8.73). Define the continuous and discrete cumulants for k > 0 by
fsrfc = d*io g *(s)
s=0
(8.77)
s=0
respectively. Note that K0 = \og
- r
(8.78) OO
/ -oo
dt(t-r)2
If (j) were a true probability distribution (i.e., (£) > 0), then K n e " s ,
(8.79)
n
and define the discrete moments (of the c n 's) by analogy with (8.76): /*fc = a f P ( e ' ) |
=|Vcnnfe.
(8.80)
A Friendly Guide to Wavelets
194
The discrete cumulants and discrete moments satisfy relations similar to (8.78), the first three of which are K0 = l o g P ( l ) = 0 P'(l) Kl =
= Ml
P(l)
(8
*81)
K2 = ^ 2 - Ml •
This allows us to compute the K'S from the c's, and all that remains is to relate these to the ICs. Theorem 8.7. The continuous cumulants are given in terms of the discrete cumulants by
Kk
k
= wb['
^ °-
(8-82)
Proof: By (8.74),
Kk=J2d*logP(e^m)\
s=0
OO
= Y^2-mkdku\ogP(eu)\ oo
= £*
-mfc „ _ Kk =
u=0
(8.83)
K
k
fc
2 -r ■
m=l
For fc = 1, (8.82) reduces to (8.70), and we can proceed with the modified diadic interpolation scheme. The results for the minimum-phase, minimum-support filters with N = 2, 3, 4, and 5 are shown in Figures 8.1 through 8.4. Figures 8.1 and 8.2 suggest that 2 and 0 3 are exact, while Figures 8.3 and 8.4 show very small deviations from exactness for 4 and (p5. Similar computations for the minimum-phase, minimum-support filters with N = 9 and 10, and other Daubechies filters with nonminimal phase or nonminimal support, show that although obviously not exact, Condition E still gives a good set of initial values leading to rapid convergence. The new derivation of the filter coefficients from cumulants is based on the following result: Theorem 8.8. Let P(z) = ■^^2ncnzn pose P{z) satisfies \P(z)\2 + \P(-z)\2
be an entire function with cn real. Sup = l
for
|s| = l
(8.84)
and has a zero of order N at z = — 1:
P{z) =(?-¥■)
W(z),
(8.85)
8. Daubechies' Orthonormal Wavelet Bases
195
with W(z) entire. Then K2k
= d2klogP(es)\
= 0,
0
(8.86)
\s—0
Proof: Since the c n 's are real, P{z) = P(z). Equation (8.84) implies that P(es)P(e-s)
+ P(-es)P(-e-s)
= 1 for all s e C.
(8.87)
(For s = —2TTIUJ with u real, (8.87) reduces to (8.84); since the left-hand side of (8.87) is an entire function of s and equals 1 on the imaginary axis, it has to equal 1 identically for all s.) But ^ ( - e
s
) = ( ^ )
W(~es).
(8.88)
Since 1
-^
= sU(s)
(8.89)
where U(s) is entire, (8.88) gives P(-es)
= sN U{s)NW(-es)
= sNV{s),
(8.90)
with V(s) entire. Therefore P{es)P{e~s) = 1 -
P{-es)P{-e~s)
= l-s2N(-l)NV(s)V(-s) = 1+
(8.91)
2N
s F(s)
for some entire function F(s). Now for |iy| < 1, log(l + w) = w-
±w3 + ±w3
= wH(w),
(8.92)
where H(w) is analytic in |iu| < 1. Therefore there exists a neighborhood D of 5 = 0 where log [P(es)P(e-s)]
= s2NF(s)
H (s2NF(s))
= s2NG{s),
(8.93)
with G(s) analytic in D. Recall that we can also define a branch of logP(e s ) in a neighborhood Dp of s = 0. We may assume that Dp is circular, so that —s£ Dp whenever s £ Dp. Then log P(es) + log P(e~s) = s2NG{s)
(8.94)
in D H Dp. Now apply dk and set s — 0. The right-hand side vanishes because it still has factors of s left after the differentiation. Hence by the definition of Kfc + (-l)knk = 0, k < 2N. (8.95) For odd fc, (8.95) is an identity; for k = 0, 2 , 4 , . . . , 2AT-2, it reduces to (8.86). ■
A Friendly Guide to Wavelets
196
Therefore, a necessary condition for orthonormality is t h a t the first N even, discrete cumulants K2k vanish. Then Theorem 8.7 implies t h a t the corresponding continuous cumulants also vanish. In particular, K2 = 0 and our "probability distribution" (£) must have a vanishing "variance"! Theorem 8.8 can be used as a basis for the derivation of orthonormal multiresolution filters. For example, if (j> is to have a minimal compact support for a given value of A7", then W(z) in (8.85) must be a polynomial of degree N — 1, which means it has TV coefficients. But (8.86) gives TV equations for these coef ficients, so we can find the c n ' s by solving (8.86). For N = 2 and N = 3, this gives the Daubechies filter coefficients. For an orthonormal, minimal-support filter, the first N even cumulants thus determine the filter coefficients (by vanishing). The odd cumulants, on the other hand, play an important role in relation to exactness. It can be shown (Kaiser [1994d]) t h a t if a filter satisfies (8.85), then 7V> 1 => ^4>(r
+ n) = 1
n
N>2
N>3
=> ^ n < / > ( r + n ) = 0
(8-96)
A
=* 22n2(l)(T + n) = 0 n
A T > 4 =► ^ n 3 ^ ( r + n) = Ar3 =
y ,
n
with similar equations for N > 5,iV > 6, etc. For any N > 1, we may regard the 2N — 1 values of (p(r + n) in the support of <j> as unknowns. Then (8.96) gives N equations for these unknowns, leaving N — 1 degrees of freedom. The case N = 1 is trivial, and implies t h a t the Haar scaling function is exact. For N = 2, we find r = .634."'' T h e first two equations in (8.96) are consistent with exactness for cf)2 but not quite enough to prove it, since there are three unknowns (r), (r + l ) , and <j)(r + 2). For N = 3 and minimum phase, we have r = .817. T h e first three equations are again consistent with exactness, but now there are five unknowns (r),..., <j>{r + 4), so we are again short of a proof. The fourth equation proves t h a t (fiN cannot be exact for N > 4, unless K3 — 0. But K3 = —.109 for (ft4 (minimum-phase), so this filter is not exact. T h e fact t h a t it is close t o being exact, as evidenced by Figure 8.3, is consistent with the small value of K3/7 = —.0156 in (8.96). Higher-order odd cumulants further spoil the exactness of higher-order filters. One might say t h a t they are obstructions to exactness. ' This is for the minimum-phase filter; for the maximum-phase filter we have (£) —* (j)(2N - 1 - t), so T -► 2JV - 1 - r = 2.366.
8. Daubechies' Orthonormal Wavelet Bases
197
Conjecture 8.9. (j)2 and (f)3 are exact. Equations (8.96) can be derived from the Poisson summation formula in com bination with (8.7); they are related to the beautiful Strang-Fix approximation theory (Strang and Fix [1973]; Strang [1989]), which has been generalized re cently by de Boor, DeVore, and Ron (1994). 8.5 Plots of Daubechies Wavelets Figures 8.1 through 8.4 show the results of computing the minimal-support, minimal-phase Daubechies scaling functions and wavelets by the cumulant algo rithm, using four iterations. The initial values of <j> are marked with o, and the first three iterations are marked with *, x , and + . The fourth iteration for <j> is a solid curve, and the wavelet ip as computed from the third iteration of <j> is a dashed curve. The vertical dotted line is t = r. Exercises 8.1. Verify that P(z), defined by (8.6), satisfies the orthonormality condition (8.2) if W(z) is a solution of (8.16), with WN{S) given by (8.15). 8.2. Prove (8.23). 8.3. Let (j> be the scaling function (8.41). For any M G N , let u^n = 1, W3n+i = — 1, n = 0 , 1 , . . . , M — 1, with all other u^ = 0. Compute and compare J2k \uk\2 and || ^2k Uk4>k\\2 ■ What happens a s M - > oo? Explain why 0 is "bad." 8.4. Find the wavelet corresponding to the scaling function (8.41). 8.5. Compute the next recursion for (j> from (8.62), i.e., find 0(.25),0(.75), ...,0(2.75). 8.6. Use a computer to find and plot 4>2{t) by the method of Section 8.2: Com pute <j> to some accuracy, then take its inverse Fourier transform. (Exper iment with the accuracy.) 8.7. Use a computer to find and plot 4>2{t) by diadic interpolation, to the scale At = 2~ 4 . 8.8. Plot the minimum-phase, minimum-support scaling function (j)10(t) by the method of cumulants, using three iterations. (The coefficients hn = c n / \ / 2 are listed in Daubechies (1992), p. 195.)
198
A Friendly Guide to Wavelets
-1.5 0
0.5
1
1.5
2
2.5
2
3
2
Figure 8.1. The scaling function
curve).
2 1.5-
-1.5 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Figure 8.2. The scaling function
5 curve).
8. Daubechies'
Orthonormal r
2i
Wavelet
Bases
1
199
1
IXlXIX I
-1.5 0
1
2
3
4
4
5
6 4
Figure 8.3. The scaling function 0 (solid curve) and wavelet ip (dashed r
T—
n
7 curve).
r
1.5
xi jycKi xfa xi k y i m xi I*CK\ m xi »o
i
.
i_
_i
i_
"1'50 1 2 3 4 5 6 7 8 Figure 8.4. The scaling function <j>5 (solid curve) and wavelet ip5 (dashed
9 curve).
PART II
Physical Wavelets
Chapter 9
Introduction to Wavelet Electromagnetics Summary: In this chapter, we apply wavelet ideas to electromagnetic waves, i.e., solutions of Maxwell's equations. This is possible and natural because Maxwell's equations in free space are invariant under a large group of symme tries (the conformal group of space-time) that includes translations and dila tions, the basic operations of wavelet theory. The structure of these equations also makes it possible to extend their solutions analytically to a certain domain T C C 4 in complex space-time, which renders their wavelet representation par ticularly simple. In fact, the extension of an electromagnetic wave to complex space-time is already its wavelet transforms, so no further transformation is needed beyond analytic continuation! The process of continuation uniquely determines a family of electromagnetic wavelets tyz, labeled by points z in the complex space-time T. These points have the form z = x + zy, where x G R 4 is a point in real space-time and y belongs to the open cone V' consisting of all causal (past and future) directions in R 4 . S&z itself is a matrix-valued so lution of Maxwell's equations, representing a triplet of electromagnetic waves. This triplet is focused at the real space-time point x — Rez £ R 4 , and its scale, helicity, and center velocity are vested in the imaginary space-time vec tor y = Imz € V. An arbitrary electromagnetic wave can be represented as a superposition of wavelets \£ z parameterized by certain subsets of T, for ex ample, the set E of all points with real space and imaginary time coordinates called the Euclidean region. Such wavelet representations make it possible to design electromagnetic waves according to local specifications in space and scale at any given initial time. Prerequisites: Chapters 1 and 3. Some knowledge of electromagnetics, complex analysis, and partial differential equations is helpful in this chapter.
9.1 S i g n a l s a n d W a v e s Information, like energy, has the remarkable (and very fortunate!) property t h a t it can repeatedly change forms without losing its essence. A "signal" usually refers to encoded information, such as a time-variable amplitude or frequency modulation imprinted on an electromagnetic wave acting as a "carrier." T h e information is taken for a "ride" on the wave, at the speed of light, to be decoded later by a receiver. Many of the most important signals do, in fact, have to ride an electromagnetic wave at some stage in their journey. But this ride may not be entirely free, since electromagnetic waves are subject to the physical laws of propagation and scattering. The effects of these laws on communication are often ignored in discussions of signal analysis and processing. But the physical G. Kaiser, A Friendly Guide to Wavelets, Modern Birkhäuser Classics, DOI 10.1007/978-0-8176-8111-1_9, © Gerald Kaiser 2011
203
204
A Friendly Guide to Wavelets
laws are, in some sense, a code in themselves - in fact, an unchangeable, absolute code. Therefore it would seem that an understanding of the interaction between the two codes (the physical laws and the encoded information) can be helpful. In the transmission phase, it may suggest a preferred choice of both carrier and code. In the reception phase, it may suggest better decoding methods that take full advantage of the nature of the carrier and use it to understand the precise impact of the "medium" on the "message." This is especially important when the medium is the message, as it can be, for example, in optics and radar. In these cases, the initial signal is often trivial (for example, a spot light) and the information sought by the receiver is the distribution and nature of reflectors, absorbers, and so on. In order to arrange a marriage between signal analysis and physics, it is first necessary to formulate both in the same language or code. For signals, this means any choice of scheme for analysis and synthesis. For electromagnetic waves, the "code" consists of the laws of propagation and scattering, so we have little choice. It therefore makes sense to begin with the physics and try to reformulate electromagnetics in a way that is most suitable for signal analysis. The two themes for signal analysis explored in this book have been time-frequency and time-scale analysis. In this chapter we will see that the propagation of elec tromagnetic waves can be formulated very naturally in terms of a generalized time-scale analysis. In fact, the laws of electromagnetics uniquely determine a family of wavelets that are specifically dedicated to electromagnetics. These wavelets already incorporate the physical laws of propagation, and they can be used to construct arbitrary electromagnetic waves by superposition. Since the physical laws are now built into the wavelets, we can concentrate on the "message" in the synthesis stage, i.e., when choosing the coefficient function for the superposition. The analysis stage consists of finding a coefficient function for a given wave; again this isolates the message by displaying the informational contents of the wave. Time-scale analysis is natural to electromagnetics for the following funda mental reason: Maxwell's equations in free space are invariant under a group of transformations, called the conformal group C, which includes space-time translations and dilations. That is, if any solution is translated by an arbitrary amount in space or time, then it remains a solution. Similarly, a solution can be scaled uniformly in space-time and still remain a solution. Since these are pre cisely the kind of operations used to construct wavelet analysis, it is reasonable to expect a similar analysis to exist for electromagnetics. The other ingredient in the construction of ordinary wavelet analysis is the Hilbert space L 2 (R), chosen somewhat arbitrarily. Translations and dilations were represented by unitary operators on L 2 (R), and the wavelets formed a (continuous or discrete) frame in L 2 (R). In the case of electromagnetics, we need a Hilbert space 7i consisting
9. Introduction to Wavelet
Electromagnetics
205
of solutions of Maxwell's equations instead of unconstrained functions (such as those in L2(R)). But we must not blindly apply the "hammer" of wavelet analysis to this new "nail" Hj since this nail actually (and not surprisingly) turns out to have much more structure than L 2 (R). It is, in fact, more like a screw than a nail! The ex tra structure is related to the other symmetries contained in the conformal group C. Besides translations and dilations, C contains rotations, Lorentz transforma tions, and special conformal transformations. In fact, these transformations taken together essentially characterize free electromagnetic waves: It is possible to begin with C and construct H as a representation space for C, where con formal transformations act as unitary operators (Bargmann and Wigner [1948], Gross [1964]). When constructing a wavelet-style analysis for electromagnetics, it is important to keep in mind that most of the standard continuous analysissynthesis schemes are completely determined by groups. In Fourier analysis, the group consists of translations (Katznelson [1976]). In windowed Fourier analy sis, it is the Weyl-Heisenberg group W consisting of translations, modulations and phase factors (Folland [1989]). In ordinary wavelet analysis, it is the affine group A consisting of translations and dilations (Aslaksen and Klauder [1968, 1969]). Since both W and A contain translations, the associated analyses are related to Fourier analysis but contain more detail. The extra detail appears in the parameters labeling the frame vectors: The (unnormalizable) "frame vec tors" ew{t) = e2ntut in Fourier analysis are parameterized only by frequency; the frame vectors gWit of windowed Fourier analysis are parameterized by fre quency and time, and the frame vectors tpSit of ordinary wavelet analysis are parameterized by scale and time. Since the conformal group C contains A, we expect that the electromagnetic wavelet analysis will be related to, but more detailed than, the standard wavelet analysis. In fact, its frame vectors will be seen to be electromagnetic pulses *&z parameterized by points z = x + iy in a complex space-time domain T C C 4 , thus having eight real parameters. The real space-time point x = (x, t) G R 4 represents the focal point of Sfrz. This means that at times prior to t, \£ 2 converges toward the space point x, and at later times it diverges away from x. At time £, \I/ z is localized around x £ R 3 . Since it is a free electromagnetic wave, it does not remain localized. But it does have a well-defined center that is described by the other half of z, the imaginary space-time vector y — (y, s). y must be causal, meaning that |y| < c|s|, where * I am thinking of the wonderful saying: If the only tool you have is a hammer, then everything looks like a nail. Until recently, of course, the only tool available to most signal analysts was the Fourier transform or its windowed version, so everything looked like either time or frequency. Now some wavelet enthusiasts claim that everything is either time or a scale.
206
A Friendly Guide to Wavelets
c is the speed of light, and it has the following interpretation: v = y / s is the velocity of the center of \£ z , and \s\ is the time scale (duration) of \I/ z . Equivalently, c\s\ is the spatial extent of \PZ at its time of localization t. The sign of the imaginary time 5 is interpreted as the helicity of *$>z, which is related to its polarization. In Section 9.2 we solve Maxwell's equations by Fourier analysis. Solutions are represented as superpositions of plane waves parameterized by wave number and frequency. We define a Hilbert space H of such solutions by integration over the light cone in the Fourier domain. In Section 9.3 we introduce the analyticsignal transform (AST). This is simultaneously a generalization to n dimensions of two objects: the ordinary wavelet transform, and Gabor's idea of "analytic signals." The AST extends any function or vector field from R n to C n , and we call the extended function the analytic signal of the original function. In general, such analytic signals are not analytic functions since there may not exist any analytic extensions. We prove that the analytic signal of any solution in H is analytic in a certain domain T c C 4 , the causal tube. This analyticity is due to the light-cone structure of electromagnetic waves in the Fourier domain, just as the analyticity of Gabor's one-dimensional analytic signals is due to their having only positive-frequency components. In Section 9.4 we use the analytic signals of solutions to generate the electromagnetic wavelets \I>Z. We derive a resolution of unity in H in terms of these wavelets, which gives a means of writing an arbitrary electromagnetic wave in H as a superposition of wavelets. In Section 9.5 we compute the wavelets explicitly, which also gives their reproducing kernel. In Section 9.6 this explicit form of ^z is used to give the wavelet parameters z =■ x + iy a complete physical and geometric interpretation, as explained above. In Section 9.7 we show how general electromagnetic waves can be constructed from wavelets, with initial data specified locally and by scale. The wavelet analysis of electromagnetics is based entirely on the homoge neous Maxwell equations. Acoustic waves, on the other hand, obey the scalar wave equation, which has a very similar structure to Maxwell's equations in that its symmetry group is the same conformal group C. Therefore a similar wavelet analysis exists for acoustics as well as for electromagnetics. In the language of groups, the two analyses are just different representations of C. Acoustic wavelet representations and the corresponding wavelets ^ J are studied in Chapter 11.
9.2 Fourier Electromagnetics An electromagnetic wave in free space (without sources or boundaries) is de scribed by a pair of vector fields depending on the space-time variables x = (x, XQ), namely, the electric field E(x) and the magnetic field B(x). (Here x is
9. Introduction to Wavelet
Electromagnetics
207
t h e position and XQ = t is t h e time.) These are subject to Maxwell's equations, V x E + <90B = 0,
V • E = 0,
V x B - 5 b E = 0,
V • B = 0,
where V is t h e gradient with respect to t h e space variables and do is t h e time derivative. We have set t h e speed of light c = 1 for convenience. T h e parameter c m a y be reinserted at any stage by dimensional analysis. In keeping with the style of this book, we are not concerned here with t h e exact nature of t h e "functions" E ( x ) and B ( x ) (they may be regarded as tempered distributions; see Kaiser [1994c]). Note t h a t t h e equations are symmetric under t h e linear mapping defined by J : E — i ► B , B H —E, and t h a t J 2 is minus t h e identity m a p . This means t h a t J is a "90°-rotation" in t h e space of solutions, analogous to multiplication by i in t h e complex plane. Such a mapping on a real vector space is called a complex structure. T h e combinations B ± i E diagonalize J , since J ( B ± « E ) = ±i ( B ± iE). They each m a p Maxwell's equations to a form in which t h e concepts of helicity and polarization become very simple. It will suffice to consider only F = B + i E , since t h e other combination is equivalent. Equations (9.1) then become d0F = zV x F ,
V • F = 0.
(9.2)
Note t h a t t h e first of these equations is an evolution equation (initial-value problem), while t h e second is a constraint on t h e initial values. Taking t h e divergence of t h e first equation shows t h a t t h e constraint is conserved by time evolution. Note also t h a t it is t h e factor i in (9.2) (i.e., t h e complex structure) t h a t couples t h e time evolution of t h e electric field E to t h a t of t h e magnetic field B . Equation (9.2) implies -<9 2 F = V x (V x F ) = V ( V • F ) - V 2 F = - V 2 F
(9.3)
in Cartesian coordinates, hence t h e components of F become decoupled and each satisfies t h e wave equation D F E E ( - d g + V 2 ) F = 0.
(9.4)
To solve (9.2), write F as a Fourier transform* F(x)
= (27T)"4 /
dApeipxF(p),
(9.5)
t In this chapter we change our convention for Fourier transforms. The four-vector p now denotes wave number and frequency in radians per unit length or unit time. Also, we are identifying R 4 with its dual R4 by means of the Lorent-invariant inner product, so the pairing between x and p is expressed as an inner product. See the discussion in Section 1.4.
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where p — (p,Po) € R 4 with p G R 3 as the spatial wave vector and po as the frequency. We use the Lorentz-invariant inner product p • x = poxo - p • x. (Note also t h a t we have reverted to the convention for the Fourier transform with the 2TT factors in front of integrals instead of in the exponents. This is the convention used in the physics literature.) T h e wave equation (9.4) implies t h a t p2F(p) = 0, where 2
2
i
i2
=P'P = P0-\P\ • Thus F must be supported on the light cone C = {p : p2 = 0,p / 0}* shown in Figure 9.1, i.e., on P
C = {(p,po) 6 R 4 : Pi = | p | 2 , P ^ 0} = C+ U C. C+ = {(p,po) : Po = | p | > 0} = positive-frequency light cone :
C_ = {(p,po) Po — ~|p|
< 0}
=
(9.6)
negative-frequency light cone.
p0 (frequency)
p-plane
F i g u r e 9 . 1 . The positive- and negative-frequency cones V± and their bound aries, and the positive- and negative-frequency light cones C± = dV±. Hence F has the form F(p) = 27r<5(p2)f(p), where f : C -* C
3
is a function on C or, equivalently, the pair of functions on
R 3 given by f±(p) = f ( p » i ^ ) j
where
u = |p|
t We assume that the fields decay at infinity, so F(p) contains no term proportional to 6{p), i.e., F(x) has no "DC component." To eliminate this possibility, we exclude p = 0 from the light cone.
9. Introduction to Wavelet Electromagnetics
209
is the absolute value of the frequency. But Stf)
= S((P0 -
w)(po
+
w))
= % ° - « H - ' ( * » + "> .
(9.7)
Hence (9.5) becomes F(x) = ( 2 ^ r 3 /
^ H . e -ip-x [ e -t
f + ( p ) + e-ict f _ ( p ) ]
(9.8) ipx
= [ Jc
dpe f(p),
where dp = (27r)" 3 d 3 p/2u; is a Lorentz-invariant measure on C. In order for (9.8) to give a solution of (9.2), f (p) must further satisfy the algebraic conditions ipof(p) = p x f ( p ) ,
p-f(p) = 0
(9.9)
for all p G C , and the first of these equations suffices since it implies the second. Let
n(p) = £ ,
so that p e C if and only if |n(p)| — 1. Define the operator T = T(p) on arbitrary functions g : C —► C 3 by (r g )(p) = - i n(p) x g(p),
pGC.
(9.10)
r(p) is represented by the Hermitian matrix T(p) = -L
0 -P3
L P2
p3 0
-p2] P! ,
-PI
o J
(9.11)
with matrix elements T m n (p) = zp^ 1 £) f c = 1 £mnkPk , where £mnfc is the totally antisymmetric tensor with £123 = 1. In terms of T(p), (9.9) becomes rf(p) = f(p).
(9.12)
r 2 g = - n x (n x g) = g - (n • g ) n ,
(9.13)
Now for any g : C -» C 3 ,
so T(p) 2 is the orthogonal projection to the subspace of C 3 orthogonal to n(p), and it follows that r3g = rg. (9.14) The eigenvalues of T(p), for each p 6 C, are therefore 1,0 and —1, and (9.12) states that f(p) is an eigenvector with eigenvalue 1. Since T(p) = —T(p), (9.12) implies that r f = — f. A similar operator was defined and studied in much
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210
more detail by Moses (1971) in connection with fluid mechanics as well as elec trodynamics. Consider a single component of (9.8), i.e., the plane-wave solution Fp(x) = e*'* f (p) = Bp(x) + i E p ( x ) ,
(9.15)
with arbitrary but fixed p G C and f (p) ^ 0. The electric and magnetic fields are obtained by taking the real and imaginary parts. Now T(p) f (p) = f (p) and r(p)f(p) = - f 5 ) imply T(p)Fp(x)
= F p (x),
r(p)F p (aO = - F p ( x ) .
(9.16)
Since T(p)* = T(p), these eigenvectors of T(p) with eigenvalues 1 and —1 must be orthogonal: Fp(x)*Fp(x) = Fp(x) • Fp(x) = 0, where the asterisk denotes the Hermitian transpose. Taking real and imaginary parts, we get Bp(x) ■ Ep{x) = 0. (9.17) |B p (x)| 2 = |E p (z)| 2 , The first equation shows that neither B p (z) nor E p (x) can vanish at any x (since f(p) zfz 0). Furthermore, (9.16) implies that pxEp(rr)
=p0Bp(x).
Thus, for any x, {p, E p (x),B p (:r)} is a right-handed orthogonal basis if po > 0 (i.e., p G C+) and a left-handed orthogonal basis if po < 0 (p G C_). Taking the real and imaginary parts of (9.15) and using f(p) = B p (0) + zE p (0), we have B p (x) = cos(p • x)B p (0) - sin(p • x)E p (0),
^
Ep(x) = cos(p • x)Ep(0) + sin(p • x)B p (0). An observer at any fixed location x G R 3 sees these fields rotating as a function of time in the plane orthogonal to p. If p G C+, the rotation is that of a righthanded corkscrew, or helix, moving in the direction of p, whereas if p G C_, it is that of a left-handed corkscrew. Hence Fp(x) is said to have positive helicity if p G C+ and negative helicity if p G C_. A general solution of the form (9.8) has positive helicity if f(p) is supported in C+ and negative helicity if f(p) is supported in C_. Other states of polar ization, such as linear or elliptic, are obtained by mixing positive and negative helicities. The significance of the complex combination F(x) — B(x) + iE(x) therefore seems to be that, in Fourier space, the sign of the frequency po gives the helicity of the solution! (Usually in signal analysis, the sign of the frequency is not given any physical interpretation, and negative frequencies are regarded as a necessary mathematical artifact due to the choice of complex exponentials over sines and cosines.) In other words, the combination B + zE "polarizes"
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Electromagnetics
211
t h e helicity, with positive and negative helicity states being represented in C+ and C- , respectively. Had we used the opposite combination B — i E , C+ and C- would have parameterized the plane-wave solutions with opposite helicities. Nothing new seems to be gained by considering this alternative.t In order to eliminate the constraint, we now proceed as follows: Let
n(p) = | [r(p) + r 2 ( P )].
(9.19)
Explicitly, -PlP2 + «P0P3
Po-Pi
n(p) =
0
o
-P1P2 - IP0P3 L ~PlP3 + ^P0P2
~PlP3 ~ IP0P2
PI-PI
-P2P3 + IPOPI
(9.20)
PI-PI
-P2P3 - *PoPl
T h e established properties T* =T = T3 imply t h a t II* = II = I I 2 and T i l = II, which proves t h a t II(p) is the orthogonal projection to eigenvectors of T(p) with eigenvalue 1. Thus, to satisfy the constraint (9.12), we need only replace the constrained function f (p) in (9.8) by II(p)f (p), where f (p) is now unconstrained. P r o p o s i t i o n 9 . 1 . Solutions tation:
of Maxwell's
have the following
represen
(9.21)
Jc with f : C —* C 3
Fourier
unconstrained.
Consequently, t h e mapping f — i ► F is not one-to-one since II is a projection operator. In fact, f is closely related to the potentials for F , which consist of a real 3-vector potential A ( x ) and a real scalar potential AQ{X) such t h a t B = V x A,
E = -d0A
-
VA0.
(9.22)
T h e combination ( A ( x ) , Ao(x)) is called a "4-vector potential" for the field. We can assume without loss of generality t h a t the potential satisfies the Lorentz condition (Jackson [1975]) V • A + d0A0 = 0. * Maxwell's equations are invariant under the continuous group of duality rotations, of which the complex structure J mapping E to B and B to —E is a special case. In the complexified solution space, the combinations B ± i E form invariant subspaces with respect to the duality rotations. That gives the choice of B + *E an interpretation in terms of group representation theory.
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212
Since A and Ao also satisfy the wave equation (9.4), they have Fourier repre sentations similar to (9.8): A(x) = f dp eipx8i{p), A0(x) = f dp eipxa0(p). (9.23) Jc Jc The Lorentz condition means that poao(p) = p • a(p), or ao(p) = n • a(p), so ao is determined by a. Equations (9.22) will be satisfied provided that the Fourier representatives e(p)1 b(p) of E , B satisfy b = - i p x a = p0ra,
e = -ipoa + i pa 0 = —ipo T2 a.
(9.24)
Hence F = B + i E is represented in Fourier space by b(p) + ie(p) = po [T(p) + T(p)2] a(p) = 2p0U(p) a(p).
(9.25)
This shows that we can interpret the unconstrained function f(p) in (9.21) as being directly related to the 3-vector potential by f(p) = 2poa(p),
(9.26)
modulo terms annihilated by II (p), which correspond to eigenvalues —1 and 0 of T(p). Viewed in this light, the nonuniqueness of f in (9.21) is an expression of gauge freedom in the B + i E representation, as seen from Fourier space. In the space-time domain, (B, E) are the components of a 2-form F in R 4 and (A, AQ) are the components of a 1-form A. Then equations (9.1) become dF = 0 and SF = 0 (where 6 is the divergence with respect to the Lorentz inner product), Equations (9.22) become unified as F = dA, the Lorentz condition reads 8A = 0, and the gauge freedom corresponds to the invariance of F under A —> A + dx, where x(x) i s a scalar solution of the wave equation. Maxwell's equations are invariant under a large group of space-time trans formations. Such transformations produce new solutions from known ones by acting on the underlying space-time variables (possibly with a multiplier to ro tate or scale the vector fields). Some trivial examples are space and time transla tions. Obviously, a translated version of a solution is again a solution, since the equations have constant coefficients. Similarly, a rotated version of a solution is a solution. A less obvious example is Lorentz transformations, which are in terpreted as transforming to a uniformly moving reference frame in space-time. (In fact, it was in the study of the Lorentz invariance of Maxwell's equations that the Special Theory of Relativity originated; see Einstein et al. (1923).) The scale transformations x —* ax, a ^ 0, also map solutions to solutions, since Maxwell's equations are homogeneous in the space-time variables. Finally, the equations are invariant under "special conformal transformations" (Bateman [1910], Cunningham [1910]), which can be interpreted as transforming to a uniformly accelerating reference frame (Page [1936]; Hill [1945, 1947, 1951]).
9. Introduction to Wavelet Electromagnetics
213
Altogether, these transformations form a 15-dimensional Lie group called the conformal group, which is locally isomorphic to 5C7(2, 2) and will here be de noted by C. Whereas wavelets in one dimension are related to one another by translations and scalings, electromagnetic wavelets will be seen to be related by conformal transformations, which include translations and scalings. (A study of the action of SU(2, 2) on solutions of Maxwell's equations has been made by Riihl (1972).) To construct the machinery of wavelet analysis, we need a Hilbert space. That is, we must define an inner product on solutions. It is important to choose the inner product to be invariant under the largest possible group of symmetries since this allows the largest set of solutions in H to be generated by unitary transformations from any one known solution. (In quantum mechanics, invari ance of the inner product is also an expression of the fundamental invariance of the laws of nature with respect to the symmetries in question.) Let f (p) satisfy (9.12), and let a(p) be a vector potential for f satisfying the Lorentz condition so that the scalar potential is determined by ao(p) — n(p) • a(p). By (9.25), |f (p)\2 = 4p2 \U(p) a(p)| 2 = 4a;2 a(p) • U(p) a(p) = 2UJ2 a(p) • T(p) a(p) + 2LJ2 a(p) • T(p)2
a(p).
(9.27)
The first term is -2icj2 a(p) • (n x a(p)) = 2iu2 n • (a(p) x a(p)),
(9.28)
which cancels its counterpart with p —> — p on account of the reality condition a (~~p) — a (p)- Thus % c w v
|f (p)| 2 = 2 f dp Kp)- [a(p) - n(n • a(p))] Jc Jc = 2 I dp
Jc
(9.29)
|a(p)| 2 -|ao(p)|^
The integrand in the last expression is the negative of the Lorentz-square of the 4-potential (a(p),ao(p)). Consequently, the integral can be shown to be invariant under Lorentz transformations. (Note that |a| 2 — |ao| 2 > 0, vanishing only when a(p) is a multiple of p, in which case f = 0. This corresponds to "longitudinal polarization.") Hence (9.29) defines a norm on solutions that is invariant under Lorentz transformations as well as space-time translations. In fact, the norm (9.29) is uniquely determined, up to a constant factor, by the requirement that it be so invariant. Moreover, Gross (1964) has shown it to be invariant under the full conformal group C. Again we eliminate the constraint by replacing f (p) with II(p) f(p). Thus, let Ji be the set of all solutions F(x)
A Friendly Guide to Wavelets
214
defined by (9.21) with f : C —► C 3 square-integrable in the sense that
||F|| 2 =/§|n(p)f( P )| 2 = (2T)_3 f |n(p)f(p)|2
*
i ; l^
-
(9-3°)
7i is a Hilbert space under the inner product obtained by polarizing the norm (9.30) and using (Ilf)*IIg = f*ITIIg = f*IIg. Definition 9.2. H is the Hilbert space of solutions (9.21) with inner product (P,G>=/^-f(prn(p)g(p).
(9.31)
7i will be our main arena for developing the analysis and synthesis of solutions in terms of wavelets. Note that when (9.12) holds and f±(p) = f (p, ±CL>) are continuous, then f(0) = f±(0) = 0 (9.32) must hold in order for (9.30) to be satisfied. This is a kind of admissibility condi tion that applies to all solutions in H: There must not be a "DC component." In fact, the measure d 3 p / | p | 3 is a three-dimensional version of the measure df/|£| that played a role in the admissibity condition of one-dimensional wavelets. To show the invariance of (9.30) under conformal transformations, Gross derived an equivalent norm expressed directly in terms of the values of the fields in space at any particular time XQ = t:
\\F\\k0ss = 4 / £ ^ J - F(x,t)'F(y,t).
(9.33)
" J R 6 I X yi The right-hand side is independent of t due to the invariance of Maxwell's equa tions under time translations (which is, in turn, related to the conservation of energy). A disadvantage of the expression (9.33) is that it is nonlocal since it uses the values of the field simultaneously at the space points x and y. In fact, it is known that no local expression for the inner product can exist in terms of the field values in (real) space-time R 4 (Bargmann and Wigner [1948]). In Section 4, we derive an alternate expression for the inner product directly in terms of the values of the electromagnetic fields, extended analytically to com plex space-time. This expression is "local" in the space-scale domain (rather than in space alone). But first we must introduce the tool that implements the extension to complex space-time.
9. Introduction to Wavelet Electromagnetics
215
9.3 The Analytic-Signal Transform We now introduce a multidimensional generalization of the wavelet transform that will be used below to analyze electromagnetic fields. To motivate the con struction, imagine that we have a (possibly complex) vector field F : R 4 —► C m on space-time. For example, F could be an electromagnetic wave (in which case m = 3 and F is given by (9.21)), but this need not be assumed. Suppose we want to measure* this field with a small instrument, one that may be regarded as a point in space at any time. Suppose further that the impulse response of the instrument is a (possibly) complex-valued function h : R —► C. If the support width of h is small, then we can assume that the velocity of the instrument is approximately constant while the measurement takes place. Then the trajectory on the instrument in space-time may be parameterized as x(r) = x + ry,
i.e.,
(
x(r) = X +
^
(9.34)
where x = (x, xo) G R 4 is the "initial" point and y = (y, yo) is "velocity." The actual three-dimensional velocity in space is v = 1- ,
(9.35)
i.e.,
(9.36)
so we require |v|
\y\
We model the result of measuring the field F with the instrument h, moving along the above trajectory, as co
/
drh(r)F(x
+ ry).
(9.37)
-co
This is a multidimensional generalization of the wavelet transform (Kaiser [1990a, b]; Kaiser and Streater [1992]). To see this, suppose for a moment that we replace space-time by time alone, and the field is scalar-valued, i.e., F : R —> C. Then x,y G R and (9.36) reduces to y ^ 0. By changing variables in (9.37), the transform becomes
Fh(x,y) = J°°duy-h (j—^\ F(u).
(9.38)
* I do not claim that this is a realistic description of electromagnetic field measure ment. It is only meant to motivate the generalized wavelet transforms developed in this section by relating them to the standard wavelet transforms.
A Friendly Guide to Wavelets
216
This is recognized as essentially identical to the usual wavelet transform, with the wavelets now represented by W « ) ~ ^ ( ~ ) -
(9-39)
(The symbol ~ instead of equality will be explained below.) In the one-dimen sional case treated in Chapter 3, we assumed, without any particular reason, that F belonged to L 2 (R). Inspired by the electromagnetic norm (9.30), let us now suppose instead that F belongs to the Hilbert space Ha = {F : \\F\\l < oo}, where \\F\\l = ± | ~ ^
|F(p)| 2
(9.40)
for some a > 0. The inner product in Ha is obtained, as usual, by polarizing the expression for ||F||^. Battle (1992) calls H.2a a "massless Sobolev space of degree a." We now derive a reconstruction formula for this case to prepare ourselves for the more complicated case of electromagnetics. The main points of interest in this "practice run" are the modifications forced on the definition of the wavelets and the admissibility condition by the presence of the weight factor \p\~a in the frequency domain. Since
jye~tpu¥\h(^)=e~ipxHpy)'
(9 4i)
-
Parseval's formula applied to (9.38) gives CO
/
_
dp e^
h(py)F(p)
-oo
= Wl fl ^ \p\ae'px h(py)F(p)
= \ 'lx,y i r /a
=
(9 42)
-
"-ay r ,
where Kv(p) = \v\ae-ipxh{vy)This shows that the wavelets are given in the time domain by 1 C°°
hXtV(u) = — y_ dp |Pre*<"-*> h(py).
(9-43)
(9.44)
If we take a = 0, then Ha = L2(R) and (9.44) reduces to (9.39). For a > 0, (9.39) is not valid. (That explains the symbol ~ in (9.39).) Now let us see how to reconstruct F from its transform Fh- Applying Plancherel's formula to (9.42) gives /»oo
oo
/ -oo
dx \Fh(x,y)f
= (27T)-1 / J — oo
dp \h(py)\2\F(p)f.
(9.45)
9. Introduction to Wavelet Electromagnetics
217
Integrating both sides over y with the weight function |?/| Q _ 1 gives rOO
OO
/
dy\yrl\Fh(x,y)\2
dx / ~oo
J — oo
= (27T)"1 f°
dp |F(p)| 2 f ° dy I J / I - 1 Iftfpy)!2
J—oo
( 9 - 46 )
J— oo
where we have changed y to £ = py, and let oo
/
<*£ K M W .
(9.47)
-oo
which now is assumed to be finite. Therefore we define an inner product for transforms by /•OO
OO
/ Ot
dx f
/
,—roo
dy\y\°-1Fh(x,y)Gh(x,y)
/
r dx /
di/M^F'^.y^G,
J — oo J — oo so that (9.46) becomes a Plancherel-type relation ||i^|| 2 = ||^||a? ization, (Fh,Gh)=F*G. This means that (9.48) gives a resolution of unity:
dx / -oo
an
d by polar (9.49)
/"OO
OO
/
(9.48)
J — oo
dy \y\a~l hXtV h^y = I
weakly in Ha .
(9.50)
As usual, this means that vectors in Ha can be expanded in terms of the wavelets hX)V , and all the theorems in Chapter 4 apply concerning the consistency con dition for transforms, least-squares approximations, and so on. Note that the admissibility condition is now Ca < oo, which reduces to the usual admissibility condition only when a = 0. Having explained the relation of (9.37) to the usual one-dimensional wavelet transform, we now return to the four-dimensional case. Inserting the Fourier representation of F, we obtain Fh(x,y) = (2TT)- 4 ^ dr h(r) f dfipeP'^+^Flp) = (2TT)"4/
4
Jn = (2TT)- 4
/ Jn4
d4peipxF{p)
[ dreiTpyh{r) J-oo
(9.51)
d4pe^xl(p-y)F(p).
That is, Fh(x,y) is a windowed inverse Fourier transform of F(p), where the window in the Fourier domain is h(p • y). Note that all Fourier components
A Friendly Guide to Wavelets
218
F ( p ) in a given three-dimensional hyperplane Pa = {p : p • y = a} are given equal weights by this window.* To construct wavelets from /i, as was done in t h e one-dimensional case by (9.44), we need an inner product on the vector fields F . This will be done in the next section, where the F ' s will be assumed to be electromagnetic waves. But now we specialize our transform by choosing h(r) = — —
.
(9.52)
iri r — %
W i t h this choice, we denote F ^ ( x , y) by ¥{x -f iy) and call it the analytic signal of F . We state the definition in the general case, where R 4 is replaced by R n . D e f i n i t i o n 9 . 3 . The analytic signal o / F : R n -> C m is the function C m defined formally by 1
C°°
F(x + iy) = — /
TTl J_00
F : C n ->
rln:
T - I
F(x + ry).
The mapping F i - > F will be called the analytic-signal transform
(9.53) (AST).
In general F is not analytic, so it may actually depend on both z = x + iy and z = x — iy. T h e reason for the above notation will become clear below. By contour integration, we find kS)
= --j^—e-«T
= 2e(t)eS,
(9.54)
where 6 is the step function
r i ?>o tf(0 =1.5 ( = 0 I 0 £ < 0.
(9.55)
T h u s we have shown the following: P r o p o s i t i o n 9.4. The analytic-signal by F{x + iy) =
(2TT)-4
f
transform
is given in the Fourier
d4p29{p • y) eip<x+iy"> F ( p ) .
domain (9.56)
T h e factor 6(p • y) plays two important roles in (9.56): * Equation (9.37) defines a one-dimensional windowed Radon transform. A similar transform can be defined modeled on field measurements where the instrument is assumed to occupy k < 3 space dimensions. (For a point instrument, k = 0; for a wire antenna, k = 1; for a dish antenna, k = 2.) This gives a k + 1-dimensional windowed Radon transform in R 4 . See Kaiser (1990a, b), and Kaiser and Streater (1992).)
9. Introduction to Wavelet
Electromagnetics
219
(a) Suppose that F(p) is absolutely integrable, so that F(x) is continuous. If the factor 9(p • y) were absent in (9.56), then e~py can grow without bounds, and the above integral can diverge unless F{p) happens to decay sufficiently rapidly to balance this growth. This option is not available for general electromagnetic waves in the Hilbert space W of Section 9.2, since there F(p) is supported in the light cone C = C+ U C_; so whenever p belongs to the support of F, then — p may possibly also belong to the support of F , and there is no reason for F to have the exponential decay required to make the integral converge. (b) The factor 6{p • y) tends to spoil the analyticity of F. If this factor were absent, and the integral somehow converged, then the resulting function depends only on z = x + iy rather than on both z and z, and F(z) is analytic (formally, at least). We will now show that under certain conditions, which hold in particular when F belongs to the Hilbert space TC, F(z) is indeed analytic, provided z is restricted to a certain domain T C C 4 in complex space-time. Again, we revert temporarily from four dimensions (space-time) to one dimension (time), in order to clarify the ideas. Suppose that F : R —► C is such that F(p) is absolutely integrable. Then (9.56) is replaced by oo
/
dp 2Q(py) eip(x+iy) F{p).
(9.57)
-oo
If y > 0, then 6{py) = 6{p) and />00
F(x + iy) = Tf-1 / dpeip{x+iy) F(p), y > 0. (9.58) Jo The integral converges absolutely and, as z = x + iy varies over the upper-half z-plane, it defines an analytic function. Therefore F(z) is analytic in the upperhalf plane C+, and for z € C+, F contains only positive-frequency components. Similarly, when y < 0, then 9(py) — 0(—p) and
f°
F(x + iy) = 7T-1 /
dpeip{x+iy)
F(p),
y<0.
(9.59)
J — oo
F is analytic in the lower-half plane C_, and for z G C _ , F(z) contains only negative-frequency components. Thus, F is analytic in C + U C _ . It is not analytic on the real axis, in general. The relation between F(z) and the original function F(x) can be summarized as follows: POO
\2 lim
y-o+
L
F(x + iy) + F(x -iy)\=
(27r) _1 /
dp eipx F(p) = F(x).
(9.60)
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220
T h u s F(x) is a "two-sided boundary value" of F(z), as z approaches the real axis simultaneously from above and below. On the other hand, the discontinuity across R is given by
\
lim F(x + iy) - F(x - iy) \= J y^o+ L =
where H : L2(H)
—► L2(H)
(2TT)~1
/ dp sign (p)eipx J_00
F(p)
(9.61)
iHF(x),
is the Hilbert transform.
W h e n F is real, then
F(z) = F(z) for all z and F(x) = HF(x)
lim ReF(x
= lim ImF(x
+ iy) + iy).
(9.62)
For example, F(x) = cosx
=> F(p)=7r8(p-l)
+ 7r6(p+l),
(9.63)
sinx. HF{x) HF(x) = smx.
(9.64)
so
elz, zeC+ F{z) = { cosz, cos z, zeK , zeK iz z€C_ e~ ,
The function F in (9.58) was introduced by Gabor (1946), who called it the analytic signal of F. It has proved to be very useful in optics, radar, and many other fields. (For optics, see Born and Wolf (1975), Klauder and Sudarshan (1968). For radar, see Rihaczek (1968); also Cook and Bernfeld (1967), where analytic signals are called "complex waves.") W h e n F is real, then the Gabor analytic signal (defined in C + only) suffices to determine F by (9.62), because t h e negative-frequency part of F is simply the complex conjugate of t h e positivefrequency part. When F is complex, the two parts are independent and both (9.58) and (9.59) are needed in order to recover F. Our definition (9.53) is a generalization of Gabor's idea t o any number of dimensions and to functions t h a t are not necessarily real-valued. T h a t explains the name "analytic-signal transform" given to the mapping F i—»• F in (9.53). To extend the relations (9.60) and (9.61) to the general case, we return to the original definition (9.53) with z = x+iey , 2 / ^ 0 , and consider its limit as £ —>• 0 + :
9. Introduction to Wavelet Electromagnetics 1
221
f°°
C\T
lim Fix + ley) = — lim / e^0+
F(x + ery)
1TI e-+0+ J_OQ
T — l
7TI 6-^0+
U — IS
1 ,. f°° = — hm /
du
J_00
„, Fix + uy)
= — \iirF(x) + P / ™ L
— F(x + uy)\
J-oo
= F{x) + —P / 7TZ
J_00
= F(x) + -P /
u
(9.65)
J
— F(z + m/ U
— Ffr-uy),
where P J • • • denotes the Cauchy principal value. The multivariate Hilbert transform in the direction of y =fi 0 is defined by 1
# y F ( x ) = -P
/*00
/
7
— F(x - wy).
(9.66)
It actually depends only on the direction of y, since # a y F = sign (a) F y F ,
a ^ 0,
(9.67)
so the usual definition (Stein [1970]) assumes that y is a unit vector with respect to an arbitrary norm in R n (which Stein takes to be the Euclidean norm). Since in our case R n = R 4 is the Minkowskian space-time with the indefinite Lorentz scalar product, we prefer the unnormalized version. Equation (9.65) gives the generalizations of (9.60) and (9.61): F(x) = - lim \F(X + ley) + F(x - iey)] J \£-*0+[ _ _ HvF(x) = — lim \F(X + iey) - Fix - iey)} . 2% e—►()+ L
(9.68)
J
What about analyticity in the general case? Recall that for free elec tromagnetic waves, F(p) = <5(p2)f(p), s o F *s supported on the light cone C — {p : p2 = 0}. More generally, suppose F is supported inside the solid double cone V = {p= ( p l P 0 ) : p2 =p20 - c 2 |p| 2 > 0} = V+ U VL , y+ = { p = ( p , p o ) : p o > c | p | } ,
(9.69)
V_ = {p= (p,po) :p0 < - c | p | } , where we have temporarily reinserted the speed of light c. (This will clarify the difference between V± and their dual cones V±, to be introduced below.)
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222
V+ is the positive-frequency cone and VL is the negative-frequency cone. The positive- and negative-frequency light cones are the boundaries of V± : C± = dV± C V±. (See Figure 9.1.) V± are both convex^ a property not shared by C±. (This means that arbitrary linear combinations of vectors in V± with positive coefficients also belong to V±.) The four-dimensional positive- and negativefrequency cones V± take the place of the positive- and negative-frequency axes in the one-dimensional case, i.e., V+ <-► {p G R : p > 0} = R+ ,
VI *-* {p G R : p < 0} = R_ .
(9.70)
The dual cones of V+ and VL are defined by V | = {y G R 4 : p • y > 0 for all p G V + } V^ = {y G R 4 : p • j/ > 0 for all p G VL}. To find V\_ explicitly, let y G V+. Then we have p • y = ujy0 - p • y > 0 for all p G V+ .
(9.72)
If y = 0, then (9.72) implies y0 > 0. If y ^ 0, taking p = ujy/(c |y|) in (9.72) gives cyo-|y|>0. (9.73) This condition, in turn, is seen to be sufficient for y to belong to V+. If it holds, then by the Schwarz inequality in R 3 , peV+
=> p ■ y = ujy0 - p • y > ujy0 - |p| |y| = \p\(cy0 - |y|) > 0.
(9.74)
Thus we have shown that Vi = {y = (y,y0)-cy0>\y\}.
(9.75)
VL = {y = (y,y0):cy0<-\y\}.
(9.76)
Similarly, V+ and VL are called the future cone and the past cone in space-time, respec tively. The reason for the name is that an event at the space-time point x can causally influence an event at x' if and only if it is possible to travel from x to x' with a speed less than c, which is the case if and only if x' — x G V+. Similarly, x can be influenced by x" if and only if x" — x G VL. For this reason we call the double cone V' = V^VL = {y.vi = c2y2o - |y| 2 > 0}, (9.77) the causal cone. This name will be seen to be justified in the case of electromag netics in particular; we will arrive at a family of electromagnetic wavelets labeled by z = x + iy with y G V, and it will turn out that v = y/yo is the velocity of their center. But |v| < c means exactly the same thing as y = (y, yo) G V'\
9. Introduction to Wavelet Electromagnetics
223
Note that C+ represents the extreme points in V+ (since C+ = dV+); hence in order to construct V^, we need only p G C+ instead of all positive-frequency wave vectors p G V+: Vl = {y:p-y>0
for all p G C + } .
(9.78)
Similarly, V'_ is "generated" by C_ . We come at last to analyticity. In view of the above, define the future tube as the complex space-time region T+ = {z = x + iy€C*:y€Vl}
(9.79)
T_ = {z = x + iy G C 4 : y G V l } .
(9.80)
and the past tube as
7%e future tube and the past tube are the multidimensional generalizations of the upper-half and the lower-half complex time planes C+ and C_ . See Stein and Weiss (1971) for a mathematical treatment of this general concept. These domains play an important role in rigorous mathematical treatments of quantum field theory; see Streater and Wightman (1964), Glimm and Jaffe (1981). They are also related to the theory of twistors; see Ward and Wells (1990). The union T = T+UT_ (9.81) will be called the causal tube. Note that the two halves of T are disjoint, like C+ and C_ . Proposition 9.5. Let F(x) be any electromagnetic wave in Ti. Then its analytic signal T?(z) is analytic in the causal tube T. Proof: Inserting F(p) = 6(p2) U(p)f(p) into (9.56) gives [ dp 20(p . y) eip^x+iy>>U(p){(p), (9.82) Jc exactly as in Section 9.2. If z 6 T+ , i.e., y G V+ , then p • y > 0 for all p 6 C+ and p • y < 0 for all p € C_ . Therefore (9.82) reduces to F(x + iy)=
¥{x + iy) = 2 /
dp eip<x+iy^U{p){{p)
= ¥+{x + iy),
(9.83)
which is analytic. Similarly, if z G 71 , i.e., 2/ G VI , then p • ?/ < 0 for p G V+ and p ■ y > 0 for p G V_ , and thus F(x + iy) = 2 f which is also analytic.
■
dp eip<x+iy">U{p)f{p) = F_(x + iy),
(9.84)
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224
Note that, for z G T+ , F(z) contains only positive-frequency components, and for z G 7 1 , F(z) contains only negative-frequency components. This is a general ization of the one-dimensional case, where the analytic signals in the upper-half time plane contain only positive frequencies and those in the lower-half time plane contain only negative frequencies. The analytic-signal transform there fore "polarizes" F(x): its positive-frequency part goes to T+ , and its negativefrequency part goes to 7 1 . We have seen that F+ and F _ have direct physical interpretations as the positive-helicity and negative-helicity components of the wave.
9.4 Electromagnetic Wavelets Fix any z = x + iy G T, and define the linear mapping Sz : H -» C 3
EZF = F{z) = f dp 26{p • y) eip'z U{p) f (p). (9.85) Jc Ez is an evaluation map, giving the value of the analytic signal F at the fixed point z. We will show that Sz is a bounded operator from the Hilbert space H to the Hilbert space C 3 (equipped with the standard inner product). Therefore, Sz has a bounded adjoint. by
Definition 9.6. The electromagnetic wavelets ^z, adjoints of the evaluation maps Sz: * 2 = S*z :C3^H.
z G 7"', are defined as the (9.86)
If we deal with scalar functions F rather than vector fields F, then F is just a complex-valued function and the wavelets are linear maps \&z : C —► TL As explained in Section 1.3, such maps are essentially vectors in H\ namely, we can identify tyz with \1>Z1 G H. (This will indeed be done for the acoustic wavelets in Chapter 11, since there we are dealing with scalar solutions of the wave equation.) To find the \I>z's explicitly, choose any orthonormal basis 111,112,113 in C 3 and let *z,k = *zu f c eH, k = 1,2,3. (9.87) This gives three solutions of Maxwell's equations, all of which will be wavelets "at" z. *&z is a matrix-valued solution of Maxwell's equations, obtained by putting the three (column) vector solutions ^fZik together. The A>th component
9. Introduction to Wavelet
Electromagnetics
225
of J?(z) with respect to the basis {u^} is
= < * * F = (* z u f c )*F = *;,*
(9.88)
F
By (9.85), u*kF(z) = [ %2u2 9(p ■ y) &* u ; n ( p ) f (p), Jc Jc & which shows that \P z ^ is given in the Fourier domain by il>Zik(p) = 2a,2 0(p • y) e-ip* U(p) uk .
(9.89)
(9.90)
Note that each ipz,k{p) satisfies the constraint since T(p) H(p) = II(p). The matrix-valued wavelet ^z in the Fourier domain is therefore ^ , ( p ) = 2u;2 tf(p • y) e-^~z n ( p ) .
(9.91)
In the space-time domain we have (using II(p) ipz{p) = ipz(p))
*,(*/)=
[ dp e^x' *pz(p)
Jc
= f dp Jc
(9.92) 2
2uj d(p-y)e
ip{x
'-^U(p).
Now that we have the wavelets, we want to make them into a "basis" that can be used to decompose and compose arbitrary solutions. This will be accomplished by constructing a resolution of unity in terms of these wavelets. To this end, we derive an expression for the inner product in H directly in terms of the values F(^) of the analytic signals. To begin with, it will suffice to consider the values of F only at points with an imaginary time coordinate ZQ = is and real space coordinates z = x. Therefore define E = {z = (x, is) : x 6 R 3 , 0 + s € R } C T.
(9.93)
In physics, E is called the Euclidean region because the negative of the indefinite Lorentz metric restricts to the positive-definite Euclidean metric on E: -z2 = -{is)2
+ |x| 2 = s2 + |x| 2 .
(9.94)
The Euclidean region plays an important role in constructive quantum field theory; see Glimm and Jaffe (1981). Later x will be interpreted as the center of
226
A Friendly Guide to Wavelets
the wavelets ^z%k , and s as their helicity and scale combined. By (9.82), F(x, is) = f dp 20{pos) e~PoS~ipxU{p) ./c = 2 f
dp e~ipx +
f(p)
\e{us)e-u,sU(p,u;){(p1u;)
0(-LJS)
e"s n ( p , - w ) f (p,
-CJ)]
(9'95)
= ^"1[o;-1^)c-w'n(p,a;)f(p,a;) + aT ^ ( - s ) e" s n ( p , - w ) f (p, -<*)] (x), where ^ r ~ 1 denotes the inverse Fourier transform with respect to p. Hence by Plancherel's formula, / d3x|F(x,is)|2 = / ^ ! P - ^2 [ ^ ) e - 2 - | n ( p , a , ) f ( P ) c ) | 2 yRs 7R3 (2TT) a; L
( 9 _ g6 )
+ »(-«)ea"|n(p,-a;)f(p,-a;)|2]) where we used 8(u)2 = 8{u) and 0(u) 0(—u) = 0 for « ^ 0. Thus
/>xds|F(x,«)|2 = /
d 3
- I PP
r , T T / _ . A r / _ ,.M2 [|n(p,o;)f(p1a;)|s
+ |n(p,-u,)f(p,-w)|2] = /^|n(P)f(P)|
2
(9.97)
= /^-f( P rn( P )f( P ) = ||F||2, Jc v
since i.e.,
r*TT — II*II = TT2 II 2 =_
II. Let H be the set of all analytic signals F with F £ H, W= { F : F G 4
(9.98)
Definition 9.7. TTie inner product in 7i is defined by ( F , G ) = / dfji(z) F(z)* G{z) = F*G,
(9.99)
where dfx(z) = d3x ds,
z = (x, is) £ J5.
(9.100)
Theorem 9.8. TC is a Hilhert space under the inner product (9.99), and the map F t-> F is unitary from H onto H, so { F , G ) =
(9.101)
9. Introduction to Wavelet Electromagnetics
227
Proof: Most of the work has already been done. Equation (9.97) states that the norms in H and H are equal, i.e., that we have the Plancherel-like identity ||F|| 2 = ||F|| 2 . By polarization, this implies the Parseval-like identity (9.101), so the map is an isometry. Its range is, by definition, all of 7i, which proves that the map F >—> F is indeed unitary. ■ Using star notation, the Hermitian transpose F(z)* : C 3 —* C of the column vector F(z) G C 3 can be expressed as the composition F(z)* = ( # 2 F ) * = F * * 2 .
(9.102)
In diagram form, C3 '—^ H Hence the integrand in (9.99) is
—
> C.
F{z)* G(z) = ( # * F)* ^ * G = F* * z V*z G, where ^z^*z
(9.103) (9.104)
: H —► H is the composition "—-> C 3
H
** ) H.
(9.105)
Therefore (9.101) reads / .E
d/x(s)F* ¥ z ¥ * G = F* G,
F,GG«.
(9.106)
T h e o r e m 9.9. (a) The wavelets \£ z with z G E give the following resolution of the identity I in H: f dv{z)*z**z = I, (9.107) JE
where the equality holds weakly in H, i.e., (9.106) is satisfied. (b) Every solution F G H can be written as a superposition of the wavelets SI/z with z — (x, is) G E, according to F - / dfi(z) *z * * F = / dfx(z) *z F{z), JE
(9.108)
JE
i.e., F{x')=
f d^{z)^z(x')F{z)
JE
a.e.
(9.109)
(9.108) holds weakly in 7i, i.e., the inner products of both sides with any member of Ji are equal. However, for the analytic signals, we have F(z') = * ; , F = / dfi(z) *z, * 2 F(z)
(9.110)
JE
pointwise, for all z' G T . Proof: Only the pointwise convergence in (9.110) remains to be shown. This follows from the boundedness of \Pz<, which will be proved in the next section. ■
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228
The pointwise equality fails, in general, for the boundary values ¥(x) because the evaluation maps (or, equivalently, their adjoints \J/2) become unbounded as y —> 0. This will be seen in the next section. : C 3 —>• C 3 , i.e.,
The opposite composition ^*z^z C3
**
> H
—>
C3,
(9.111)
is a matrix-valued function on T x T: K(z' \z) = **z,*z=
f %u 4CJ4 9(p • y') 6{p • y)
(9.112) = 4 f dp LU2 0(p • y') 0{p • y) eip
In the next section we compute ^z(x') and therefore, by analytic continuation, also K(z' | z). Let us examine the nature of the matrix solution \&z. Recall that ^ z may be regarded as three ordinary (column vector) wavelet solutions ^Zjk combined. Since II(p) is the orthogonal projection to the eigenspace of T(p) with the nondegenerate eigenvalue 1, all the columns (as well as the rows) of II(p) are all multiples of one another. But the factors are p-dependent, and the algebraic linear dependence in Fourier space translates to a differential equation in spacetime. For the rows, this differential equation is just Maxwell's vector equation (9.1) relating the different components of each wavelet \I/2)fc- (Recall that the scalar Maxwell equation is then implied by the wave equation.) Since II(p) is Hermitian, the same argument goes for the columns. Explicitly, (9.91) together with T i l = I I = I i r implies
i W z ( p ) = V*(P) = ^ ( P ) F ( P ) .
(9.ii4)
When multiplied through by po and transformed to space-time, these equations read V x *z{x') = -id'0Vz{x') = ¥z(x') x V', (9.115)
9. Introduction to Wavelet Electromagnetics
229
where d'0 denotes the partial with respect to x'0, V' the gradient with respect to x', and V' indicates that V' acts to the left, i.e., on the column index. This states that not only the columns, but also the rows of ^z are solutions of Maxwell's equations. The three wavelets ^Zjk Q-^e thus coupled. Note also that since &z(x') = ^ - ^ ' ( O ) , equation (9.115) can be rewritten as Vx*
2
= -id0*z
=*,xV,
(9.116)
where now do and V are the corresponding operators with respect to the labels XQ — Re ZQ and x = Re z . We will see in Section 7 that the reconstruction of F from F can be obtained by a simpler method than (9.108), using scalar wavelets tyz associated with the wave equation instead of the matrix wavelets \I/2 associated with Maxwell's equations. However, that presumes that we already know F(x, is), and without this knowledge the simpler reconstruction becomes meaningless, since no new solutions can be obtained this way. The use of matrix wavelets will be necessary in order to give a generalization of (9.108) through (9.110), where F(z) can be replaced with an unconstrained coefficient function. In other words, we need matrix wavelets in space-time for exactly the same reason that II (p) was needed in Fourier space (9.21): to eliminate the constraints in the coefficient function. 9.5 Computing the Wavelets and Reproducing Kernel To obtain detailed information about the wavelets, we need them in explicit form. Since the wavelets are boundary values of the reproducing kernel, our computation also gives an explicit form for that kernel as a byproduct. Note first of all that = [ dp J1 26(p • y) eip<*-*') n ( p ) = *iy(x' - x). (9.117) Jc That is, all the wavelets can be obtained from ^iy by space-time translations. Therefore it suffices to compute &iy(x), for y E V. But *z(x')
= [ dp LJ2 20(p • y) e - ^ - t e ) U(p) (9.118) Jc is analytic in w = y — ix, for iw E T. Thus we need only compute ^ ^ ( 0 ) , since then ^iy(x) can be obtained by analytic continuation. Now *iy(x)
*iy (0) = f dp u2 29(p • y) e~p-y U(p) = * _ i y (0), (9.119) Jc since II(—p) = II(p). Therefore we can assume that y G V+ . Then the integral extends only over C+ and *iy(0)=
[ dp 2cj2e-pyU(p). Jc+
(9.120)
A Friendly Guide to Wavelets
230
Now the matrix elements of
2LJ2 II(p)
are given by (9.20): 3
2
2 J Umn{p) = 8mnpl - pmPn + i ] P emnk popk .
(9.121)
fc=i
(Recall that emnk is the completely antisymmetric tensor with £123 = 1.) To compute ^ ^ ( 0 ) , it is useful to write the coordinates of y in contravariant form: y° = 2/0 1 _ _
3
2/ 2 = "2/2
^
0
2/ 3 = - 2 / 3
(This is related to the Lorentz metric; p is really in the dual R4 of the space-time R 4 , as explained in Sections 1.1 and 1.2. By writing the pairing beween p € R4 and y € R 4 as p • y, we have implicitly identified y with y* 6 R 4 , so that the pairing can be expressed as an inner product. The above "contravariant" form reverts y to R 4 . See the discussion below (1.85).) To evaluate (9.120), let S(y) = f dp e~p-y, yGVj. (9.123) Jc+ We will evaluate S(y) below using symmetry arguments. S(y) acts as a gener ating function for the matrix elements of ^ y ( 0 ) , as follows: Let da denote the partial derivative with respect to ya for a = 0,1,2,3. Then
L
dp paPpe-py
= dadpS(y)
= SaP(y),
a,/? = 0,1,2,3.
(9.124)
By (9.120) and (9.121), the matrix elements of ^ ^ ( 0 ) are therefore 3 0
[*«/( )L,n = ^mnSoo(y)-Srnn(y)+i^2emnkSok(y),
m , n = 1,2,3. (9.125)
fc=i
It remains only to compute S(y). For this, we use the fact that S(y) is invariant under Lorentz transformations, since p • y and dp are both invariant and C+ is a homogeneous space for the proper Lorentz group. Since y G V+, there exists a Lorentz transformation mapping y to (0, A), where \(y) = V ^ = \ A o - lyl2 > 0.
(9.126)
The invariance of S implies that S(y) = S(0, A). Therefore
S(y) = I
- ^ 3c - ^ l = - L2
yR3
16TT |P|
1 47T 2 A 2
_
4TT
1 47T 2 y 2 *
J0
r^due^ (9.127)
9. Introduction to Wavelet Electromagnetics
231
Taking partials with respect to ya and y& gives ^
^
w
1
'
'
'
a,/? = 0,1,2,3,
(9.128)
where # a 0 = diag(l, — 1 , - 1 , - 1 ) is the Lorentz metric. It follows that rlT/ / n v, _ &mn{yl + Vi + V* + vl) ~ ^VrnVn + 2% ^ L l ^mnkVoyk L*tylu;jm,n 7T2(y2)3 '
Theorem 9.10. The matrix elements [^z(x')]mn <5mn(^0 +
W
where
l +
W
2 + ^ 1 ) - 2 ™mW n + 2z £ ^ 7T2(W2Y
of^z{x') = 1
WQ
. '
are
£
rnnk™0Wk
w = i(z — x') = y + i(x — x') W2 = W - W =
, ^
— W2 — W2 — W%
(9.130)
(9.131)
TTie reproducing kernel is given in terms of the wavelets by K(z' I z) = 2 % ' • y) *,_*.(<)). T/ie matrix elements of the wavelets ^z{x') whenever z' and z belong to T', since 2
z
(9.132)
and the kernel K(z' \ z) are finite
^ 0 for all zeT.
(9.133)
Proof: By analyticity, to find Sm?z(x') we need only replace y by w = y-\-i{x — x') in (9.129). (It can be shown that the analytic continuation is unique, i.e., that all paths from y to w give the same result. See Streater and Wightman (1964).) This proves (9.130). To prove (9.132), note that K(z' | z) vanishes when y' and y are in opposite halves of V , since p • y' and p • y have opposite signs in (9.112). If y' and y belong to the same half of V , then p ■ y' and p • y have the same sign, and we may replace 9(p • y')9(p • y) by 9{p • y) in (9.112). But then K(z' \z)=
[dp
Au20{p • y)eip
= 2¥ z _*/(0).
(9.134)
For general z',z £ T, the kernel is therefore obtained by multiplying (9.134) by 6(y' • y). Finally, to show that all the matrix elements are finite, it suffices to prove (9.133). Suppose that z2 = 0 for some z E T. Then x2 - y2 = 0,
x-y = x0y0 - x • y = 0.
The second equation implies i rn i < ixi lyi < ixi
A Friendly Guide to Wavelets
232
since y G V. (The second inequality is strict unless x = 0.) But then x2 = XQ — |x| 2 < 0, hence y2 > 0 implies t h a t x2 — y2 < 0. This contradicts the first equation. ■ Note t h a t if z' and z are in the same half of T , then 2 — z' = (x — x') + i(y + y') is also in t h a t same half, since V+ and VZ. are each convex. Therefore 2 — z' E T , and ^z-z' exists as a bounded m a p from C 3 to 7i. (This has been tacitly assumed for all ^5fz with z G T , and will be proved below.) If 2/ and z are in opposite halves of 7", then y' + y may not belong to V7 (since V' = V | U V^ is not convex); thus z — z' may not belong to T and therefore ^ - ^ ( O ) need not exist. But then we are saved by the factor 9(y' • y) in (9.132), since it vanishes! In Section 4 we stated t h a t the evaluation maps Sz are bounded, and t h a t they become unbounded as y —> 0. T h e boundedness of Sz is of fundamental importance, since otherwise \ P z = £* does not exist as a (bounded) m a p from C 3 to H and we have no (normalizable) wavelets.* The statement t h a t Sz : H —► C 3 is bounded means t h a t there is a constant iV, (which turns out to depend on z) such t h a t ||fzF||2 = ||F(z)||23<JV(z)||F||2. (9.135) But in terms of an orthonormal basis {u^} in C 3 and the corresponding (vector) wavelets ^ z , k = ^zUfc, the Schwarz inequality gives iiPWii|3 = ^ i * : , l t F | 2 k
<^||*z,fc||2||F||2 * =
(9.136)
2
||F|| ^u^:*zufc k 2
= | | F | | trace ^ * * z , so (9.135) holds with N(z)
= trace ^*z^z
In fact, the proof shows t h a t N(z),
= trace K ( z | z).
(9.137)
as given by (9.137), is the best such bound.
T h e o r e m 9 . 1 1 . For any z £ T , £/ie evaluation map 8Z : H —» C 3 is bounded. It becomes unbounded whenever z approaches the 7-dimensional boundary of T, dT={x
+ iy:y2
= y2-
\y\2 = 0}.
(9.138)
■ If our solutions were scalar valued, as will be the case for acoustics in Chapter 11, then Ez would be a linear functional on H and ^ z would be its unique representative in 7^, as guaranteed by the Riesz representation theorem. That theorem fails for unbounded linear functionals!
9. Introduction to Wavelet Electromagnetics
233
Therefore, the electromagnetic wavelets *&z exist as bounded linear maps from C 3 to H, for every z G T . They satisfy | | * : P | | 2 < JV(*)||F|| 2 ,
(9.139)
where N(z) is the trace of the reproducing kernel, N(z) = t r a c e * : * , = ^
1
3;/02 + |y| 2 , 7 ,!'L2 3 •
&r2 (vl ~ |y| )
(9-140)
For £/ie wavelets labeled by the Euclidean region, K(x,is|x,zs) = g ^ J ,
(9.141)
where I is the 3 x 3 identity matrix. Proof: By (9.132), K(* | z) = 26(y2)*2iy{0)
= 2* 2 i y (0),
(9.142)
since y2 > 0 in V . Therefore by (9.129),
1
3^02 + |yl 2
(9.143)
8^ 2 feo2 - l y l 2 ) 3 ' so (9.139) and (9.140) hold in view of (9.136). If y2 = 0, then all the matrix elements of ~K(z \ z) diverge, and (9.136) shows that \I/ z is unbounded. Finally, (9.141) follows from (9.142) and (9.129) with y = 0 and y0 = s. m
9.6 Atomic Composition of Electromagnetic Waves The reproducing kernel computed in the last section can be used to construct electromagnetic waves according to local specifications, rather than merely to reconstruct known solutions from their analytic signals on E. This is espe cially interesting because the Fourier method for constructing solutions (Sec tion 9.2) uses plane waves and is therefore completely unsuitable to deal with questions involving local properties of the fields. It will be shown in Section 9.7 that the wavelets ^x+iy(x/) are localized solutions of Maxwell's equations at the "initial" time x/0 = XQ. Hence we call the composition of waves from wavelets "atomic." (See Coifman and Rochberg (1980) for "atomic decomposi tions" of entire functions; see also Feichtinger and Grochenig (1986, 1989 a, b), and Grochenig (1991), for other atomic decompositions.)
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234
Suppose F is the analytic signal of a solution F G H of Maxwell's equations. Then according to Theorem 9.8, / " ^ ( * ) | F ( * ) | 2 = HF|| 2
(9.144)
JE
Let C2(E) be the set of all measurable functions 3> : E —► C 3 for which the above integral converges. C2(E) is a Hilbert space under the obvious inner product, obtained from (9.144) by polarization.'1' Define the map RE :H —► C2(E) by (RE F)(Z) = **ZF = F(z),
z = (x,is) G E.
(9.145)
That is, RE F is the restriction F \E to E of the analytic signal F of F. Then (9.144) implies that the range AE of RE is a closed subspace of £2(E), and i? B maps % isometrically onto AE- The following theorem characterizes the range of RE and gives the adjoint R*E. T h e o r e m 9.12. (a) The range of RE is the set AE of all & G C2(E) satisfying the "consistency condition" $(zf) = f dfj.(z) K(z' | z) &(z),
(9.146)
JE
pointwise in z' G E. (b) The adjoint operator R*E : C2 (E) —> H is given by R*E*=
[ dii(z)*z3>(z),
(9.147)
JE
where now the integral converges weakly in H. That is, R*E*(x') = / dn(z)*z(x')&(z)
a.e.
(9.148)
JE
Proof: If & e AE, then &(z) = F(z) for some F e H, and (9.146) reduces to (9.110), which holds pointwise in z' G E by Theorem 9.9. On the other hand, given a function 3> G C2(E) that satisfies (9.146), let F denote the right-hand side of (9.146). Then for any GeH, G*F = / dfi(z) G(z)**(z) JE f
= {REG,&
)C2,
(9.149)
We could identify E with R 4 w {(x,is) G C 4 : x G R 3 ,s G R} and £2(E) with L (R 4 ), since the set H4\E = {(x,0) : x G R 3 } has zero measure in R 4 . But this could cause confusion between the Euclidean region E and the real space-time R 4 . 2
9. Introduction to Wavelet Electromagnetics
235
where we have used G * ^ z = (**G)* = G(z)*. Hence the integral in (9.147) converges weakly in ft. The transform of F under RE is (RE F)(zf) = **Z,F = f dfi(z) * ; , * , # ( « ) JE
(9.150)
= / dn(z)K{z'\z)*(z)
= Q(z')i
JE
by (9.146). Hence 3? E AE as claimed, proving (a). Equation (9.149) states that ( G , F ) n = ( R E G, $ >£3 for all G <E ft. This shows that F = R%&, hence (b) is true. ■ According to (9.147), R% constructs a solution R%& G ft from an arbitrary coefficient function G C2 (E). When is actually the transform RE F of a solution F £ ft, then $(z) = * * F and #* RE F = R*E& = f dfi(z) *z * 2 F = F ,
(9.151)
JE
by (9.108). Thus R*ERE = / , the identity in ft. (This is equivalent to (9.144).) We now examine the opposite composition. Theorem 9.13. The orthogonal projection to AE in C2(E) is the composition P = RER*E : C2(E) -> C2(E), given by P${z')
= f dn{z) K(z' | z) *(z).
(9.152)
JE
Proof: By (9.147), (RBR*B*)(z')
= *;,RE*
= / dp(z) 9*g,9z*(z)
= P*(z'),
(9.153)
JE
since ^z,^z = K(z'\z). Hence RER*E = P. This also shows that P* = P. Furthermore, RERE — I implies that P2 = P; hence P is indeed the orthogonal projection to its range. It only remains to show that the range of P is AE ■ If # = REF G AE, then RER*E& = RER*EREF = REF = 3>. Conversely, any function in the range of P has the form <£ = RERE® for some 0 G C2(E); hence $ = REF where F = R*ES G ft. ■ When the coefficient function <1> in (9.147) is the transform of an actual solution, then RE reconstructs that solution. However, this process does not appear to be too interesting, since we must have a complete knowledge of F to compute F(z). For example, if z = (x, is) G E, then (9.53) gives F(x, is) = -
/
i r = — /
:
*
F((x, 0) + T ( 0 , «))
Pr
,
;F(x,rs).
(9 154)
-
A Friendly Guide to Wavelets
236
So we must know F(x, t) for all x and all t in order to compute F on E. There fore, no "initial-value problem" is solved by (9.108) and (9.109) when applied to <1> = F G AE. However, the option of applying (9.147) and (9.148) to arbitrary G C2{E) is a very attractive one, since it is guaranteed to produce a solution without any assumptions on
(9.155)
JE
directly with its Fourier counterpart (9.21): F(z') = f dp eipx
n ( p ) f(p).
(9.156)
Jc In both cases, the coefficient functions ( $ and f) are unconstrained (except for the square-integrability requirements). The building blocks in (9.155) are the matrix-valued wavelets parameterized by E, whereas those in (9.156) are the matrix-valued plane-wave solutions etp'x H(p) parameterized by C. E plays the role of a wavelet space, analogous to the time-scale plane in one dimension, and replaces the light cone C as a parameter space for "elementary" solutions. 9.7 Interpretation of the Wavelet Parameters Our goal in this section is twofold: (a) To reduce the wavelets &z to a sufficiently simple form that can actually be visualized, and (b) to use the ensuing picture to give a complete physical and geometric interpretation of the eight complex space-time parameters z G T labeling *&z. That the wavelets can be visualized at all is quite remarkable, since ^ z(x') is a complex matrix-valued function of x' G R 4 and z € T. However, the symmetries of Maxwell's equations can be used to reduce the number of effective variables one by one, until all that remains is a single complex-valued function of two real variables whose real and imaginary parts can be graphed separately. We begin by showing that the parameters z G T can be eliminated entirely. Note that since II(p)* = II(p) = II(—p), the Hermitian adjoint of &z(x') is *z{x'Y
= f dp Jc
2u2d(p-y)e-ilp<x'-z)U{p)
= [ dp 2u>2e(-p . y)ei^x'-^U(p) Jc
(9'157)
9. Introduction to Wavelet Electromagnetics
237
Equation (9.157) shows that it suffices to know ^!z for z G T+ , since the wavelets with z G T_ can be obtained by Hermitian conjugation. Thus we assume z G T+ . Now recall (9.117), yx+iy(x') = Viy(xf - x). (9.158) It shows that it suffices to study \I/j y with y G V\_, since ^x+iy can then be obtained by a space-time translation. We have mentioned that Maxwell's equa tions are invariant under Lorentz transformations. This will now be used to show that the wavelet ^iy with general y G V+ can be obtained from ^o.tAj where A = y # . (Recall that a similar idea was already used in computing the integral S(y) defined by (9.123).) A Lorentz transformation is a linear map A : R 4 —+ R 4 (i.e., a real 4 x 4 matrix) that preserves the Lorentz metric, i.e., (Ax) • (Ax') = x>x\
x,x' G R 4 .
(9.159)
Equation (9.159) can be obtained by "polarizing" the version with x' = x. Letting x = Ax, we have C2XQ - |x| 2 = x 2 = (Ax) • (Ax) = x 2 = C2XQ -
|x| 2 ,
where the speed of light has been inserted for clarity. An electromagnetic pulse originating at x — x = 0 has wave fronts given by x 2 = 0, and the above states that to a moving observer the wave fronts are given by x 2 = 0. That means that space and time "conspire" to make the speed of light a universal constant. If xo = xo, then |x| 2 = |x| 2 and A is a rotation or a reflection (or both). But if A involves only one space direction, say xi, and the time xo, then X2 = X2 and x 3 = x 3 , so 2-2 C XQ
~2 2 2 X^ — C XQ
2 X-^.
This means that A is a "hyperbolic rotation," i.e., one using hyperbolic sines and cosines instead of trigonometric ones. Such Lorentz transformations are called pure. Since time is a part of the geometry, these transformations are as geometric as rotations! A transforms the space-time coordinates of any event as measured by an observer "at rest" to the coordinates of the same event as measured by an observer moving at a velocity vA relative to the first. There is a one-to-one correspondence betweeen pure Lorentz transformations A and velocities v A with |v A | < c. (In fact, vA is just the hyperbolic tangent of the "rotation angle"!) If the observer "at rest" measures an event at x, then the observer moving at the (relative) velocity vA measures the same event at A _ 1 x. But how does A affect electromagnetic fields? The electric and magnetic fields get mixed together and scaled! (See Jackson (1975), p. 552.) In our complex representation, this means F —> M A F, where MA is some complex 3 x 3 matrix determined by A. The total effect of A on the field F at x is F(x) HH. F'(x) = M A F(A" 1 x).
(9.160)
238
A Friendly Guide to Wavelets
F'{x) represents the same physical field but observed by the moving observer. Thus a Lorentz transformation, acting on space-time, induces a mapping on solutions according to (9.160). We define the operator UA:H^H
by
UAF = F',
(9.161)
with F ' given by (9.160). It is possible to show that UA is a unitary operator (Gross [1964]). In fact, our choice of inner product in 7i is the unique one for which UA is unitary. Our definition (9.53) of analytic signals implies that 1
f{z) = - .
rli-
r°°
—.
M A F (A-'(x + ry))
= -MAJ_^ — F{K-^ = MAF (A^x
+
+
T^y)
{9M2)
ik-lTy)
= M A F(A - 1 *). This shows that the analytic signals F ' and F stand in the same relation as do the fields F ; and F. But note that now A acts on the complex space-time by z' = Az = A(x + iy) = Ax + iAy = x' + iy\
(9.163)
with a similar expression for A~*z. In order for (9.162) to make sense, it must be checked that A preserves V, so that z' € T whenever z € T. This is indeed the case since y,.y'
= {Ay)-(Ay)=y.y>0,
for all
y G V\
(9.164)
by (9.159). By the definition of the operator UAj (9.162) states that ^ F ' = ¥*ZUAF = MA**A-lzF
(9.165)
for all F 6 H, hence *;i/A = MA¥X-lz,
=►
U:*z
= *A-lzM*A.
(9.166)
But U£ = U~x because UA is unitary. Furthermore, U^1 = UA-x. (This is intuitively clear since UA-i should undo UA , and it actually follows from the fact that the C/'s form a representation of the Lorentz group.) Similarly, M~x — M A -i. Substituting A -> A - 1 in (9.166), we get W ^ A X - . .
(9.167)
That is, under Lorentz transformations, wavelets go into other wavelets (modi fied by the matrix M*_x on the right, which adjusts the field values). How does
9. Introduction to Wavelet Electromagnetics
239
(9.167) help us find *$> Az if we know * z ? Use the unitarity of UA\ For z,w G T, we have
*;*, = **wu*AuA*z = (uA*wyuA*z
(9.168)
= (*A«,M;_ 1 )*(^ A Z M;_ 1 ) =
Using MA
1
MA-I**AW*AZM*-1
— M A -i, we finally obtain
K(Aw | A*) = **AW*AZ
= MA**W¥ZMZ
= MAK(w | z)M*A .
(9.169)
This shows that the reproducing kernel transforms "covariantly." To get the kernel in the new coordinate frame, simply multiply by two 3 x 3 matrices on both sides to adjust the field values. (See also Jacobsen and Vergne (1977), where such invariances of reproducing kernels are used to construct unitary representations of the conformal group.) To find \I/Az{x') from ^z(x'), we now take the boundary values of K(Azf \ Az) as y' —► 0, according to (9.68), and use (9.113): $A?(AI') = M
A
*
Z
( I X ,
=> *AZ(X')
= MA*Z(A-1X')M:.
(9-170)
Returning to our problem of finding ^iy with J / 6 V J , recall that we can write y = Ay', where y' = (0, A) with A= V ^ = \ A o - | y | 2 > 0
and vA = ^ .
(9.171)
Then (9.170) gives Viytf) = M^O^A-'X^M: . (9.172) Since MA is known (although we have not given its explicit form), (9.172) enables us to find i&iy explicitly, once we know M*o,tAfc>rA > 0. So far, we have used the properties of the wavelets under conjugations, translations and Lorentz transformations. We still have one parameter left, namely, A > 0. This can now be eliminated using dilations. From (9.130) it follows that, for any z G T, *az(ax')
= a-4*z{x%
a^0.
(9.173)
Therefore *o,iA(*) - A - ^ o ^ A - 1 * ) . (9.174) We have now arrived at a single wavelet from which all others can be obtained by dilations, Lorentz transformations, translations and conjugation. We therefore denote it without any label and call it a reference wavelet:
*& = ^o,i •
(9.175)
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240
Of course, this choice is completely arbitrary: Since we can go from *$? to tyz , we can also go the other way. Therefore we could have chosen any of the \P z 's as t h e reference wavelet, t T h e wavelets with y = 0, i.e., z = x + i(0, 5) have an especially simple form, reminiscent of the one-dimensional wavelets: *x.t+i.(x') = s - 4 * { ^ )
= *-4* (
^
.^
)
•
(9-176)
In particular, this gives all the wavelets ^o,is labeled by E, which were used in the resolution of unity of Theorem 9.9. The point of these symmetry arguments is not only t h a t all wavelets can be obtained from \£. We already have an expression for all the wavelets, namely (9.130). T h e main benefit of the symmetry arguments is t h a t they give us a geometrical picture of the different wavelets, once we have a picture of a single one. If we can picture \I/, then we can picture ^fx+iy as a scaled, moving, translated version of \£. Thus it all comes down to understanding *&. T h e matrix elements of *&(x) (we drop the prime, now t h a t z = (0, is)) can be obtained form (9.130) by letting w = — ix. and w$ = 1 — it: [*(x,t)]mn < W ( 1 - it)2 ~ r2} + 2xmxn 2
+ 2(1 - it) £ L i emnkxk 2
TT [(1 - it)
+
(9-177)
2 3
r ]
where r = |x|. But \£(x) is still a complex matrix-valued function in R 4 , hence impossible to visualize directly. We now eliminate the polarization degrees of freedom. Returning to the Fourier representation of solutions, note t h a t if f (p) already satisfies the constraint (9.12), then H(p) f(p) = f (p) and (9.101) reduces to
/ d^z) \F(z)\2 = I % \i{p)\2 = ||F||2. JE
Define the scalar wavelets by ^z(x')
(9.178)
JC &
= [ dp 2u26{p • y) eip<x'~^ Jc
,
(9.179)
and the corresponding scalar kernel K : T x T —*• C by K(z' \z)=
f dp 4uj29(p • y') 6(p ■ y) eip
(9.180)
Jc * For this reason we do not use the term "mother wavelet." The wavelets are more like siblings, and actually have their genesis in the conformal group. The term "reference wavelet" is also consistent with "reference frame," a closely related concept.
9. Introduction to Wavelet
Electromagnetics
241
Then (9.178), with essentially the same argument as in the proof of Theorem 9.9, now gives the relations F(z') = / dfi(z) K(z' | z) F(z) JE
F = JE
d^{z) Vz F(z)
F{x') = f dn{z)Vz(x')F(z) JE
pointwise in z' e T,
weakly in H,
(9.181)
a.e.
The first equation states that K{z' \ z) is still a reproducing kernel on the range AE of RE ' *H —»• C2 (E). The second and third equations state that an arbitrary solution F € H can be represented as a superposition of the scalar wavelets, with F = RE F as a (vector) coefficient function. Thus, when dealing with coefficient functions in the range AE of RE, it is unnecessary to use the matrix wavelets. The main advantage of the latter (and a very important one) is that they can be used even when the ceofficient function 3>(x, is) does not belong to the range AE of RE, since their kernel projects <£ to AE, by Theorem 9.13. The scalar wavelets and scalar kernel were introduced and studied in Kaiser (1992b, c, 1993). They cannot, of course, be solutions of Maxwell's equations precisely because they are scalars. But they do satisfy the wave equation, since their Fourier transforms are supported on the light cone. We will see them again in Chapter 11, in connection with acoustics. To see their relation to the matrix wavelets, note that II(p) is a projection operator of rank 1, hence trace U(p) = 1. Taking the trace on both sides of (9.92) and (9.112) therefore gives Vz(x') = trace * z ( x ' ) ,
K(z' | z) = trace K(z' | z).
(9.182)
Taking the trace in (9.182) amounts, roughly, to averaging over polarizations. From (9.130) we obtain the simple expression
*.EO = ^,yi~ W '7,. {^ s »+f»-,f 2
2
s
ir [c wfi+w-w] [ w = y + 2(x-x'). The trace of the reference wavelet is therefore 1 3(1 — it)2 — r2 trace g ( x , t) = ^ [(\ _ .^ + ^ = * ( r , t ) , r = |x|,
(9.183) (9.184)
where we have set c — 1. Because it is spherically symmetric, the scalar reference wavelet ^(r, i) can be easily plotted. Its real and imaginary parts are shown at the end of Chapter 11, along with those of other acoustic wavelets. (See the graphs with a = 3.) These plots confirm that ^ is a spherical wave converging towards the origin as t goes from — oo to 0, becoming localized in a sphere of radius \/3 around the origin at t — 0, and then diverging away from the origin as t goes from 0 to oo. Even though \£(r,0) does not have compact support (it
A Friendly Guide to Wavelets
242
decays as r ~ 4 ) , it is seen to be very well localized in |x| < \ / 3 . Reinserting c, the scaling and translation properties tell us t h a t at t = 0, SE^is is well localized in the sphere |x' — x | < y/Sc \s\. Also shown are the real and imaginary parts of
*(0'i) = ^ r ^ '
(9 185)
-
which give the approximate duration of the pulse \&. An observer fixed at the origin sees most of the pulse during the time interval —y/S < t < \ / 3 , although the decay in time is not as sharp as the decay in space at t — 0. By the scaling property (9.176), ^ X ) i s has a duration of order 2\/S\s\. These pulses can therefore be made arbitrarily short. Recall t h a t in t h e one-dimensional case, wavelets must satisfy the admissibility condition f dtip(i) = 0, which guarantees t h a t ip has a certain amount of oscillation and means t h a t the wavelet transform ip* t f represents detail in / at the scale s (Section 3.4). The plots in Chapter 11 confirm t h a t \I> does indeed have this property in the time direction, but they also show t h a t it does not have it in the radial direction. In the radial direction, ^ acts more like an averaging function t h a n a wavelet! This means t h a t in the spatial directions, the analytic signals F(x + iy) are blurred versions of the solutions F(x) rather t h a n their "detail." This is to be expected because F is an analytic continuation of F . In fact, since y is the "distance" vector from R 4 , y controls the blurring: y0 controls the overall scale, and y controls the direction. This can be seen directly from t h e coefficient functions of the wavelets in the Fourier domain. Consider the factor Ry(p)
= 2u26(p • y)e~py
(9.186)
occurring in the definitions of ^z and tyz. Suppose t h a t y G V+. Then Ry(p) for p G C-, and for p G C+ we have P • 3/> w(yo - |y|/c) > 0,
= 0
(9.187)
with equality if and only if either y = 0 or p = c j ^ ,
y/0.
(9.188)
If y = 0, p cannot be parallel to y (except for the trivial case p = 0, which was excluded from the beginning). Then all directions p ^ O get equal weights, which accounts for the spherical symmetry of ^ ( r , t). If y ^ 0, then Ry(p) decays most slowly in the direction where p is parallel to y. Along this direction, Ry(p)
= 2u2e-^yo-^
= py{uj).
(9.189)
To get a rough interpretation of Ry , regard py(uj) as a one-dimensional weight function on the positive frequency axis. Then it is easily computed t h a t the
9. Introduction to Wavelet
Electromagnetics
243
mean frequency and the standard deviation ("bandwidth") are
*(y) =
rvr >
2/o - | y | / c
a
(9-19°)
^ =—Vrr > 2/0 -
|y|/c
respectively. A similar argument applies when y G V'_ , showing t h a t Ry(p) favors the directions in C_ where p is parallel to y. The factor 6(p • y) in (9.186) provides stability, guaranteeing t h a t Fourier components in the opposite direction (which would give divergent integrals) do not contribute. Therefore we interpret Ry(p) as a ray filter on the light cone C, favoring the Fourier components along the ray Ray (y) = {peC:
sign p0 = sign y0, p = u ; y / | y | } ,
y + 0.
(9.191)
Now t h a t we have a reasonable interpretation of \I/o,is with s > 0, let us return to the wavelets ^iy with arbitrary y G V+ . By (9.172), such wavelets can be obtained by applying a Lorentz transformation to ^o,is- To obtain V — (y? 2/0)5 we must apply a Lorentz transformation with velocity v(2/) - 2/o
(9.192)
to the four-vector (0, s). To find 5, we apply the constraint (9.164): c2yl - | y | 2 = c V
=► a = ^ 0 2 - | y l 2 / c 2 = W l - "
2
/c
2
,
(9.193)
where v = |v| and we have reinserted the speed of light for clarity. T h e physical picture of ^iy is, therefore, as follows: ^fiy is a Doppler-shifted version of the spherical wavelet ^o,is- Its center moves with the velocity v = y/yo, so its spherical wave fronts are not concentric. They are compressed in the direction of y and rarified in the opposite direction. This may also be checked directly using (9.183). To summarize: All eight parameters z = (x, xo) + i(y,yo) G T have a direct physical interpretation. The real space-time point (x, XQ) is the focus of the wavelet in space-time, i.e., the place and time when the wavelet is localized; y/yo is the velocity of its center; the quantity
R(y) = VW
(9-194)
is its radius at the time of localization, in the reference frame where the center is at rest; and the sign of yo is its helicity (Section 9.2). Since all these parameters, as well as the wavelets t h a t they label, were derived simply by extending the electromagnetic field to complex space-time, it would appear t h a t the causal t u b e T is a natural arena in which to study electromagnetics. There are reasons to believe t h a t it may also be a preferred arena for other physical theories (Kaiser [1977a, b, 1978a, 1981, 1987, 1990a, 1993, 1994b,c]).
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9.8 Conclusions The construction of the electromagnetic wavelets has been completely unique in the following sense: (a) The inner product (9.31) on solutions is uniquely determined, up to a constant factor, by the requirement that it be Lorentzinvariant. (b) The analytic extensions of the positive- and negative-frequency parts of F to T+ and 71 , respectively, are certainly unique, hence so is ¥(z). (c) The evaluation maps 8ZF = F(z) are unique, hence so are their adjoints \J>Z = £* with respect to the invariant inner product. On the other hand, the choice of the Euclidean space-time region £^asa parameter space for expanding solutions is rather arbitrary. But because of the nice transformation properties of the wavelets under the conformal group, this arbitrariness can be turned to good advantage, as explained in the below. Although we have used only the symmetries of the electromagnetic wavelets under translations, dilations and Lorentz transformations, their symmetries un der the full conformal group C, including reflections and special conformal trans formations, will be seen to be extremely useful in the next chapter, where we apply them to the analysis of electromagnetic scattering and radar. The Eu clidean region E, chosen for constructing the resolution of unity, may be re garded as the group of space translations and scalings acting on real space-time by gx,isx' — sx' + (x>0)- As such, it is a subgroup of C. E is invariant un der space rotations but not under time translations, Lorentz transformations or special conformal transformations. This noninvariance can be exploited by applying any of the latter transformations to the resolution of unity over E and using the transformation properties of the wavelets. The general idea is that when g G C is applied to (9.107), then another such resolution of unity is ob tained in which E is replaced by its image gE under g. If gE = E, nothing new results. If g is a time translation, then the wavelets parameterized by gE are all localized at some time t ^ 0 rather than t = 0. If g is a Lorentz transformation, then gE is "tilted" and all the wavelets with z G gE have centers that move with a uniform nonzero velocity rather than being stationary. Finally, if g is a special conformal transformation, then gE is a curved submanifold of T and the wavelets parameterized by gE have centers with varying velocities. This is consistent with results obtained by Page (1936) and Hill (1945, 1947, 1951), who showed that special conformal transformations can be interpreted as mapping to an accelerating reference frame. The idea of generating new resolutions of unity by applying conformal transformations to the resolution of unity over E is another example of "deformations of frames," explained in the case of the Weyl-Heisenberg group in Section 5.4 and in Kaiser (1994a).
Chapter 10
Applications to Radar and Scattering Summary: In this chapter we propose an application of electromagnetic wave lets to radar signal analysis and electromagnetic scattering. The goal in radar, as well as in sonar and other remote sensing, is to obtain information about ob jects (e.g., the location and velocity of an airplane) by analyzing waves (electro magnetic or acoustic) reflected from these objects, much as visual information is obtained by analyzing reflected electromagnetic waves in the visible spec trum. The location of an object can be obtained by measuring the time delay r between an outgoing signal and its echo. Furthermore, the motion of the ob ject produces a Doppler effect in the echo amounting to a time scaling, where the scale factor s is in one-to-one correspondence with the object's velocity. The wideband ambiguity function is the correlation between the echo and an arbitrarily time-delayed and scaled version of the outgoing signal. It is a maxi mum when the time delay and scaling factor best match those of the echo. This allows a determination of s and r, which then give the approximate location and velocity of the object. The wideband ambiguity function is, in fact, noth ing but the continuous wavelet transform of the echo, with the outgoing signal as a mother wavelet! When the outgoing signal occupies a narrow frequency band around a high carrier frequency, the Doppler effect can be approximated by a uniform frequency shift. The wideband ambiguity function then reduces to the narrow band ambiguity function, which depends on the time delay and the frequency shift. This is essentially the windowed Fourier transform of the echo, with the outgoing signal as a basic window. The ideas of Chapters 2 and 3 therefore apply naturally to the analysis of scalar-valued radar and sonar sig nals in one (time) dimension. But radar and sonar signals are actually waves in space as well as time. Therefore they are subject to the physical laws of propagation and scattering, unlike the unconstrained time signals analyzed in Chapters 2 and 3. Since the analytic signals of such waves are their wavelet transforms, it is natural to interpret the analytic signals as generalized, multi dimensional wideband ambiguity functions. This idea forms the basis of Section 10.2. Prerequisites: Chapters 2, 3, and 9.
10.1 A m b i g u i t y F u n c t i o n s for T i m e S i g n a l s Suppose we want to find the location and velocity of an object, such as an airplane. One way to do this approximately is to send out an electromagnetic wave in the direction of the object and observe the echo reflected to the source. As explained below, the comparison between the outgoing signal and its echo allows an approximate determination of the distance (range) R of the object
G. Kaiser, A Friendly Guide to Wavelets, Modern Birkhäuser Classics, DOI 10.1007/978-0-8176-8111-1_10, © Gerald Kaiser 2011
245
A Friendly Guide to Wavelets
246
and its radial velocity v along the line-of-sight. This is the problem of radar in its most elementary form. For a thorough treatment, see the classical books of Woodward (1953), Cook and Bernfeld (1967), and Rihaczek (1969). The wavelet-based analysis presented here has been initiated by Naparst (1988), Auslander and Gertner (1990), and Miller (1991), but some of the ideas date back to Swick (1966, 1969). Although we speak mainly of radar, the considerations of this section apply to sonar as well. The main difference is that the outgoing and returning signals represent acoustic waves satisfying the wave equation instead of electromagnetic waves satisfying Maxwell's equations. In Section 10.2 we outline some ideas for applying electromagnetic wavelets to radar. The outgoing radar signal begins as a real-valued function of time, i.e., ■ip : R —»• R, representing the voltage fed into a transmitting antenna. The antenna converts ip(t) to a full electromagnetic wave and beams this wave in the desired direction. (In the "search" phase, all directions of interest are scanned; once objects are detected, they can be "tracked" by the method described here.) The outgoing electromagnetic wave can be represented as a complex vectorvalued function F(x,t) = B(x,£) + £E(x,£) on space-time, i.e., F : R 4 —► C 3 , as explained in Chapter 9. (Of course, only the values of ip and F in bounded regions of R and R 4 , respectively, are of practical interest.) The exact relation between F ( x , t ) and ip(t) does not concern us here. For now, the important thing is that the returning echo, also a full electromagnetic wave, is converted by the same apparatus (the antenna, now in a receiving mode) to a real time signal /(£), which again represents a voltage. We will say that ip(i) and f(t) are proxies for the outgoing electromagnetic wave F(x, t) and its echo, respectively. Although it might appear that a detailed knowledge of the whole process (transmission, propagation, reflection and reception) might be necessary in order to gain range and velocity information by comparing / and ip, that process can, in fact, be largely ignored. The situation is much like a telephone conversation: a question ip(t) is asked and a response f(t) is received. To extract the desired information, we need only two facts, both of which follow from the laws of propagation and reflection of electromagnetic waves: (a) Electromagnetic waves propagate at the constant speed c, the speed of light ( « 3 x 108 m/sec). Consequently, if the reflecting object is at rest and the reflection occurs at a distance R from the radar site, then the proxy / suffers a delay equal to the time 2R/c necessary for the wave to make the round trip. Thus OD
f(t)=axl>(t-d),
where
d=—
if
v = 0.
(10.1)
Here a is a constant representing the reflectivity of the object and the attenuation (loss of amplitude) suffered by the signal.
10. Applications to Radar and Scattering
247
(b) An electomagnetic wave reflected by an object moving at a radial velocity v suffers a Doppler effect: the incident wave is scaled by the factor s(v) = (c + v)/(c — v) upon reflection. To see this, assume for the moment that the object being tracked is small, essentially a point, and that the outgoing signal is sufficiently short that during the time interval when the reflection occurs, the object's velocity v is approximately constant. Then the range varies with time as R(t) = R0 + vt. (10.2) Since the range varies during the observation interval, so does the delay in the echo. Let d(t) be the delay of the echo arriving at time t. Then cd{t) = 2R(t - d(t)/2) = 2R0 + 2vt - vd(t),
(10.3)
which gives m
v J
=
Ro_±j*
c+v generalizing (10.1). When this is substituted into (10.1), we get t
m=a^(t-^-t-^.)=an,(
V
c+v
c+ vJ
-^), \ SQ J
v
J
(10.5)
where s0 =
2R0 s0 ~ 1 c+ v D CTO , r0 = , =» v = — c, R0 = —. c—v c—v so + 1 so + 1
(10.6)
All material objects move with speeds less than c, hence so > 0 always. If v > 0 (i.e., the object is moving away), then / is a stretched version of rp. (A simple intuitive explanation is that when v > 0, successive wave fronts take longer to catch up with the object, hence the separation of wave fronts in the echo is larger.) Similarly, when v < 0, the reflected signal is a compressed version of ip. The value of a clearly depends on the amount of amplification performed — 1/2
on the echo. We choose a — s0 2
, so that / has the same "energy" as xfj, i.e.,
2
II/II = IWI For any r € R and any s > 0, introduce the notation 1>t,T(t) = s-1/2^
(—) ,
(10.7)
already familiar from wavelet theory, so that f(t)=A„,r0(t)-
(10.8)
In the frequency domain, this gives /(w) = «S/2 e-2^^
iP(sou,).
(10.9)
If v > 0, then so > 1 and all frequency components are scaled down. If v < 0, then so < 1 and all frequency components are scaled up. This is in accordance
A Friendly Guide to Wavelets
248
with everyday experience: the pitch of the horn of an oncoming car rises and that of a receding car drops. The objective is to "predict" the trajectory of the object, and this is ac complished (within the limitations of the above model) by finding RQ and v. To this end, we consider the whole family of scaled and translated versions of ip: {ips>T : 5 > 0 , r G R } .
(10.10)
We regard ipSjT as a test signal with which to compare / . A given return f(t) is matched with ipSiT by taking its inner product with ipSjT : f(s,r) =
dt s-1^
(^-\
f{t).
(10.11)
(Recall that ip is assumed to be real. If it were complex, then ip would have to be used in (10.11).) / ( s , r) is called the wideband ambiguity function of / (see Auslander and Gertner [1990], Miller [1991]). If the simple "model" (10.8) for / holds, then /(S,r) = <^,T,^0,T0>. (10.12) By the Schwarz inequality,
I / ( * , T ) | < US,T\\ N W „ I I = M I 2 ,
(io.i3)
with equality if and only if s = SQ and T — TQ. Thus, we need only maximize | / ( S , T ) | in order to find SQ and TQ, assuming (10.8) holds. (Note that since / and ip are real, it actually suffices to maximize / ( s , r ) instead of |/(s,r)|.) But in general, (10.8) may fail for various reasons. The reflecting object may not be rigid, with different parts having different radial velocities (consider a cloud). Or the object, though rigid, may be changing its "aspect" (orientation with respect to the radar site) during the reflection interval, so that its different parts have different radial velocities. Or, there may be many reflecting objects, each with its own range and velocity. We will model all such situations by as suming that there is a distribution of reflectors in the time-scale plane, described by a density D(s,r). Then (10.8) is replaced by
f{t) = JJ^^^So,Toit)Diso,To).
(10.14)
(We have included the factor SQ~2 in the measure for later convenience. This simply amounts to a choice of normalization for D, since the factor can always be absorbed into D. It will actually prove to be insignificant, since only values so ~ 1 will be seen to be of interest in radar.) D(so,ro) is sometimes called the reflectivity distribution. In this generalized context, the objective is to find D(S,T). This, then, is the inverse problem to be solved: knowing ip and given / , find D.
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Scattering
249
As the above notation suggests, wavelet analysis is a natural tool here. /(.s, r ) is just the wavelet transform of / with ip as the mother wavelet. If 'ip is admissible,t i.e., if C=
p||^H|2
/
(10.15)
M
then / can be reconstructed by /(*) = C~l
II
^
lk,T(t)/>, r)
a.e..
(10.16)
Comparison of (10.14) and (10.16) suggests t h a t one possible solution inverse problem is = C-1f(s,T),
D(s,r)
of the (10.17)
l
i.e., D is the wavelet transform of C~ f. However, this may not be true in general. (10.17) would imply t h a t D is in the range of the wavelet transform with respect to ip: D e ^
= {§: 9(8, r ) = ^Tg
: g G L2(R)}.
(10.18)
But we have seen in Chapters 3 and 4 t h a t (a) T$ is a closed subspace of L2(dsdr/s2)} and (b) a function D(s,r) can belong to T^ if and only if it satisfies the consistency condition with respect to the reproducing kernel K associated with i\). For the reader's convenience, we recall the exact statement here. T h e o r e m 1 0 . 1 . Let D e L2(dsdr/s2), dsdr
i.e., D(S,T)\2
Then D belongs to T^ if and only if it satisfies the integral D(s, T) = !f
^
^
(10.19)
equation
K(s, r | so, r0) D(s0, r 0 ),
(10.20)
where K is the reproducing kernel associated with ip: K(S,T\S0,T0)
=
C~l{ipSiT,il)So,To).
t It t u r n s out t h a t t h e "DC component" of i{), i.e., "0(0) = J^°
dt tp(t),
has no
effect on t h e radiated wave. T h a t is, ip(t) — ^ ( 0 ) has t h e same effect as if)(t), so we m a y as well assume t h a t V>(0) = 0. This is related to t h e fact t h a t solutions in 7i have no D C component; see (9.32). If ip has rapid decay in the time domain (which is always t h e case in practice), t h e n V>(o;) is smooth near OJ = 0 and t/;(0) = 0 implies t h a t i/> is admissible. Hence in this case, the physics of radiation goes hand-in-hand with the mathematics of wavelet analysis!
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A Friendly Guide to Wavelets
If (10.20) holds, then D(s,r)
where g G L2(R)
= g(s,r),
S(«) = C - 1 \ \ — - $
t t T
(t)D(s, T)
is given by a.e.
(10.21)
Incidentally, (10.20) also gives a necessary and suffficient condition for any given function D(s,r) to be the wideband ambiguity function of some time signal. (For example, / ( s , r ) must satisfy (10.20).) In radar terminology, K(s, T\SQ, TQ) is the (wideband) self-ambiguity function of ?/>, since it matches translated and dilated versions of ip against one another. (By contrast, / ( s , r ) is called the wideband cross-ambiguity function.) Thus (10.20) gives a relation that must be satisfied between all cross-ambiguity functions and the self-ambiguity function. Returning to the inverse problem (10.14), we see t h a t (10.17) is far from a unique solution. There is no fundamental reason why t h e reflectivity D should satisfy the consistency condition (10.20). As explained in Chapter 4, the or thogonal projection P^ to T^ in L2(ds dr/s2) is precisely the integral operator in (10.20): //
(PJ,D)(S,T)=
° 2 ° if(s,Tlso,To)£>(so,T 0 ).
Equation (10.20) merely states t h a t D belongs to T^ if and only if P^D = D. A general element D G L2(dsdr/s2) can be expressed uniquely as D(s,r)
= DVi(s,T) + D±{s1T),
(10.22)
where D y belongs to T^ and D± belongs to the orthogonal complement T± of T^ in L2{dsdr/s2). Then (10.14) implies t h a t v-i 2,„ „ \ __ n-\1nlt*
C-1f(81T)
= C- tl>lrf=
r
ff
ds0dr0
-^V^^(S^lS0,ro)D(5o,To)
//
JJ = (P^D)(S,T) =
4
(10.23)
D„(S,T).
This is the correct replacement of (10.17), since it does not assume t h a t D 6 T^. Equations (10.22) and (10.23) combine to give the following (partial) solution to the inverse problem: T h e o r e m 1 0 . 2 . The most general solution is given by D(S,T) = C-lf{S)r) where h(s,r)
is any function d
^PL
D(S,T)
of (10.14) in
+ /i(S,r),
L2(dsdr/s2) (10.24)
in T^ , i.e., K
^
T
I *o, TO) h(s0l ro) = 0.
(10.25)
T h e inverse problem has, therefore, no unique solution if only one outgoing signal is used. We are asking too much: using a function of one independent variable,
1 0 . Applications
to Radar and
251
Scattering
we want to determine a function of two independent variables. It has therefore been suggested that a set of waveforms ip1, ip2, tp3,... be used instead. The first proposal of this kind, made in the narrow-band regime, is due to Bernfeld (1984, 1992), who suggested using a family of chirps. In the wideband regime, similar ideas have been investigated by Maas (1992) and Tewfik and Hosur (1994). Equation (10.23) then generalizes to P^D
= C^fk,
A; = 1,2,3,...,
(10.26)
where fk(s,r) = {ip* , fk ) is the ambiguity function of the echo fk(t) of ipk(t) and P^k is the orthogonal projection to the range of the wavelet transform asso ciated with ipk. However, such schemes implicitly assume that the distribution function D(s, r) has an objective significance in the sense that the same D(s, r ) gives all echos fk from their incident signals ipk. There is no a priori reason why this should be so. Any finite-energy function and, in particular, any echo /(£), can be written in the form (10.14) for some coefficient function D ( S , T ) , namely, D = Crlf. The claim that there is a universal coefficient function giving all echos amounts to a physical statement, which must be examined in the light of the laws governing the propagation and scattering of electromagnetic waves. I am not aware of any such analyses in the wideband regime. The term "ambiguity function," as used in the radar literature (see Wood ward [1953], Cook and Bernfeld [1967], and Barton [1988]), usually refers to the narrow-band ambiguity functions, which are related to the wideband versions as follows.^ The latter can be expressed in the frequency domain by applying Parseval's identity to its definition (10.11): oo
/
dwV5e 2 , r f a "-^(sw)/(w).
(10.27)
-OO
All objects of interest in radar travel with speeds much smaller than light. Thus \v\/c
£±£«l
+
^,
TS
.?**?*>.
(10.28)
c—v c c— v c If the maximum expected speed is 0 < v m a x <^ c and the maximum range of the radar is -Rmax, then we have, to first order in c _ 1 : |
s
_ ! | < : ^
?
|
T
|<^=S.
(10.29)
t Various schemes exist for going from the wideband to the narrow-band regimes. T h e procedure given here is m e a n t t o be conceptually as simple as possible, a n d no a t t e m p t is m a d e t o be mathematically rigorous or invoke subtle group-theoretical con siderations. For more detailed analyses, see Cook and Bernfeld (1967), Auslander and Gertner (1990), and Kalnins and Miller (1992).
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252
Therefore, in practice only r w 0 and s « 1 are of interest, and we may assume that D(S,T) is small outside of a neighborhood of (1,0). (The factor s^~2 in (10.14) is therefore merely of academic interest in radar.) Suppose now that 4>(LJ) is concentrated mainly in the double frequency band 0 < a < \u\ < 7,
where
iJlSL <^ i.
(10.30)
The bandwidth j3 and the central frequency uc are defined by P=1 - a ,
UJC = ^ ^ ,
(10.31)
and (10.30) means that (3
(10.32)
Now su « ( 1 + — j u = u + —
.
(10.33)
If ip is a narrow-band signal, the second term can be approximated by letting " - { Thus su^
(LJ + 4> i f a ; > 0 \ , [UJ — (p it to < {)
"C ( -uc where
i
!">° if u) < 0.
2ucv 2v <> / = 4>(v) = —— = —, c Ac
(10-34) v ' ., (10.35)
hnQ
where Ac is the carrier wavelength. That is, for narrow-band signals, the effect of Doppler scaling can be approximated as a "Doppler shift" that translates all positive-frequency components by <j>{v) and all negative-frequency components by —<j>[v). We can then express the ambiguity function in terms of
10. Applications to Radar and Scattering
253
the numerical analysis will be much simpler if the analytic signal is demodulated, i.e., if t h e high carrier frequency is removed, so t h a t only the "message" remains. This amounts to translating the spectrum to the left by uc. Thus we define a new (complex) signal ty(t) by /•OO
V(t) = 2e-27viu,ct
dLje27riujt^(u).
/
(10.37)
Jo T h e echo f(t) has a band structure similar to t h a t of ip(t), since the Doppler shift is very small compared to the carrier frequency. Thus we perform the same operations on it: filter out the negative-frequency band, then demodulate. This gives the complex signal /•OO
F(t) = 2e~27TiuJct
dLje27riujtf(u).
/
(10.38)
Jo \I> and F carry the same information as do ^ and / , but have the advantage t h a t they oscillate much more slowly and undergo uniform Doppler shifts (rather t h a n Doppler scalings, which act oppositely on the negative frequencies). If / = ipSiT , then for u > 0 (10.35) gives
« e-27tiuJTil>(uj
(10.39)
+ (f)).
Hence /•OO
F{t) « 2 e~2n
iUct
/ Jo
duj e27r iuj{t~T)
j>(u + )
OO
dM e2*iuj{t-T^(u)
/
(10.40)
/•OO =
2 e -27T
iu,ct
e -27T
iftt-T)
/
^
e27T
iu,{t-r)
^ )
?
Jo since i>(uj) vanishes for \LJ\ < \(j>\. It follows t h a t F(t)» e"27r^^,T(t),
(10.41)
where ^ , r ( t ) = e~2vi(f,t^(t-T),
0 = 0(0,T) = (U;C-0)T.
(10.42)
T h e phase factor e - 2 7 " 6 ' is time-independent, hence it does not affect the anal ysis and can be ignored. Therefore, the echo from a point reflector can be represented by ^ T in the narrow-band approximation. For a given return f(t),
A Friendly Guide to Wavelets
254
we define t h e narrow-band
cross-ambiguity
function
in terms of F(t) by*
oo
/ If / = ^ S o , T o , then F = e-2"i9° s0 = 1 + — - , C
dt e2ni(t>t^{t-r)F(t).
(10.43)
-OO
^ 0 , T o , where <j)Q = — - , Ac
6>0 = (wc - 0)T0 .
(10.44)
Therefore |F(^T)| = |(^,
T
,^0,
T 0
)|<||*||2,
(10.45)
with equality if and only if r = r 0 and (j> =
F(t) = JJd4>dr^,T(t)Dm(4>,r).
(10.46)
(The narrow-band distribution JD N B (^>, r ) can be related t o t h e wideband dis by noting t h a t it contains t h e phase factor e~2lT%e and t h a t tribution D(S,T) d(p = ucds.) Equation (10.46) poses an inverse problem: given F , find DNB. J u s t as wavelet analysis was used t o analyze and solve t h e wideband inverse problem, so can time-frequency analysis be used t o do t h e same for t h e narrow band problem. This is based on t h e observation t h a t F(<^, r) is just the windowed Fourier transform of F with respect to the complex window \l/(£) (Chapter 2). T h e corresponding self-ambiguity function K(4>,r\4,0,T0)
= C-1(<5>4„T,<S>
CV = | | * | | 2 ,
(10.47)
is t h e reproducing kernel associated with t h e frame { ^ > T } . All t h e observations made earlier about t h e existence and uniqueness of solutions t o t h e wideband inverse problem now apply t o the narrow-band case as well, since the mathe matical structure in both cases is essentially identical, as explained in Chapter 4. In particular, t h e most general solution of (10.46) is £>NB (4>, T) = C^F(
(10.48)
where H((j), r) is any function in t h e orthogonal complement T^ of t h e range T^ of t h e windowed Fourier transform (with respect t o \I>) in L 2 ( R 2 ) . (See t In many texts, the narrow-band ambiguity function is defined as |F(, r ) | 2 rather than F(,r). The wideband ambiguity functions are real when the signals are real, therefore they can be maximized directly to find the matching delay and scale. The narrow-band versions, on the other hand, are necessarily complex and therefore it is their modulus that must be maximized.
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Chapters 2 and 4.) Again, the solution is not unique and it is necessary to use a variety of outgoing signals to obtain uniqueness. Here, too, the question of the "objectivity" of D NB (its independence of ^ ) must be addressed (see Schatzberg and Deveney [1994]). The narrow-band ambiguity function was first defined by Woodward (1953) in his seminal monograph on radar. Wideband ambiguity functions were stud ied by Swick (1966, 1969). The renewed interest in the wideband functions (Auslander and Gertner [1990], Miller [1991], Kalnins and Miller [1992], Maas [1992]) seems to have been inspired by the development of wavelet analysis. One might argue that since radar signals are usually narrow-band, there is no point in considering the wideband ambiguity functions. However, there are some good reasons to do so. (a) Since acoustic waves satisfy the wave equation, they also propagate at a constant speed. Therefore, ambiguity functions play as fundamental a role in sonar as they do in radar. However, whereas radar usually involves narrow-band signals, this is no longer true for sonar since the frequencies involved are much lower and the speed of sound (in water, say) is much smaller than the speed of light. Acoustic wavelets similar to the electromagnetic wavelets are constructed in Chapter 11. (b) Even for narrow-band signals, the formalism associated with the wideband ambiguity functions is conceptually simpler and more intuitive than the one associated with the narrow-band version. The relation of the velocity to the scale factor is direct and clear, whereas its relation to the Doppler frequency is more convoluted, depending as it does on transforming to the frequency domain. Also, the elimination of negative frequencies and subsequent demodulation make F((j),r) a less intuitive and direct object than f(s,r). (c) Accelerations are simple to describe in the time domain but very complicated in the frequency domain. Therefore an extension of the ambiguity function concept to include accelerations is much more likely to exist in the time-scale representation than in the time-frequency representation. Yet, most treatments of radar use the narrow-band ambiguity functions without even referring to the wideband versions. One reason is practical: the demodu lated signals oscillate much more slowly than the original ones, hence they are easier to handle. (Lower sampling rates can be used, for example.) Another reason may be that wavelet analysis is still new and little-known. A third rea son (suggested to me by M. Bernfeld, private communication) could be that before the advent of fast digital computers, ambiguity functions were routinely computed by analog methods using filter banks, which are better able to handle windowed Fourier transforms than wavelet transforms since the former require modulations while the latter require scaling. Given all this, a case can be made
A Friendly Guide to Wavelets
256
that modern radar analysis ought to explore the wideband regime, even when all the signals are narrow-band.
10.2 The Scattering of Electromagnetic Wavelets When the detailed physical process of propagation and scattering is ignored, the net effect of a point object on the proxy ip(t) representing the outgoing electromagnetic wave is the transformation m
.- ^,rW = s '
1
^ ( ^ )
,
(10.49)
where r is the delay and s = (c + v)/(c — v) is the scaling factor induced by the velocity of the object. In the narrow-band regime, this reduces to *(t) •-> *0, r (*) = e~2vi(f,t V(t - r ) ,
(10.50)
where <j> = 2v/Xc is the Doppler frequency shift induced on the positive-frequency band by the velocity and we have ignored the phase factor e - 2 7 r t ^(^ T ) hi (10.41). *&(i) is the demodulated Gabor analytic signal of ip(t). In each of these cases, we have a set of operations acting on signals: in the first case, it is translations and scalings, while in the second it is translations and modulations. For any finite-energy signal /(£), define Us,Tf and W
= s-1"/
(j^j
,
(WV/X*) = *-*"*
/ ( * ~ r),
(10.51)
so that, in particular, *Ps,r = Us,Ti),
V^r
= W^r*.
(10.52)
UatT and WfaT are invertible operators on L 2 (R) which preserve the energy: WUs.rff = ll/ll2
and
||W^, T /|| 2 = H/ll2.
(10.53)
By polarization, it follows that they preserve inner products, hence they are unitary: UZr = tf.Tr1, W?,r = W£ . (10.54) Furthermore, it is easily verified that in each case, applying two of the operators consecutively is equivalent to applying a single one, uniquely determined by the two. Namely, US,,T,US,T = US<S,S,T+T>,
WpyW^
= e2vi+T' Wp+^+r
.
(10.55)
Since Uito is the identity operator, C/i/ S) _ T / s is seen to be the inverse of U8fT . Hence the operators UST form a group: the inverses and products of t/'s are
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257
U's. In fact, t h e index pairs (s,r) themselves form a group, called t h e affine group or ax + b group, which we denote by A. I t consists of t h e half-plane {(s, r ) € R 2 : s > 0}, acting on time variable by affine transformations:
11—> (s,T)t = st + r.
(10.56)
We write g = (s, r ) for brevity, and (10.56) defines it as a m a p g : R —> R . Two consecutive transformations give (s', T ' ) ( S , r ) t = (s', r ' ) ( s t + r ) = (s's)t + ( s ' r + r ' ) ,
(10.57)
which implies t h e multiplication rule in A, (s\ r ' ) ( s , r ) = (s's, s ' r + r ' ) .
(10.58)
Equation (10.55) therefore shows t h a t the C/'s satisfy t h e same t h e group law as the elements of A: U9'Ug = Ug,g,
g\geA.
(10.59)
T h a t is, t h e action of (s, r ) on t h e time t G R i s now represented by a n induced action of US,T on signals / £ L 2 ( R ) . One therefore says t h a t t h e correspondence U : g *-> Ug (mapping t h e action on time t o t h e action on signals) is a rep resentation of A on L 2 ( R ) . Furthermore, since each Ug is a unitary operator, this representation is said t o b e unitary. Another important property of t h e representation is t h a t , according t o (10.52), all t h e proxy wavelets are obtained from t h e mother wavelet by applying C/'s:
Consequently, Ug^g = Ug'Ugil = Ug>g^ = ^g .
(10.60)
2
T h a t is, affine transformations, as represented on £ ( R ) by U, map wavelets to wavelets simply by transforming their labels. Note t h a t since (s, r)~lt = (t—r)/s, Ug acts on functions by acting inversely on their argument: (U„m) = s-^fig-H),
g=
(S,T).
(10.61)
(For example, t o translate a function t o t h e right by r , we translate its argument t t o t h e left by r . ) T h e phase factor in t h e second equation of (10.55) spoils t h e group law for the Ws. For example, t h e inverse of W^, T does not belong t o t h e set of Ws unless <jyr is an integer, since t h e phase factor prevents W-^^-r (the only possible candidate) from being t h e inverse. From this point of view, t h e W's are more complicated t h a n t h e C/'s. This can b e attributed t o t h e repeated "surgery" performed on signals in going from t h e wideband t o t h e narrow-band regimes. We will not be concerned here with these fine points, except t o note t h a t this
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A Friendly Guide to Wavelets
further supports the idea t h a t the wideband regime is conceptually simpler t h a n its narrow-band approximation.* The operators USjT and W^^ act on unconstrained functions, i.e., on signals whose only requirement is to have a finite energy. However, they owe their sig nificance to the fact t h a t these signals are "proxies" for electromagnetic waves t h a t are constrained (by Maxwell's equations) and undergo physical scattering. In fact, US,T and W^ )T are the effects on the proxies of the propagation and scat tering suffered by the corresponding waves: r is the delay due to the constant propagation speed, while s and <j> are the effects of "elementary" scattering in the wideband and narrow-band regimes, respectively. In this section we a t t e m p t to make the physics explicit by dealing with the electromagnetic waves them selves rather t h a n their proxies. Thus we consider solutions F ( x , t) of Maxwell's equations in the Hilbert space H (Section 9.2) rather t h a n finite-energy time signals ij)(t) in L 2 ( R ) . Suppose we know the electric and magnetic fields at t — 0. Then there is a unique solution F e H with those initial values. To find the fields at a later time t > 0, we simply evaluate F at t. Therefore, the propagation of the waves is already built into H. It can be represented simply by space-time translations of solutions, generalizing the time delay suffered by the proxies: F(x) t-» F(x - b), be R 4 . (Causality requires t h a t r > 0; in spacetime, this corresponds to 6 G V^, the future cone; see Chapter 9. However, we must consider all b € R 4 in order to have a group.) W h a t about scattering? T h a t is a difficult problem, and we merely pro pose a rough model t h a t remains to be developed and tested in future work. No a t t e m p t is made here to derive this model from basic theory, although t h a t presents a very interesting challenge. We take our clue from the effect of scatter ing on t h e proxies. Since the wideband regime is both more general and simpler, it will be our guide. The scaling transformation representing the Doppler effect on t h e proxies was derived by assuming t h a t the scatterer is small (essentially a point) and the signal is short. In this sense, it represents an "elementary" scattering event. More general scattering is modeled by superposing such elementary events, as in (10.14). But the Doppler effect is actually a space-time phenomenon rather t h a n just a time phenomenon. It results from the coordinate transformation * The set of W's can be made into a group by including arbitrary phase factors e27TlX (thus also enlarging the labels to (A, cj), r ) ) . This expanded set of labels forms a group known as the Weyl-Heisenberg group W, which is related to A by a process called "contraction." See Kalnins and Miller (1992). The narrow-band approximation is very similar to the procedure for going from relativistic to nonrelativistic quantum mechanics, with UJC playing the role of the rest energy mc2 divided by Planck's constant U. See Kaiser (1990a), Section 4.6.
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259
of space-time R 4 representing a "boost" from any reference frame (Cartesian coordinate system in space-time R 4 ) to another reference frame, in motion rel ative to the first. If the relative velocity is uniform, such a transformation is represented by a 4 x 4 matrix A called a Lorentz transformation. Because A preserves the indefinite Lorentzian scalar product in R 4 , it has the structure of a rotation through an imaginary angle. (This hyperbolic "rotation" is in the two-dimensional plane in R 4 determined by the time axis together with the ve locity vector; see Chapter 9.) This suggests that elementary scattering should be represented by Lorentz transformations. But these transformations do not form a group: the combination of two Lorentz transformation along different directions results in a rotation as well as a third Lorentz transformation. If we want to have a group of operations (and this is indeed a thing to be desired!), we must include rotations. Lorentz transformations and rotations together do form a group called the proper Lorentz group Co. Rotations are represented in space-time as linear maps g : R 4 —>• R 4 defined by x = (x, t)\-*gx=
(i?x, t),
(10.62)
where R is the 3 x 3 matrix implementing the rotation. Thus R is an orthogonal matrix with determinant 1. Reflections in R 3 are represented similarly, but now R has determinant —1. Clearly we want to include reflections in our group since they are the sim plest scattering events. The group consisting of all Lorentz transformations, ro tations and space reflections will be denoted by C. It is called the orthochronous Lorentz group (see Streater and Wightman [1964]). Like Co, £is& six-parameter group. Elements are specified by giving three continuous parameters represent ing a velocity, and three more representing a rotation. Reflections are specified by the discrete parameter "yes" or "no," once a plane for a possible rotation has been selected. ("Yes" means reflect in that plane.) We denote arbitrary el ements of £ by A and call them "Lorentz transformations," even if they include rotations and space reflections. Therefore, the minimum set of operations needed to model propagation and elementary scattering consists of all space-time translations (four parameters) and orthochronous Lorentz transformations (six parameters). Together, these form the 10-parameter Poincare group, which we denote by V. As the name implies, V is a group. A Lorentz transformation A followed by a translation I H X + I) gives g = (A, b) : x *-* gx = Ax + b, (10.63) generalizing the action (10.56) of the affine group on the time axis. If g' = (A', b') is another such transformation, then g'gx = A'gx + b' = A'(Ax + b) + b' = (A'A)x + (A'6 + b'),
(10.64)
A Friendly Guide to Wavelets
260
so g'g = (A7,&')(A, &) = (A'A, A'b + &')•
(10.65)
This is the law of group multiplication in V, which generalizes the law (10.58) for the affine group. The Poincare group is a central object of interest in relativistic quantum mechanics. It might be argued that it is unnecessary to consider relativistic ef fects in radar since all velocities of interest are much smaller than c. This is true, but it misses the point: the electromagnetic wavelets were constructed using the symmetries of Maxwell's equations, which form the conformal group C. Conse quently, these wavelets map into one another under conformal transformations, much as the proxy wavelets tps^T map into one another under affine transforma tions by (10.60). But C contains V as a subgroup, so the wavelets also map into one another under Poincare transformations. If relativistic effects are neglected, the electromagnetic wavelets no longer transform to one another under boosts, and the theory becomes much more involved from a mathematical and concep tual point of view. The nonrelativistic approximation is quite analogous to the narrow-band approximation, and it leads to a similar mutilation of the group laws. Indeed, to make full use of the symmetries of Maxwell's equations, we should include all conformal transformations. Aside from the Poincare group, C contains the uniform dilations x >—> ax (a ^ 0) and the so-called special con formal transformations, which form a four-parameter subgroup. A simple way to understand these transformations is to begin with the space-time inversion J:X
i e
"^~x'
J(X t)E
--
'
(1
AHxp-
°-66)
If translations are denoted by T^x = x + 6, then special conformal transforma tions are compositions C& = JT^J, i.e.,
cbX
=
j(^-+b) \x-x
)
= ^±±-2 (_j*L_ +
fr)2
=
,
x+{x x
l-\-2b-x
;t
+ (b-b){x-x)
,,
'
b£R\
(10.67) Since Cb is nonlinear, it maps flat subsets in R into curved ones. Depending on the choice of 6, this capacity can be used in different ways. Let Rg = {(x,0) : x G R 3 } be the configuration space at time zero. If b = (b, 0), then C& maps RQ into itself. Thus Cb maps flat surfaces in space at time zero to curved surfaces in the same space. This will be used to study the reflection of wavelets from curved surfaces. For some different choices of 6, Cb can be interpreted as boosting to an accelerating reference frame (Page [1936], Hill [1945, 1947, 1951]), just as the (linear) Lorentz transformations boost to uniformly moving reference frames. This could be used to estimate the acceleration of an object, just as Lorentz transformations are used to estimate the velocity. 4
10. Applications to Radar and Scattering
261
Thus we have arrived at the conformal group C as a model combining prop agation and elementary scattering. In order for this model to be useful, we need to know how conformal transformations affect electromagnetic waves. That is, we need a representation of C on H. Furthermore, we want the action of C on H to preserve the norms ||F|| of solutions, which means it must be unitary. A unitary representation of C analogous to the above representation of the affine group can be constructed as follows. Every conformal transformation is a map g : R 4 —> R 4 , not necessarily linear because g may include a special confor mal transformation like (10.67). g can be extended uniquely to a map on the complex space-time domain T, i.e., g:T
-* T.
(10.68)
For example, a Poincare transformation acts on z = x + iy £ T by (A, b)z = Kz + b = (Ax + b) + iAy,
(10.69)
and a special conformal transformations looks just like (10.67) but with x re placed by z. In fact, the action of g on T is nicer than its action on R 4 , since the latter is singular due to the inversion J, which maps x to infinity when x is light-like (x • x = 0). The inversion in T is Jz = z/z • z, and it is nonsingular since z2 cannot vanish on T (see Theorem 9.10). Thus all conformal transfor mations are nonsingular in T. We define the action of C on solutions F £ H by first defining it on the space of analytic signals of solutions, i.e., on H = {F(z) : F £
ft}.
(10.70)
Namely, the operator Ug representing g £ C on H is (C/ 9 F)( 2 ) = M S ( 2 ) F ( 3 - 1 2 ) ,
(10.71)
where Mg(z) is a 3 x 3 matrix representing the effect of g on the electric and magnetic fields at z (see Section 9.7). Mg(z) generalizes the factor s - 1 / 2 that was necessary in order to make scaling operations on proxies unitary. Indeed, when g is the scaling transformation z — i > az with o ^ 0 , it can be easily seen 2 from the definition of ||F|| that unitarity implies (UgF)(z) = a-2F{z/a),
(10.72)
so Mg(z) = a~2I, where I is the identity matrix. Note that Mg is independent of z in this case. In fact, Mg is independent of z whenever g is linear, which means that g is a combination of a Poincare transformation and scaling. Translations do not affect the polarization, so Mg = I. But a space rotation should be accompanied by a uniform rotation of the fields, so Mg is a real orthogonal matrix. A Lorentz transformation results in a scaling and mixing of the electric and magnetic fields (Jackson [1975]), so Mg must be complex (in order to mix E
A Friendly Guide to Wavelets
262
and B ) . Finally, when g is a special conformal transformation, Mg(z) depends on z due to the nonlinearily of g. T h e explicit form of Mg (z) is known, although we will not need to use it here; see Ruhl (1972), and Jacobsen and Vergne (1977). T h e properties of Mg(z) ensure t h a t g — i ► Ug is a unitary representation of C on H. Finally, the action of C on H is obtained by taking boundary values as y —► 0 simultaneously from the past and future cones, as explained in Chapter 9. We write it formally as (UgF)(x)
= Mg(x)F(g-1x),
(10.73)
although Mg(x) (unlike Mg(z)) may have singularities due to the singular nature of special conformal transformations on R 4 . (The singular set has zero measure; thus (10.73) still defines a unitary representation.) T h a t makes the representa tion g \-+ Ug unitary on H as well. This means t h a t from an "objective" point of view, there is absolutely no difference between the original Cartesian coordi n a t e system in R 4 and its image under g\ To an observer using the transformed system, UgF looks just as F does to an observer using the original system. This is an extension of special relativity theory t h a t is valid when only free electro magnetic waves are considered. See Cunningham (1910), B a t e m a n (1910), Page (1936), and Hill (1945, 1947, 1951). We can now derive the action of C on the wavelets. (This was already done in Section 9.7 for the special case of Lorentz transformations.) Setting g —> g~l in (10.71) and using F(z) = * * F , we have **xUg-i F = Mg-i (z)V*gxF.
(10.74)
Dropping F and taking adjoints gives U*g-.i*z = *gzMg-i{zy. B u t Ug is unitary, so U*1
(10.75)
= Ug. This gives the simple action Ug*z = *gzMg-i{z)*,
(10.76)
which generalizes (10.60) from proxies to physical wavelets. Note t h a t the po larization matrix M operates on the wavelets from the right. Aside from it, Ug acts on ^lz simply by transforming the label from z to gz. T h a t means t h a t Ug changes not only the point of focus x of ^x-\-iy » but also its center velocity v = y / s , scale |s| and helicity (sign of s). For example, if g = A is a pure Lorentz transformation, then gz = Ax + iAy, so *$?gz is focused at Ax and has velocity and scale determined by Ay. Thus Ug "boosts" ^z by transforming the coordinates of its point of focus and simultaneously boosting its velocity/scale vector. Translations, on the other hand, act only on x since z + b = (x + b) + iy. In order for a scattering event to be "elementary," both the wave and the scatterer must be elementary in some sense. In t h e case of the proxies, the
10. Applications to Radar and Scattering
263
elementary waves were the wavelets ipSyT(t). If the mother wavelet ip{t) is emitted at t — 0, then causality requires t h a t t/j(i) = 0 for t < 0. This implies t h a t ij)ST{t) = 0 for t < r. We would like to say t h a t the wavelets *&z are elementary electromagnetic waves. But recall t h a t the wavelets with stationary centers, ^fXjis(x.'Jt'), are converging to x when t' < t and a diverging from x when t' > t. For arbitrary z = x + iy G T , SP Z (V) is a converging wave when re' — x belongs to t h e past cone V'_ and a diverging wave when x' — x G V^. We will think of SE^ as being emitted^ at time t' — t from its support region near x ' = x. For this to make sense, it is necessary to ignore the converging part. T h a t , along with some other necessary operations, will be done below. Now t h a t we have defined elementary waves, we need to define elementary scatterers. For the V'S.T'S, these were point reflectors. T h a t was enough because the only parameters were time and scale. The \I>z's have many more param eters: space-time and velocity-scale on the side of the independent variables, and polarization on the side of the dependent variables. In order not to lose the potential information carried by these parameters, an elementary scatterer must have more structure t h a n a point. Since we are presently interested in radar, our scatterers will be idealized as "mirrors" in various states. (More general situations may be contemplated by assuming t h a t t h e scatterers reflect a part of the incident wavelet and absorb, transmit or refract the rest. Here we consider reflection only.) We begin by defining an elementary reflector in a s t a n d a r d state, which will be called the standard reflector, and then transform it t o arbitrary states using conformal transformations. T h e standard reflector is a flat, round mirror at rest at the origin x = 0 at time t = 0, facing the positive xi-axis. It is assumed to have a certain radius and exist only during a certain time interval around t = 0, so it occupies a compact space-time region R centered at the origin x = 0 in R 4 . This region is cylindrical in the time direction since the mirror is assumed to be at rest during its entire interval of existence. A general elementary reflector is obtained by applying any conformal transformation g to R. We will order the different operations in g as follows: first scale R by s (assume s > 0), then curve it by a special conformal trans formation Cb with b = ( b , 0 ) , rotate it to an arbitrary orientation, and boost it to an arbitrary velocity and acceleration (using Lorentz and special conformal transformations); finally, translate it to an arbitrary point b in space-time. T h e resulting scaled, curved, rotated, moving mirror at b will be denoted by gR. All of its parameters are conveniently vested in g. We now ask: W h a t happens when an elementary wave encounters an ele* The natural splitting of electromagnetic wavelets into absorbed and emitted parts is discussed in Chapter 11.
264
A Friendly Guide to Wavelets
mentary scatterer? As with all wave phenomena, the rigorous details are difficult and complicated, and so far have not been worked out. Fortunately, it is possi ble to give a rough intuitive account, since we are completely in the space-time domain. (Scattering phenomena are often analyzed for harmonic waves, i.e., in the frequency domain, where much of common-sense intuition is lost. An even more serious problem is t h a t nonlinear operations, such as special conformal transformations, cannot be transferred to the frequency domain since the Fourier variables are related to space-time by duality, which is a linear concept. See Section 1.4.) T h e simplest situation occurs when the reflector is in the stan dard state, since we then have a pulse reflected from a flat mirror. If we can guess how an arbitrary wavelet \I/ z is reflected from R, we can apply conformal transformations to this encounter to see how wavelets are reflected from an ar bitrary gR. This method derives its legitimacy from the invariance of Maxwell's equations under C. Suppose, therefore, t h a t an emitted wavelet \I/ z encounters R.^ We claim t h a t z = x + iy must satisfy certain conditions in order t h a t a substantial reflection take place. First of all, if t = XQ > 0, very little reflection can take place because the standard mirror exists only for a time interval around t = 0. Thus, causality requires (approximately) t h a t t < 0 for reflection. But even when t < 0, the reflection is still weak if x\ < 0, since then most of the wavelet originates behind the mirror. This gives two conditions on x. There are also conditions on y. T h e duration of \I/ Z is of the order \s\ ± | y | / c . If the "lifetime" of the mirror is small compared to this, then most of the wavelet will not be reflected even if the conditions on x are satisfied. Altogether, we assume t h a t there is a certain set of conditions on z = x-\-iy t h a t are necessary in order for \I/ z to produce a substantial reflection from R. We say t h a t z is admissible if all these conditions are satisfied, inadmissible otherwise. (Note t h a t this admissibility is relative to the standard reflector R\) We assume t h a t the strength of the reflection can be modeled by a function Pr(z) with 0 < Pr(z) < 1, such t h a t Pr(z) « 1 when z is admissible and Pr(z) & 0 when it is inadmissible. We call Pr(z) the reflection coefficient function. Actually, Pr(z) = 1 only if the reflection is perfect. Imperfect reflection (due to absorption or transmission, for example) can be modeled with 0 < Pr(z) < 1. * Actually, since *&z is a matrix solution of Maxwell's equations, this means (if taken literally) that three waves are emitted. The return F then also consists of three waves, i.e., it is a matrix as well. This has a natural interpretation: The matrix elements of F give the correlations between the outgoing and incoming triplets. We can avoid this multiplicity by emitting any column of *&z instead or, more generally, a linear combination of the columns given by ^ z u € H, where u G C 3 . Then F is also an ordinary solution. But it seems natural to work with * z a s a whole.
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265
To model the reflection of \I/z in R, define the linear map p : R 4 —> R 4 by p(xljx2,x3,t)
= (-xi,X2,x 3 ,t).
(10.77)
p is the geometrical reflection in the hyperplane {x\ = 0} in R 4 . Note that p belongs to the Lorentz group C C C. We expect that the reflection of ^x+iy in R seems to originate from x = px, and its center moves with the velocity v = y/s, where y = py. This is consistent with the assumption that the reflected wave is itself a wavelet parameterized by z = px + ipy = pz. Since p = p~x G £, (10.76) gives Up*z = *pzM*p (10.78) for the reflected wave when Pr{z) = 1. For general z G T , we multiply Up^fz by the reflection coefficient Pr{z). Thus our basic model for the reflection of an arbitrary wavelet \I>z by the standard reflector R is given by SVZ = Pr(z)UpVz
.
(10.79)
This model might be extended to include transmission as follows: The trans mitted wave is modeled by Pt{z)i5lzi where Pt(z) is a "transmission coefficient function" similar to Pr{z). The total effect of the scattering is then modeled by Pr(z)Up^z -f Pt(z)^z. (It may even be possible to model refraction in a similar geometric way.) If no absorption occurs, then Pr(z) and Pt{z) are such that the scattering preserves the energy. Since we are presently interested in radar, only reflection will be considered here. A general elementary scattering event is an encounter between a general wavelet ^z and a general elementary reflector gR. We assume that the reflection is obtained by transforming everything to the standard reference frame (where the reflector is a flat mirror at rest near the origin), viewing the reflection there, and transforming back to the original frame. This gives
-.pr(j-i?)v^,w,M;v(M"M'. The right-hand side of (10.80) is seen to be (by the group property of the C/'s) Pr(g-1z)UgUpUg-i^z
= Pr{g~lz)Ugpg-^z
.
(10.81)
Let p9 = gpg-1 = p;1 •
(10.82)
Here pg represents the geometrical reflection in the (not necessarily linear) threedimensional hypersurface {gx : x\ — 0} in R 4 , i.e., the image under g of {x : X\ = 0}. Thus our model for the elementary reflection of SPZ from gR is given by 5 S * Z = Pr(g-lz)Up^z = P . . ^ - 1 * ) * , , , , MPg{zY . (10.83)
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266
Equation (10.83) generalizes the model (10.8) for proxies. We can now extend the method of ambiguity functions to the present set ting. Suppose we want to find the state of an unknown reflector. We assume that the reflector is elementary, and "illuminate" it with \£ z . The parame ters z € T are fixed, determined by the "state" of the radar. For example, if z = (x, is) € E, then the radar is at rest at x and &z is emitted at t = 0. But this need not be assumed, and z G T can be arbitrary. Then the return is given by F = Sg^z as in (10.83). The problem is to obtain information about the state g of the reflector, given only F. For simplicity, we assume for the moment that F(x, t) can be measured everywhere in space (at some fixed time t), which is, of course, quite unrealistic. Define the conformal cross-ambiguity function as the correlation between F and UPh^Z) with h G C arbitrary: Az(h) = (UPk*z)*F
= *;u;j.
(10.84)
That is, Az(h) matches the return (which is assumed to have the form Sg^z, with g unknown) with UPh^z. If indeed F = Sg&z , then Az(h) = PT{g-lz)(UPhVzy(UPg*z).
(10.85)
By (10.76), (10.82), and (10.83), Pr(g-lz)M,,h(z)*;kZ*P3ZMPl(zy
A,(fc) =
= Pr(g-1z)MPk(z)K(phz\p9z)MPg(zr
,
where K(z\w)
= ^*z^w
(10.87)
is the reproducing kernel for the electromagnetic wavelets, now playing the role of conformal self-ambiguity function. Recall that for the proxies, we used Schwarz's inequality at this point to find the parameters of the reflector (10.13). Since Az(h) is a matrix, this argument must now be modified. Let Ui, 112,113 be any orthonormal basis in C 3 , and define the vectors 3>h)2,n, <&g,z,n £ "H by ®h,z,n = UPhVzun,
&g^n = UPg*zun .
(10.88)
The trace of Az(h) is defined as trace Az (h) = ^ u* Az (h) u n n
= PAg^z) ] T *J,z>n*s,*,„ n
= Prig^z) ]T( * M , n , &glZ%n ). n
(io.89)
10. Applications to Radar and Scattering
267
Applying Schwarz's inequality and using the unitarity of UPh and UPg , we obtain |traceA 2 (/i)| < Pr{g~lz) ^
l( ®h,z,n , ®g,z,n )|
n
= P r (
(10.90)
= Pr(9-^)^<***zun n 1
- P^" *) t r a c e * ^ =
Pr(g-1z)N(z),
where N(z) is given by (9.140). Equality is attained in (10.90) if and only if &h,z,n = &g,z,n for a ll n> i-e-> if a n d only if ^k** = ^,*»-
(10-91)
The right-hand side of the last equation in (10.90) is independent of h, and so represents the absolute maximum of |trace Az(h)\. It can be shown that (10.91) holds if and only if (10.92) Phz = P9z. Equation (10.92) states that the reflection of z in hR is the same as the reflection of z in gR, which is in accord with intuition. By (10.82) and (10.92), PhPgZ = z.
(10.93)
This means that phpg belongs to the subgroup Kz = {k € C : kz = z}, which is called the isotropy group at z and is isomorphic to the maximal compact subgroup K = C/(l) x SU(2) x 51/(2) of C. This gives a partial determination of the state g of the reflector, and it represents the most we can expect to learn from the reflection of a single wavelet. It may happen that a reflector that looks elementary at one resolution (choice of s = yo in the outgoing wavelet) is "resolved" to show fine-structure at a higher resolution (smaller value of s). Since conformal ambiguity functions depend on many more physical parameters than do the wideband (affine) ambi guity functions of Section 10.1, they can be expected to provide more detailed imaging information. A complex reflector is one that can be built by patching together elementary reflectors gR in different states g G C. There may be a finite or infinite number of gR's, and they may or may not coexist at the same time. We model this "building" process as follows: There is a left-invariant Haar measure on C, which we denote by dv{g). This measure is unique, up to a con stant factor, and it is characterized by its invariance under left translations by elements of C\ du(gog) = dv(g), for any fixed go G C. dv(g) is, in fact, precisely
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268
the generalization to C of the measure du(s,r) = s~2dsdr used on the affine group in Section 10.1. We model the complex reflector by choosing a density function w{g), telling us which reflectors gR are present and in what strength. The complex reflector is then represented by the measure da(g)=w(g)dv{g).
(10.94)
The measure da(g) is a "blueprint" that tells us how to construct the complex reflector: Apply each g in the support of w{g) to the standard reflector R (giving the scaled, rotated, boosted and translated reflector gR), then assemble these pieces. Suppose this reflector is illuminated with a single wavelet &z. Then the return can be modeled as a superposition of (10.83), with da(g) providing the weight:
S*z = Jd*(g)Sg*z = Jda(g)Pr(g-1z) UPg*z .
(10.95)
Note that da is generally concentrated on very small, lower-dimensional sub sets of C. If the reflector consists of a finite number of elementary reflectors giR, g
(S,T) eA-+ ips,T-^UPg*z
g £C
s~2ds dr -► dv{g)
(10.96) 1
D(s,r)-^w{g)Pr(g- z).
In particular, Dz(g) = w(g)Pr(g-1z) (10.97) is interpreted as the "reflectivity distribution" of the complex reflector with re spect to the incident wavelet *&z . w(g) tells us whether gR is present, and in what strength, while Pr(g~1z) gives the (predictable) reflection coefficient of gR with respect to \I>z , if gR is present. Just as D(S,T) was defined on the affine group A, which parameterized the possible states (range and radial velocity) of the point reflector for the proxies, so is Dz(g) defined on the conformal group C, which parameterizes the states of the elementary reflectors gR. Note that Dz(g) = 0 if g~~lz is not admissible. For g~lz to be admissible, the time compo nent otg~lx must be negative, which means that g translates to the future. This corresponds to D(s, r) = 0 for r < 0, and represents causality. The delayed and Doppler-scaled proxies ^SyT in (10.14) are now replaced by UPg&z, as indicated in (10.96). Equation (10.95) represents an inverse problem: given S^z , find Dz(g). Similar considerations can be applied toward the solution of this problem as were applied to the corresponding problem for proxies in Section 10.1, since the
10. Applications to Radar and Scattering
269
underlying mathematics is similar, although more involved. For example, the reproducing kernel K(2/ | z) gives the orthogonal projection to the range of the analytic-signal transform F —* F, as explained in Chapter 9. As an example of complex reflectors, consider the following scheme. Suppose we expect the return to be a result of reflection from one or more airplanes with more or less known characteristics. Then we can first build a "model airplane" P in a standard state, analogous to the standard reflector R. P is to represent a complex reflector at rest near the origin x = 0 at time t = 0 in a standard orientation. Hence it is built from patches gR that do not include any time translations, boosts or accelerations. Recall that the special conformal transformations C& with b = (b, 0) leave the configuration space RQ = {t = 0} invariant but curve flat objects in RQ. For example, letting x = (0,X2,X3,0) be the position vector in Ro = {x G R : t = 0} and choosing b = (a, 0,0,0), (10.67) gives C t (0, x2,x3,0)
= ( ~ a f' X2 2 ' X 2 3 ' 0) = (r', 0), J-
I
(10.98)
Lb I
where r 2 = x\ + x\. Hence the image of the flat mirror Ro at t = 0 is a curved disc C&i^o, also at t = 0, convex if a < 0 and concave if a > 0. Choosing b = (b, 0) with b<2 / 0 or 63 ^ 0 gives lop-sided instead of uniform curving. Thus by scaling, curving, rotating and translating R in space, we can build a "model airplane" P in a standard state. This can be done ahead of time, to the known specifications. Possibly, only certain characteristic parts of the actual reflector are included in P , either for the sake of economy or because of ignorance about the actual object. Then, in a field situation, this model can be propelled as a whole to an arbitrary state by applying a conformal transformation h € C, this time including time translations, boosts and accelerations. Thus hP is a version of P flying with an arbitrary velocity, at an arbitrary location, with an arbitrary orientation, etc. If the weight function for P is w(g), then that for hP is w(h~lg). If hP is illuminated with ^z, the expected return is j dv{g)w(h-1g)Pr{g-1z)UPg^z,
(10.99)
where now the reflectivity distribution w(h~lg)Pr(g~1 z) is presumed known. This can be matched with the actual return to estimate the state h of the plane. If several such planes are expected, we need only apply several conformal transformations hi, /12,. • • to P. Note that when imaging a complex object like hP, the scale parameter in the return may carry valuable information about the local curvature (for example, edges) of the object. Usually this kind of information is obtained in the frequency domain where (as we have mentioned before) it may be more complicated.
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A Friendly Guide to Wavelets
In summary, I want to stress the importance of the group concept for scat tering. The conformal group provides a natural set of intuitively appealing parameters with which to analyze the problem. This is not mere "phenomenol ogy." It is, rather, geometry and physics. All operations have a direct physical interpretation. Moreover, the group provides the tools as well as the concepts. It is a well-oiled machine, capable of automation. An important challange will be to discretize the computations and to develop fast algorithms that can be performed in real time. But a considerable amount of development needs to take place before this is possible. The simple model presented here will no doubt need many modifications, but I hope it is a beginning.
Chapter 11
Wavelet Acoustics Summary: In this chapter we construct wavelet representations for acoustics along similar lines as was done for electromagnetics in Chapter 9. Acoustic waves are solutions of the wave equation in space-time. For this reason, the same conformal group C that transforms electromagnetic waves into one another also transforms acoustic waves into one another. Hence the construction of acoustic wavelets and the analysis of their scattering can be done along similar lines as was done for electromagnetic wavelets. Two important differences are: (a) Acoustic waves are scalar-valued rather than vector-valued. This makes them simpler to handle, since we do not need to deal with polarization and matrix-valued wavelets, (b) Unlike the electromagnetic wavelets, acoustic wavelets are necessarily associated with nonunitary representations of C. This means, for example, that Lorentz transformations change the norm energy of a solution, and relatively moving reference frames arc not equivalent. There is a unique reference frame in which the energy of the wavelets is lowest, and that defines a unique rest frame. The nonunitarity of the representations can thus be given a physical interpretation: unlike electromagnetic waves, acoustic waves must be carried by a medium, and the medium determines a rest frame. We construct a one-parameter family of nonunitary wavelet representations, each with its own resolution of unity. Again, the tool implementing the construction is the analytic-signal transform. We then discover that all our acoustic wavelets * 2 split up into two parts: an absorbed part \Pj , and an emitted part "PJ. These parts satisfy an mlwmogencous wave equation, where the inhomogeneity is a "current" representing the response to the absorption of "P* or the current needed to emit ^ + . A similar splitting occurs for the electromagnetic wavelets of Chapter 9, and this justifies one of the assumptions in our model of electromagnetic scattering in Chapter 10. Prerequisite: Chapters 9.
11.1 C o n s t r u c t i o n o f A c o u s t i c W a v e l e t s In this section we use t h e analytic-signal transform to generate a family of scalar wavelets dedicated to t h e wave equation. It is hoped t h a t these wavelets will be useful for modeling acoustic scattering and sonar. We construct a one-parameter family \I>£ of such wavelets, indexed by a continuous variable a > 2. For any fixed value of a , t h e set {\&^ : z € T } forms a wavelet family similar to the electromagnetic wavelet family {*f?z : z G T } . T h e wavelets with a = 3 coincide with t h e scalar wavelets already encountered in Chapter 9 in connection with electromagnetics. Those wavelets have already found some applications in acoustics.
G. Kaiser. A Friendly Guide lo Wavelets, Modern Birkhauser Classics. DOI 10.1007/978-0-8176-8111 -l_l 1. C Gerald Kaiser 2011
271
272
A Friendly Guide to Wavelets
Modified from spherical waves to nondiffracting beams, they have been proposed for possible ultrasound applications (Zou, Lu, and Greenleaf [1993a, b]). Most of the results in this section are straightforward extensions of ideas developed in Chapter 9 and Section 10.2, and the. explanations and proofs will be rather brief. I have tried to make the presentation self-contained so that readers interested exclusively in acoustics can get the general idea without referring to the previous material, doing so only when they decide to learn the details. At a minimum, the reader should read Section 9.3 on the analytic-signal transform before attempting to read this section. As explained in Section 9.2, solutions of the wave equation can be obtained by means of Fourier analysis in the form
F
<*>- f 4^-f e t M " P X ) /(P^) + « i( - U;t - px) /(p,-a;) (11.1)
J R 3 107T\J .
= J dp e**/(p), where X — (x, t) — space-time four-vector P — (P»Po) — wavenumber-frequency four-vector po = ± | p |
by the wave equation
u) = |p| = absolute value of frequency
(112^
p • x = pot — p • x = Lorentz-invariant scalar product C = {p= (p,po) e R 4 : p2 =p0-
|p| 2 = 0} = C + U C_ = light cone
dpH = ^ .,3 = Lorentz-invariant measure on C. 167r a; In order to define the wavelets, we need a Hilbert space of solutions. To this end, fix a > 1 and define
(11.3)
Let Ha be the set of all solutions (11.1) for which the above integral converges, i.e.,
Then 7ia (11.3):
12 Ha = {F:\\F\\2
F*G=(F,G) = f JL-J(p)g(p),
F'F=\\F\\2.
(11.5)
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11. Wavelet Acoustics
Recall that a similar Hilbert space was defined in the one-dimensional case in Section 9.3. The meaning of the parameter a is roughly as follows: For large values of a, f(p) must decay rapidly near u> = 0 in order for F to belong to HaThis means that F(x) necessarily has many zero moments in the time direction, and therefore it must have a high degree of oscillation along the time axis. On the other hand, a large value of a also means that f(p) may grow or slowly decay as w - » o o , and this would tend to make F(x) rough. That is, a large value of a simultaneously forces oscillation and permits roughness. The wavelets we are about to construct will be seen to have the oscillation (as they must) but not the roughness. They decay exponentially in frequency. To construct the wavelets, extend F(x) from the real space-time R 4 to the causal tube T = {x+iy E C 4 : y2 > 0} by applying the analytic-signal transform (Section 9.3): F(x + iy) = — /
:
7
F(x + ry)
r 2
7 -~ = J dp20(p-yyr(*+^f(p),
(11.6)
where $ is the unit step function. By multiplying and dividing the integrand in the last equation by u/* - 1 , we can express the integral as an inner product of F with another solution $x+iy, which will be the acoustic wavelet parameterized by z = x + iy:
3
where
L
dp
-
if>z(p) = 2ua-l0(p-y)erip*.
(1L7)
(11.8)
i})z is the Fourier coefficient function for the solution » , ( a 0 = [ dpei**'il>x(j>)= f dp2u,n-l8(p-y)Jrt*'-*\ Jc Jc
(11.9)
which we call an acoustic wavelet of order a. For a — 3 we recognize the scalar wavelets of Section 9.7. By the definition (11.5) of the inner product, F{z) = $lF=(Vz,F)
for all F C 7ia and z e T.
(11.10)
For the acoustic wavelets to be useful, we must be able to express every solution F(x) as a superposition of ^ 2 ' s . That is, we need a resolution of unity. We follow a procedure similar to that used for the electromagnetic wavelets. The restriction of the analytic signal F(z) to the Euclidean region E = {(x, is) : x €
A Friendly Guide to Wavelets
274 R V ^ O } in T is F(x,is)
2e(p0s)e-*>3e-ipxf(p)
m f dp
Jc
(11.11) , ,
• /
CTT«"**w«)«" " /(p.«)+^-»y"/(p.-«)]-
Therefore by Plancherel's theorem, c/ 3 x|F(x,is)| 2
/
(11.12)
./R3
Note that for a > 2, (11.13) In order to obtain ||F|j 2 on the right-hand side of (11.12), we integrate both sides with respect to s with the weight factor | s | Q _ 3 , assuming that a > 2: /
rf3X(i.s|.9|a-3|F(x,i.9)|2
_r( A -2) | ( F | | 2 2a~3 Let 7in be the space of all analytic signals of solutions in 7ia, i.e.,
na =
{F:Fena}.
(11.15)
Equation (11.14) suggests that we define an inner product on Ha by
F*G={F,G)= f dp,a{z)J\z)G{z),
(11.16)
JE
where dp,a(z) is the measure on the Euclidean region defined by 2
dp,a(x. is) =
«-3
r(o-2)
(11.17)
With this definition, we have the following result. The heart of the theorem is (11.18), but we also state some important consequences for emphasis. See Chapter 9 for parallel results and terminology in the case of electromagnetics.
1 1 . Wavelet Acoustics
275
Theorem 11.1. If a > 2, then (a) The inner product defined on HQ by (11.16) satisfies F*G = F'G,
i.e.,
f dna(z) Fm*z*mzG
= F'G.
(11.18)
JE
In particular, ||F|| = ||F|| and 7{Q is a closed subspace of L2(dfia). (b) The acoustic wavelets of order a give the following resolution of unity in «•••
/ dna(z) *V&* « J, weakly in HQ .
(11.19)
(This means that (11.18) holds for all F,G €HQ •) (c) Every solution G € "%« can be written G=
J dfia{z) * Z * * G = / dna(z) * 2 G(z), weakly in HQ . JE
(11.20)
JE
(This means that (11.18) holds for all F € Ha .) Therefore, G(x') = f dfia(z) ^2{xf) G{z)
a.e.
(11.21)
(d) The orthogonal projection in L (dfiQ) to the closed subspace 7l!a is given by $€L 2 (rf/x a ),
( P » ) ( « 0 - / dfia(z)K(z'\z)*(z),
(11.22)
JE
where K(z' \z) s * * , ¥ , = 4 / dp uja-{ 0{p • y')6(p • y) Jc = 4 % • y') f dp a;"" 1 9{p • y) Jc
ip z s) e
< '-
(11.23)
&&-*>
is the associated reproducing kernel. For a < 2, (11.13) shows that the expression for the inner product in HQ diverges and Theorem 11.1 fails. We therefore say that the \I/Z's are admissible if and only if a > 2. Only admissible wavelets give resolutions of unity over E. 11.2 Emission, Absorption, and Currents In order for the inner products ^*ZF to converge for all F € Ha, ^z must itself belong to Hlx. The norm of ^z is \\**Hy\\2=
^^2"-2e(p-y)e-*™
f
(11.24)
= 4 [ dp w Jc
a_1
0(p •
y)e-2py
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276
where we used 0(p • y)2 = 6{p • y) a.e. Clearly ||* x -»y|| assume t h a t y belongs to the future cone V!. Then
|*.|a . 4 /
dp uP-le-**
= ll*x+»y|| , so we can
21-QEa(y),
=
(11.25)
Jc+ where /; .,••
Jc+
<
(11.26)
Since py
= UJS- p - y > ws - w | y |
(11.27)
and s — |y| > 0 for y € V+ , the integrand in (11.26) decays exponentially in the radial (w) direction and the integral converges. (It decays least rapidly along the direction where p is parallel to y, which means t h a t $?z favors these Fourier components; see t h e discussion of ray filters at the end of Chapter 9.) This proves t h a t \£ z € W a > f ° r a n Y <* > 1 and all z € T. We now compute Ea(z) when a is a positive integer. T h e resulting expression will be valid for arbitrary a > 1, and it will be used to find an explicit form for Vz(x'). According t o (9.127), dp e~PV
S(y) = f
I
=
!
A-K2y2
Jc+
I
4TT2 S2 — \y\2
(11.28)
We split S(y) into partial fractions, assuming u = j y | f& 0:
S(y)
=
8TT2U
S—U
(11.29)
8+U
Then
Ea(y) =
{-dsY-'Siy) 1
I
I
8TT2W
8 —U
8+U
Y{«) 2
8?r « To get Ea(y)
i. .(s-u)
a
1 {s + u)a
(11.30)
, t* = |y| ^ 0.
when y = 0, take the limit u —> 0 + by applying L'Hospital's rule:
«M-2??&
s>o
-
(11.31)
For general y € V, we need only replace s by \s\ in (11.30)- (11.31). Although these formulas were derived on t h e assumption t h a t a £ N , a direct (but more
1 1 . Wavelet Acoustics
277
tedious) computation shows that they hold for all a > 1. In view of (11.26), this proves that Wz G He, for a > 1.* Equation (11.30) can also be used to compute the wavelets. Recall that it suffices to find ^iy(x) since *x+iy(x')
= Viy(x'-x).
(11.32)
But Viy(x) = 2 [ dp^-ie-p(v-ix) Jc+ Theorem 9.10 tells us that = -{x + iy)'2 ? 0
(y - ixf
= 2Ex(y
for
_ ixy
y 6 V.
( n
33)
(11.34)
As a result, it can be shown that Ea(y) has a unique analytic continuation to E0(y - ix). This continuation can be obtained formally by substituting 3 - 3 - it,
u = ( y 2 ) 1 ' 2 -> ((y - ix) 2 ) 1 / 2
(11.35)
in (11.30). We are using the notation (y - ix) 2 = (y - ix) • (y - ix) = y 2 - x 2 - 2ix • y
(11.36)
because |y — ix| 2 could be confused with the Hermitian norm in C 3 . We find the explicit form of the reference wavelet for Ha, i.e., of * ( r , t ) = #o,i(x,*) -2
f
^^-ic-O-aj-ip*
r
=|x].
(1L37)
Jc+ As the notation implies, ty is spherically symmetric. T h e substitution (11.35) is now
s->l-it,
u-*(-x2)1/2sir,
(11.38)
giving
•M-3*
IT
t(o,i)-
1 {\-it-ir)<*
(l-it
1 , r^O + ir)a
(11.39)
r ( a + 1) 27T2(l-it)°+1 '
* Since tyj depends on a, we should have indicated this somehow, e.g., by writing ty°; to avoid cluttering the notation, we usually leave out the superscript, inserting it only when the dependence on a needs to be emphasized. Also, the integral for H^zll2 actually converges for all a > 0. Although only wavelets with a > 2 give resolutions of unity, it may well be that wavelets with 0 < a < 2 turn out to have some significance, and therefore they should not be ruled out.
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278
We choose t h e branch cut for the square-root function in (11.38) along t h e positive real axis, and the branch cut of g(w) = wa along the negative real axis. Note t h a t ^ is independent of the choice of branches. Equation (11.39) can be written as * ( r , * ) = «'"(r,*) + * + ( r , t ) J
(11.40)
where * - ( M ) ,
r W
4n2ir
l (1 - it — ir)a
(11.41)
t*(r,t) ••"(-!•,*). ^ * ( r , t) are singular at r = 0, and the singularity cancels in # ( r , t). Aside from t h a t singularity, # ~ ( r , t) is concentrated near the set { ( x , i ) e R 4 : t = -r = - | x | < 0} = dV'_ ,
(11.42)
i.e., near the boundary of the past cone in space-time. (Remember t h a t we have set c = 1.) This is the set of events from which a light signal can be emitted in order to reach the origin x = 0 at t — 0. Thus • # " ( r , t) represents an incoming x = 0 in space-time.
wavelet that gets absorbed near the
origin
Similarly, ty+(r, t) is concentrated near {(x,t) € R 4 : t = r = | x | > 0} = d V | ,
(11.43)
the boundary of t h e future cone. This is the set of all events t h a t can be reached by light emitted from the origin. • ^ + ( r , t) represents
an outgoing wavelet that is emitted near x = 0.
But these are just the kind of wavelets we had in mind when constructing the model of radar signal emission and scattering in Section 10.2! Upon scaling, boosting, and translating, \I>~(r, t) generates a family of wavelets ^ j ( x ' ) t h a t are interpreted as being absorbed at x' = x and whose center moves with velocity y/yo- Since ty~ is not spherically symmetric unless y = 0, its spatial localization at t h e time of absorption is not uniform. It can be expected to be of the order of ya ± |y|, depending on the direction. (This has to do with the rates of decay of 0(p • y)e~py along the different directions in p . See "ray filters" in Chapter 9.) Similarly, 9*{r% t) generates a family of wavelets ^ (#') t h a t are interpreted as being emitted at x and traveling with v = y/yoT h e wavelets ^ f a r e n o t global solutions of the wave equation since they have a singularity a t the origin. In fact, using V2 - =-4TT<5(X),
r
(11.44)
11. Wavelet Acoustics
279
it can be easily checked that ^ (r, t) and
- a?*-(r,t) + v»«-(r,t) = - _ J M ^ W (11.45)
The second equation is an analytic continuation of the first equation to the other branch of ir = (—x 2 ) 1 / 2 . These remarkable equations have a simple, intuitive interpretation. In view of the above picture, the second equation states that to emit ofV*, we need to produce an " elementanj current" at the origin given by
J(X,t) =
i j \ - L)« *(X) S -*tf*MX>»
(1L46)
and the first equation states that £/ie absorption of^~ creates the same current with the opposite sign. The sum of the two partial wavelets satisfies the wave equation because the two currents cancel. By "elementary current" I mean a rapidly decaying function of one variable, in this case
4>{t)= J(a\
,
(11.47)
concentrated on a line in space-time, in this case, the time axis. Thus (£) is just an old-fashioned time signal, detected or transmitted by an "aperture" (e.g., an antenna, a microphone, or loudspeaker) at rest at the origin! This brings us back to the "proxies" of Section 10.2. Evidently, equations (11.45) give a precise relation between the waves W* and the signal
(11.48)
This relation is not an accident.
0 = **M(0 *(*' ~ x ) = -J*MU&)
a'•£**.(*•0 - -**..t(0*(x' - x) = JX(t+is(x'),
,„AM (11.49)
A Friendly Guide to Wavelets
280 where n' = -d?. + V2x,
and
(11-50)
By applying a Lorentz transformation to (11.49) we obtain a similar equation for a general z eT (with v = y/.s not necessarily zero), relating • ' 3Jf (x') to a moving source Jz(x'). Jx+iy is supported on the line in R 4 parameterized by X'(T)
= x +
TV.
This extends the above relation between waves and signals to the whole family of wavelets tyf. Equations (11.49) suggest a practical method for analyzing and synthesizing with acoustic wavelets: T h e analysis is based on the recognition of the received signal i
(n 5i)
wi-jeSS^i-
-
For a G N , it is easy to compute the Fourier transform of 0 by contour integration: dt e"*'
*>-¥£d *
J-oo
t)«
U-lt
«=i
(11.52)
= -(-0ur~l [{-2wi)0(S)e-&] where we have closed the contour in the upper-half plane if £ < 0 and in the lower-half plane if £ > 0. But comparison of (11.52) with (11.08) shows t h a t
1 1 . Wavelet Acoustics
281
0 ( 0 is the Fourier representative
ipo,i of the reference wavelet ty = ^o,i > i»©«j
4>(po) - 2 ^ ( p o ) b o | a ~ 1 e - p o - ^o,i(p,Po). Referring back to (9.47), this shows t h a t d£ l ^ - ' W O I 2 = 2*-2{*-Pr(2a-rp-
C0 = [^
2) < oo
whenever 2a + (3 > 2. Therefore 4> is admissible in t h e sense of (9.47), and it can be used as a mother wavelet for the proxy Hilbert space W p
y
= {/ : R -
C :j T
| ^
| / ( 0 P < oo},
/? > 2 -
2a.
We can now return to Section 10.2 and fix a gap (one of many) in our discussion of electromagnetic scattering there. We postulated that the electromagnetic wavelets ^z can be split up into an absorbed part ^~ and an emitted part SP+. This can now be established. Since the matrix elements of the electromagnetic wavelets were obtained by differentiating S(y), t h e same splitting (11.29) t h a t resulted in the splitting of Wz also gives rise to a splitting of SP z . T h e partial wavelets *£?* are fundamental solutions of Maxwell's equations, t h e right-hand sides now representing elementary electrical charges and currents. (In electromagnetics, t h e charge density and current combine into a four-vector transforming naturally under Lorentz transformations.) Superpositions like (10.95) and (10.99) may perhaps be interpreted as modeling absorption and reemission of wavelets by scatterers. In t h a t case, the "elementary reflectors" of Section 10.2 ought to be reexamined in terms of elementary electrical charges and currents.
1 1 . 3 N o n u n i t a r i t y a n d t h e R e s t F r a m e of t h e M e d i u m We noted t h a t the scalar wavelets (9.179) associated with electromagnetics are a special case of (11.9) with a = 3. T h e choice a = 3 was dictated by the requirement t h a t the inner product in H be invariant under the conformal group, which implies t h a t the representation of C on "H is unitary. It is important for Lorentz transformations to be represented by unitary operators in free-space electromagnetics since there is no preferred reference frame in free space. If the operators representing Lorentz transformations are not unitary, they can change t h e norm ("energy") of a solution. We can then choose solutions minimizing the norm, and t h a t defines a preferred reference frame. Let us now sec how the norm of tyz depends on the speed v = |v| = |y/.s| = u/s < 1. By (11.25) and (11.30), IT
,,2
KM2 =
r(«) 2ir (2e)a+1v 2
1 .(!-«)"
1 0 +v)am
'
zeT+.
(11.53)
282
A Friendly Guide to Wavelets
Now
Hy) = \fl? = *Vl - v2 (11.54) is invariant under Lorentz transformations. Therefore we eliminate s in favor of A: 1 1 r(c*)(l - v2Ya+1^2 f |2 l* : l v 2TT 2 (2A)«+ 1 V X ~ )° 0- + v)a. (11.55) T{a)y/l-v2 1+v l-v 27r 2 (2A) rt+, v \ l - v j \l + vt For the statement of the following theorem, we reinsert c by dimensional analysis. The measure dp has units [ I t ] , where [t] and [/] are the units of time and length, respectively; hence ||^ 2 || 2 has units [Z~1t-0tJ. Since A has units of length, we multiply (11.55) by c a , and replace v by v/c. P r o p o s i t i o n 11.2. The unique value a > 1 for which ||^z|| 2 W invariant under Lorentz transformations is a = 1. For a > 1, !|^ a || 2 is a monotonically increasing function of the speed v(y) = |y/t/o| of the center of the wavelet, given by T(ct) ca sinh (aO) 2 (11.5G) l l * * l l = 7T2(2A) a+l sinh 9 where 9 is defined by
- = tanh 9. (11.57) c Proof: We have mentioned that v is the hyperbolic tangent of a "rotation angle" in the geometry of Minkowskian space-time. This suggests the change of variables (11.57) in (11.55), which indeed results in the reduction to (11.56). Since A is Lorentz-invariant, only 9 depends on v in (11.56). Hence the expression is independent of v if and only if a = 1. For a > 1, let sinh (a9) (11.58) sinh a We must show that f(9) is a monotonically increasing function of 9. The derivative of / is a sinh 9 cosh {a9) — cosh 9 sinh {cx9) (11.59) sinh2 9 Therefore it suffices to prove that /(*) =
m=
tanh (a9) < a tanh 9,
>0,a>l.
(11.60)
9 > 0,
(11.61)
Begin with the fact that cosh (a9) > cosh 9,
which follows from the monotonicity of the hyperbolic cosine in [0, oo). Thus sech 2 (»0') < sechV,
9' > 0.
Integrating both sides over 0 < 9' < 9 gives (11.60).
•
(11.62)
11. Wavelet Acoustics
283
Equation (11.56) can be expanded in powers of v/c. T h e first two terms are „T
ll2
V(a + 1)
a 2 - 1 v2
(11.63)
This is similar to t h e formula for the nonrelativistic approximation to the energy of a relativistic particle, whose first two terms are the rest energy rac2 and the kinetic energy mv2/2. T h e wavelets tyz behave much like particles: their parameters include position and velocity, t h e classical phase space parameters of particles. T h e time and scale parameters, however, are extensions of the classical concept. Classical particles are perfectly localized (i.e., they have vanishing scale) and always "focused," making a time parameter superfluous. (The time parameter used to give the evolution of classical particles corresponds to t' in tyx+iy(x'), and not to the time component t of x\) T h e wavelets are therefore a bridge between the particle concept and the wave concept. It is this particle-like quality t h a t makes it possible to model their scattering in classical, geometric terms. Suppose t h a t a > 1. Fix any z = (x, is) £ E, and consider t h e whole family of wavelets t h a t can be obtained from tyz by applying Lorentz transformations, i.e., viewing ^ z from reference frames in all possible states of uniform motion. According to Proposition 11.2,
11.4 E x a m p l e s of A c o u s t i c W a v e l e t s Figures 11.1—11.8 show various aspects of ^ ( r , t) and 4 ' ± ( r , t) for a = 3,10,15, and 50.
284
A Friendly Guide to Wavelets X10
xlO
x10
Figure 11.1. Clockwise, from top left: «I»(r, 0) with a — 3,10, 15, and 50, showing excellent localization at t = 0. (Note that 4»(r, 0) is real.) x10
-0.2 -0.4
Figure 11.2. Clockwise, from top left: The real part (solid) and imaginary part (dashed) of ^ ( 0 , t) with « = 3, 10,15, and 50, showing increasing oscillation and decreasing duration.
1 1 . Wavelet Acoustics
285
0.15
-0.05
F i g u r e 11.3. The real part (top) and imaginary part (bottom) of ^ ( r , t ) with a = 3.
A Friendly Guide t<> Wavelets
280 0,2 0.15 0.1 0,05
o -0,1 -0, 15 -0,2
0.1
0 -1
-2
0.05
0
2
°
-0,05
-0,1 -0.15
·0.2 -0,25
-0.3 ~
2
~
~
... tim e
0
-1
-2
0
2
t (bottom ) of the abs orb ed (or rcoJ par t (top) and ima gina ry par Fig ure 11.4_ The'l'3. (r, t) with Q dete cted ) wavelet
=
1 1 . Wavelet Acoustics
287
-0.1
Figure 11.5. The real part (top) and imaginary part (bottom) of the emitted wavelet * + ( r , 0 with Q = 3 .
A Friendly Guide to Wavelets
288 X10
1.5
2
Figure 11.6. The real part (top) and imaginary part (bottom) of *(r, t) with a = 10.
11 . £Va vei et Acoustics X
28 9
10 "
1.5
-0.5
-1
0.5
1.5
-1
-0.5
o
0.5
0.5
o -0.5 -1
-1 .5
0.5
time
o -1
-0.5
o
0.5
(r, t ) wi th " = 15. ry pa r t (bo tto m) ofl lna i ag im d an p) (to rt l pa Fi gu re 11 .7. T hc rca
A Friendly Guide to Wavelets
290
x10
62
-0.5 Figure 11.8. The real part {top) and imaginary part (bottom) of * ( r , t) with a = 50.
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Feichtinger HG and Grochenig K, Banach spaces related to integrable group represen tations and their atomic decompositions, I, J. Funct. Anal. 86(1989), 307-340, 1989a. Feichtinger HG and Grochenig K, Banach spaces related to integrable group represen tations and their atomic decompositions, II, Monatsch. f. Mathematik 108(1989), 129-148, 1989b. Feichtinger HG and Grochenig K, Gabor wavelets and the Heisenberg group: Gabor expansions and short-time Fourier transform from the group theoretical point of view, in Chui CK (1992), ed., Wavelets: A Tutorial in Theory and Applications, Academic Press, New York, 1992. Feichtinger HG and Grochenig K, Theory and practice of irregular sampling, in Bene detto JJ and Frazier MW (1993), eds., Wavelets: Mathematics and Applications, CRC Press, Boca Raton, 1993. Folland GB, Harmonic Analysis in Phase Space, Princeton University Press, Princeton, NJ, 1989. Gabor D, Theory of communication, J. Inst. Electr. Eng. 93(1946) (III), 429-457. Gel'fand IM and Shilov GE, Generalized Functions, Vol. I, Academic Press, New York, 1964. Glimm J and Jaffe A (1981) Quantum Physics: A Functional Point of View, SpringerVerlag, New York. Gnedenko BV and Kolmogorov AN, Limit Distributions for Sums of Independent Ran dom Variables, Addison-Wesley, Reading, MA, 1954. Grochenig K, Describing functions: atomic decompositions versus frames, Monatsch. f Math. 112(1991), 1-41. Grochenig K, Acceleration of the frame algorithm, to appear in IEEE Trans. Signal Proc, 1993. Gross L, Norm invariance of mass-zero equations under the conformal group, J. Math. Phys. 5(1964), 687-695. Grossmann A and Morlet J, Decomposition of Hardy functions into square-integrable wavelets of constant shape, SI AM J. Math. Anal. 15(1984), 723-736. Heil C and Walnut D, Continuous and discrete wavelet transforms, SI AM Rev. 31 (1989), 628-666. Hill EL, On accelerated coordinate systems in classical and relativistic mechanics, Phys. Rev. 67(1945), 358-363. Hill EL, On the kinematics of uniformly accelerated motions and classical electromag netic theory, Phys. Rev. 72(1947), 143-149; Hill EL, The definition of moving coordinate systems in relativistic theories, Phys. Rev. 84(1951), 1165-1168. Hille E, Reproducing kernels in analysis and probability, Rocky Mountain J. Math. 2(1972), 319-368. Jackson JD, Classical Electrodynamics, Wiley, New York, 1975. Jacobsen HP and Vergne M, Wave and Dirac operators, and representations of the conformal group, J. Fund. Anal. 24(1977), 52-106. Janssen AJEM, Gabor representations of generalized functions, J. Math. Anal. Appl. 83(1981), 377-394.
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Kaiser G, Phase-Space Approach to Relativistic Quantum, Mechanics, Thesis, Univer sity of Toronto Department of Mathematics, 1977a. Kaiser G, Phase-space approach to relativistic quantum mechanics, Part I: Coherentstate representations of the Poincare group, J. Math. Phys. 18(1977), 952-959, 1977b. Kaiser G, Phase-space approach to relativistic quantum mechanics, Part II: Geomet rical aspects, J. Math. Phys. 19(1978), 502-507, 1978a. Kaiser G, Local Fourier analysis and synthesis, University of Lowell preprint (unpub lished). Originally NSF Proposal #MCS-7822673, 1978b. Kaiser G, Phase-space approach to relativistic quantum mechanics, Part III: Quantiza tion, relativity, localization and gauge freedom, J. Math. Phys. 22(1981), 705-714. Kaiser G, A sampling theorem in the joint time-frequency domain, University of Lowell preprint (unpublished), 1984. Kaiser G, Quantized fields in complex spacetime, Ann. Phys. 173(1987), 338-354. Kaiser G, Quantum Physics, Relativity, and Complex Spacetime: Towards a New Syn thesis, North-Holland, Amsterdam, 1990a. Kaiser G, Generalized wavelet transforms, Part I: The windowed X-ray transform, Technical Reports Series #18, Mathematics Department, University of Lowell. Part II: The multivariate analytic-signal transform, Technical Reports Series #19, Mathematics Department, University of Lowell, 1990b. Kaiser G, An algebraic theory of wavelets, Part I: Complex structure and operational calculus, SIAM J. Math. Anal. 23(1992), 222-243, 1992a. Kaiser G, Wavelet electrodynamics, Physics Letters A 168(1992), 28-34, 1992b. Kaiser G, Space-time-scale analysis of electromagnetic waves, in Proc. of IEEE-SP Internat. Symp. on Time-Frequency and Time-Scale Analysis, Victoria, Canada, 1992c. Kaiser G and Streater RF, Windowed Radon transforms, analytic signals and the wave equation, in Chui CK, ed., Wavelets: A Tutorial in Theory and Applications, Academic Press, New York, pp. 399-441, 1992. Kaiser G, Wavelet electrodynamics, in Meyer Y and Roques S, eds., Progress in Wavelet Analysis and Applications, Editions Frontieres, Paris, pp. 729-734 (1993). Kaiser G, Deformations of Gabor frames, J. Math. Phys. 35(1994), 1372-1376, 1994a. Kaiser G, Wavelet Electrodynamics: A Short Course, Lecture notes for course given at the Tenth Annual ACES (Applied Computational Electromagnetics Society) Conference, March 1994, Monterey, CA, 1994b. Kaiser G, Wavelet electrodynamics, Part II: Atomic composition of electromagnetic waves, Applied and Computational Harmonic Analysis 1(1994), 246-260 (1994c). Kaiser G, Cumulants: New Path to Wavelets? UMass Lowell preprint, work in progress, (1994d). Kalnins EG and Miller W, A note on group contractions and radar ambiguity functions, in Grunbaum FA, Bernfeld M, and Bluhat RE, eds., Radar and Sonar, Part II, Springer-Verlag, New York, 1992. Katznelson Y, An Introduction to Harmonic Analysis, Dover, New York, 1976. Klauder JR and Surarshan ECG, Fundamentals of Quantum Optics, Benjamin, New York, 1968.
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Index
T h e keywords cite the main references only.
accelerating reference frame, 212, 244 acoustic wavelets, 273 absorbed, 278 emitted, 278 adjoint, 13, 24-25 admissibility condition, 67 admissible, 264 affine group, 205, 257 transformations, 257 aliasing operator, 162 almost everywhere (a.e.), 22 ambiguity function, 250, 254 conformal cross, 266 self, 266 narrow-band, 254 cross, 254 self, 254 wide-band, 248 cross, 250 self, 250 analytic signal, 68, 206, 218, 219 lower 73 upper 73 analyzing operator, 83 antilinear, 12 atomic composition, 233 averaging operator, 142 property, 141 Balian-Low theorem, 116 band-limited, 99 bandwidth, 99, 252 basis, 4 dual, 9 reciprocal, 16 Bernstein's inequality, 101
biorthogonal, 15 blurring, 144, 242 bounded, 26 linear functional, 27 bra, 14 carrier frequency, 252 causal, 205 cone, 222 tube, 206, 223 characteristic function, 40 chirp, 47 complete, 4, 80 complex space-time, 206 structure, 207 conformal cross-ambiguity function, 250, 254, 266 group, 204 self-ambiguity function, 266 transformations, 260 special, 260, 269 consistency condition, 57, 87, 234, 249 counting measure, 40, 106 cross-ambiguity function, 250, 254 cumulants, 191-196 de-aliasing operator, 163, 164 dedicated wavelets, 204 demodulation, 253 detail, 63 diadic interpolation, 189 rationals, 188, 190 differencing operator, 152 dilation equation, 142 Dirac notation, 14 distribution, 26
Index
298 Doppler effect, 243, 247 shift, 252 down-sampling operator, 149 duality relation, 9 duration, 206, 242 electromagnetic wavelets, 224 elementary current, 279 scatterers, 263 scattering event, 265 Euclidean region, 225 evaluation map, 224 exact scaling functions, 191 filter, coefficients, 142 extremal-phase, 181 linear-phase, 181 maximum-phase, 181 minimum-phase, 181 finite impulse response (FIR), 155, focal point, 205 Fourier coefficients, 27 series, 26 transform, 29 fractal, 183 frame, 82 bounds, 82 condition, 82 deformations of, 117-218 discrete 82, 91 generalized, 82 normalized 83 operator, 83 reciprocal, 86 snug 90 tight 82 vectors, 82 future cone, 222 Gabor expansions, 114 gauge freedom, 212 Haar wavelet, 159 harmonic oscillator, 118 helicity, 206, 210
high-pass filter, 147 sequence, 154 Hilbert space, 23 transform, 251, 252 Holder continuous, 187-188 exponents, 188 hyperbolic lattice, 124 ideal band-pass filter, 167 ideal low-pass filter, 110 identity operator, 10 image, 6 incomplete, 4 inner product, 12 standard, 12 instantaneous frequency, 47 interpolation operator, 148 inverse Fourier transform, 33 image, 36 problem, 248 isometry, 69 partial 69 ket, 14 Kronecker delta, 5 least-energy representation, 101 least-squares approximation, 88 Lebesgue integral, 21 measure, 35 light cone, 208 linear, 6 functional, 24 bounded 24 linearly dependent, 4, 80 independent, 4 Lipschitz continuous, 187 Lorentz condition, 211 group, 259 invariant inner product, 208 transformations, 237 low-pass filter, 147
Index massless Sobolev space, 216 matching, 248 Maxwell's equations, 207 mean time, 191 measurable, 21, 35 measure, 21, 35 space, 40 zero, 22 metric operator, 18 Mexican hat, 75 Meyer wavelets, 167 moments, 193 multiresolution analysis, 144
299 reflector complex, 267 elementary, 263 standard, 263 regular lattice, 117 representation, 206, 261 reproducing kernel, 70, 228-233 resolution, 145 resolution of unity, 10 continuous, 55 rest frame, 281-283 Riesz representation theorem, 24
quadrature mirror filters, 148 quaternionic structure, 171
sampling, 92, 99 rate, 101 sampling operator, 161 scale factor, 63 scaling function, 140 invariant, 126 periodic, 127 Schwarz inequality, 18, 23 Shannon sampling theorem, 99 wavelets, 167 space-time domain, 31 inversion, 260 span, 9 special conformal transformations, 212 spectral factorization, 180 square-integrable, 23 square-summable, 20 stability, 243 condition, 128 star notation, 17 subband, 111 filtering, 153-155 subspace, 4 support, 23 compact, 23 synthesizing operator, 85, 236
range, 245 ray filter, 243 redundancy, 108 reference wavelet, 239, 240n., 241 reflectivity distribution, 248
time-limited, 102 transform, analytic-signal, 218 triangle inequality, 18, 23 trigonometric polynomial, 143
narrow-band signal, 252 negative-frequency cone, 222 nondiffracting beams, 272 norm, 12 numerically stable, 104 Nyquist rate, 101 operator, 23 bounded, 23 orthogonal, 13 orthonormal, 13 oscillation, 273 over complete, 4 oversampling, 88, 101 Parseval's identity, 33 past cone, 222 periodic, 26 Plancherel's theorem, 29, 32 plane waves, 206 Poincare group, 259 polarization identity, 33 positive-definite operators, 18 potentials, 211 proxies, 246, 256, 280
300
tube, causal, 206, 223 future, 223 past, 223 uncertainty principle, 52 up-sampling operator, 148 vector space, 4 complex 4 real 4 wave equation, 207, 272 vector, 30 wavelet, 62 basic 62 mother 62 packets, 172 space, 236 wave number-frequency domain, 31 Weyl-Heisenberg group, 51, 205 window, 45 Zak transform, 117 zero moments, 178 zoom operators, 148
Index
