Bases for time frequency analysis

Transcription

1 4 Bases for time frequency analysis In the first part of this chapter we construct various bases having good time frequency localization. The construction of wavelet packets, for example, is presented in Section 4.3. Wavelets are, at least metaphorically, localized in the upper time frequency plane on rectangles whose heights are inversely proportional to the lengths of their time intervals. Wavelet packets arise when a pair of wavelets living on adjacent time intervals is replaced by a pair of packets whose time interval is the union of the wavelet time intervals and whose frequency intervals are the lower and upper halves of the wavelet frequency intervals. Further wavelet packets can be obtained by repeating this type of recombination. When suitably defined, such packets give rise to a general class of basis functions associated with certain rectangles in the plane. An alternative approach is to start with the Gabor tiling. The Balian Low theorem asserts that Gabor functions cannot have good time frequency localization. However, Wilson basis functions in which the role of exponentials in Gabor functions is replaced by cosines can have good time frequency localization. They can be regarded as living on rectangles in the upper half plane having a unit time interval (support) and a unit frequency interval corresponding to the frequency of the cosine. Wilson bases having optimal time frequency localization are constructed in Section 4.1. Local trigonometric bases (LTBs) generalize Wilson bases in the sense of being localized on tiles whose time intervals are determined by a partition of R into intervals of not necessarily uniform length. They are constructed explicitly in Section 4.2. In a certain sense, the splittings in time leading to LTBs are dual to the splittings in frequency leading to wavelet packets (the analogy is made explicit in the chapter notes). Neither lead to arbitrary tilings. It is natural to consider the possibility of tiling phase space by mixing wavelet packet and LTB based recombinations. However, the separate recombinations are possible only because of particular conditions on the starting basis functions: the recombination transformations are not arbitrary. In Section 4.4 we consider uncertainty issues that must be addressed in order to build bases adapted to given tilings.

2 164 4 Bases for time frequency analysis In Section 4.5, we begin with a review of the Walsh functions. If one replaces the concept of frequency of a trigonometric function by sequency (rate of sign change) of a Walsh function, one can think of Walsh packets shifted and dilated Walsh functions alternately as wavelet packets which they truly are or as discrete analogues of LTBs. In this way one is equipped with an uncertainty-free Walsh model of the time sequency plane in which Walsh packets live precisely on time sequency rectangles analogues of Heisenberg tiles. Following Thiele s thesis [347] we then discuss combinatorial consequences of this Walsh picture, some of which will be applied to operator theory in Chapter 7. One particularly appealing result is the following: if a region in the plane can be expressed as a finite pairwise disjoint union of tiles, then there is a well-defined orthogonal projection onto the subspace of L 2 (R) spanned by the Walsh packets of those tiles (Section 4.5.1). What is important here is that any other covering by pairwise disjoint tiles defines the same projection. This becomes particularly useful when one associates a measure of information cost to a tiling: if the measure satisfies certain properties then one can define a best tiling or best basis to a region, along with a fast algorithm for computing a best basis expansion (Section 4.5.2). In Section 4.6 we consider a finite version of the Walsh plane for working with finite data and a generalization of it to the phase plane for finite Abelian groups. 4.1 Wilson bases and the Zak transform According to the Balian Low theorem, if the functions g kn (x) = e 2πinx g(x k) form a Riesz basis for L 2 (R) then g cannot be well localized both in time and in frequency. Wilson (see [99] for history) observed that matters are not as drastic as they seem: good localization is possible if the exponentials e 2πinx are replaced by sines and cosines. Given a suitable window function w, the Wilson basis construction of Daubechies et al. [105] yields functions 2 w(x k/2), if n = 0, k even, θk n (x) = 2 w(x k/2) cos(2πnx), if n {1, 2, 3,... }, k even, 2 w(x (k + 1)/2) sin(2π(n + 1)x), if n {0, 1, 2,... }, k odd. The alternating polarities of the contiguous shifts arise naturally as we will see, also in the case of local trigonometric bases in Section 4.2. Requiring that the θk n be orthogonal imposes a strong constraint on the class of admissible windows. The constraint is not so severe in the case of Riesz bases and allows for modified Gaussian windows as Coifman and Meyer [90] showed. They considered specific Gaussians of the form exp( ζ(x 1/4) 2 ) in which Re(ζ) > 0. The reason for this particular shift has to do with a folding operation introduced below. Bittner [54] used the Zak transform to construct a general family of biorthogonal Wilson bases on R, including the Gaussian bases. We consider here a slightly simpler case of his construction.

3 4.1 Wilson bases and the Zak transform 165 Let w be real-valued and symmetric with respect to x = 1/4. The Zak transform of w therefore satisfies ( 1 ) Zw 2 x, ξ = Zw(x, ξ). (4.1) We will also want to assume that w is well localized. A natural hypothesis is that both w and ŵ belong to the Wiener space W defined in Chapter 3. Then the sum defining Zw(x, ξ) converges uniformly to a continuous function. One defines the folding matrix M w (x, ξ) by [ ] Zw(x, ξ) Zw( x, ξ) M w (x, ξ) = (4.2) Zw( x, ξ) Zw(x, ξ) for (x, ξ) Q + = [0, 1/2) [ 1/2, 1/2). Notice that M(x, ξ) has determinant det M(x, ξ) = Zw(x, ξ) 2 + Zw( x, ξ) 2. Define the folding operator T w on L 2 (R) by [ ] [ ] ZTw f(x, ξ) Zf(x, ξ) = M ZT w f( x, ξ) w (x, ξ), (x, ξ) Q +. (4.3) Zf( x, ξ) To fix some notation, set 2, for n = 0, ε = 0, e n ε (x) = 2 cos 2πnx, for n {1, 2, 3,... }, ε = 0, 2 sin(2(n + 1)πx), for n {0, 1, 2,... }, ε = 1. (4.4) More generally, for integers n and k, n 0, set e n k = en k mod 2 and define ( ψk n (x) = e n k(x) w x k ). (4.5) 2 Since the functions e n k are periodic, Zψ2k(x, n ξ) = e 2πikξ e n 0 (x) Zw(x, ξ) and Zψ2k+1(x, n ξ) = e 2πikξ e n 1 (x) Zw (x 1 ) 2, ξ. (4.6) Boundedness of Zw is enough to ensure boundedness of T w on L 2 (R) and invertibility of M w ensures invertibility of T w as the following result demonstrates. Proposition If Zw is bounded, then T w is bounded on L 2 (R) and T w sup det M w (x, ξ). (x,ξ) Q + Further, if inf (x,ξ) Q + det M w (x, ξ) > 0, then T w is invertible and ( ) 1 Tw 1 inf det M w(x, ξ). (x,ξ) Q +

4 166 4 Bases for time frequency analysis Proof. From (4.3), (4.2), and the unitarity of the Zak transform, T w f 2 2 = ZT w f(x, ξ) 2 dx dξ Q ( = ZTw f(x, ξ) 2 + ZT w f( x, ξ) 2) dx dξ Q + ( = Zw(x, ξ) Zf(x, ξ) + Zw( x, ξ) Zf( x, ξ) 2 Q + = from which the result follows. Q + Zw(x, ξ) Zf( x, ξ) Zw( x, ξ) Zf(x, ξ) 2) dx dξ det M w (x, ξ) Zf(x, ξ) 2 dx dξ The upper and lower norm bounds of Proposition are sharp. To see this, observe that det M w ( x, ξ) = det M w (x, ξ). One can choose a bounded continuous function E(x, ξ) on Q +, concentrated near the set on which det M w (x, ξ) attains its supremum, in such a way that for any η > 0, det M w (x, ξ) E(x, ξ) 2 dx dξ (1 η) det M w E(x, ξ) 2 dx dξ. Q + Q + For 1/2 < x < 0, let E(x, ξ) = E( x, ξ). Upon extending E quasiperiodically to the plane (E(x + k, ξ + l) = e 2πikξ E(x, ξ)), E becomes the Zak transform of a function g L 2 (R), namely g(x) = 1 E(x, ξ) dξ. Then 0 T w g 2 2 = det M w (x, ξ) ( E(x, ξ) 2 + E(x, ξ) 2) dx dξ Q + = 2 det M w (x, ξ) E(x, ξ) 2 dx dξ Q + 2(1 η) sup det M w (x, ξ) E(x, ξ) 2 dx dξ (x,ξ) Q + Q + = (1 η) sup det M w (x, ξ) Zg(x, ξ) 2 dx dξ (x,ξ) Q + = (1 η) sup (x,ξ) Q + det M w (x, ξ) g 2 2. Sharpness of the lower bound is similarly proved. The connection between the folding operator and the basis functions e n k and ψk n is provided by the following result. Proposition Let T w, e n k and ψn k be as above. Then for all f L2 (R), Q f, ψ n k = (k+1)/2 k/2 T w f(x) e n k(x) dx.

5 4.1 Wilson bases and the Zak transform 167 Proof. The result hinges on the simple fact that { Z(χ [k,k+1/2) e n e 2πikξ e n 0 (x), for x [0, 1/2), 0 )(x, ξ) = 0, for x [ 1/2, 0); { Z(χ [k+1/2,k+1) e n e 2πi(k+1)ξ e n 1 (x), for x [ 1/2, 0), 1 )(x, ξ) = 0, for x [0, 1/2). The unitarity of the Zak transform, even symmetry of e n 0, (4.3) and (4.6) then imply that f, ψ2k n = Zf(x, ξ) Zψ2k n (x, ξ) dx dξ Q = Zf(x, ξ) Zw(x, ξ) e 2πikξ e n 0 (x) dx dξ Q ( ) = Zf Zw(x, ξ) + Zf Zw( x, ξ) e 2πikξ e n 0 (x) dx dξ Q + = ZT w f(x, ξ) Z(χ [k,k+1/2) e n 0 )(x, ξ) dx dξ = Q k+1/2 k T w f(x) e n 0 (x) dx. A similar calculation, using the fact that e n 1 is odd, applies to f, ψ n 2k+1. The uniform upper and lower bounds on det M w was shown in Proposition to be equivalent to the boundedness and invertibility of T w. The following result shows that this is in turn equivalent to the system {ψk n } (n 0, k Z) forming a Riesz basis. Theorem Let the functions {ψ n k } be defined as in (4.5). Then {ψn k } forms a Riesz system with lower Riesz bound A = inf (x,ξ) Q + det M(x, ξ) and upper Riesz bound B = sup (x,ξ) Q + det M w (x, ξ). Proof. Suppose 0 < A B <. Then T w and its adjoint T w are bounded with bounded inverse. Since the collection e n k χ [k/2,(k+1)/2) (n 0, k Z) is an orthonormal basis for L 2 (R), for each f L 2 (R) we have (T w) 1 f = k,n (T w ) 1 f, e n k χ [k/2,(k+1)/2) e n k χ [k/2,(k+1)/2). Hence, f admits the expansion f = Tw(T w) 1 f = (T w ) 1 f, e n k χ [k/2,(k+1)/2) T w (e n k χ [k/2,(k+1)/2) ). (4.7) k,n

6 168 4 Bases for time frequency analysis However, by Proposition 4.1.2, for all g L 2 (R) g, ψ n k = T w g, χ [k/2,(k+1)/2) e n k = g, T w(χ [k/2,(k+1)/2) e n k), so that T w(χ [k/2,(k+1)/2) e n k ) = ψn k and from (4.7) we have f = k,n (T w) 1 f, e n k χ [k/2,(k+1)/2) ψ n k so that the system {ψk n } is complete. To check the Riesz bounds, let {c kn } be a square-summable sequence. Then since Tw = T w, k,n c kn ψk n = T w ( ) c kn e n k χ [k/2,(k+1)/2) k,n Tw c kn e n k χ [k/2,(k+1)/2) k,n = T w k,n c kn 2 = sup det M w (x, ξ) c l 2 (x,ξ) Q which is the desired upper Riesz bound. Also, Tw is invertible and (Tw) 1 = (Tw 1 ) = Tw 1 = (inf (x,ξ) Q det M w (x, ξ)) 1. Hence c kn 2 = c kn e n k χ [k/2,(k+1)/2) k,n k,n = c kn (Tw) 1 (ψk n ) k,n (Tw) 1 ( ) c kn ψk n 1 = inf det M w(x, ξ) c kn ψ n k (x,ξ) Q k,n which gives the desired lower Riesz bound. We now compute the dual basis for {ψk n }. Define a dual window w by k,n w(x) = 1/2 1/2 Zw(x, ξ) det M(x, ξ) dξ and in analogy with (4.5), a system { ψ n k } by Then we have: ( ψ k n (x) = e n k(x) w x k ). 2 Theorem { ψ n k } forms a Riesz basis biorthogonal to {ψn k }.

7 4.1 Wilson bases and the Zak transform 169 Proof. With these definitions, Z w is continuous and bounded since Z w(x, ξ) = Zw(x, ξ) det M w (x, ξ). Therefore, det M w (x, ξ) = (det M w (x, ξ)) 1, so that { ψ l m } forms a Riesz basis with lower Riesz bound (sup (x,ξ) Q M w (x, ξ)) 1 and upper Riesz bound (inf (x,w) Q M w (x, ξ)) 1. The biorthogonality condition ψk n, ψ l m = δ nm δ kl will be verified by considering the relative parity of the indices k, l. First, ψ2k, n ψ 2l m = e 2πi(k l)ξ e n 0 (x) e m Zw(x, ξ) 2 0 (x) dx dξ Q det M w (x, ξ) = e 2πi(k l)ξ e n 0 (x) e m 0 (x) dx dξ = δ kl δ nm Q + where we have used the orthogonality of {e 2πikξ } k= on [ 1/2, 1/2] and the orthogonality of {e n 0 } n=0 on [0, 1/2]. Next, ψ2k+1, n ψ 2l+1 m = e 2πi(k l)ξ e n 1 (x) e m Zw(x 1/2, ξ) 2 1 (x) dx dξ Q det M w (x, ξ) = e 2πi(k l)ξ e n 1 (x) e m 1 (x) dx dξ = δ kl δ nm. Q + Finally, we have ψ2k, n ψ 2l+1 m = = Q e 2πi(k l)ξ e n 0 (x) e m Zw(x, ξ)zw(x 1/2, ξ) 1 (x) dx dξ det M w (x, ξ) en 0 (x)e m 1 (x) [ Zw(x, ξ) Zw (x 1 ) det M w (x, ξ) 2, ξ ( Zw( x, ξ) Zw x 1 )] 2, ξ dx dξ = 0 Q + e 2πi(k l)ξ since the symmetry condition (4.1) and the quasiperiodicity of Zak transforms forces the integrand to vanish. Consider now the problem of identifying the dual basis in the case where the window is the shifted Gaussian w(x) = exp( α(x 1/4) 2 ) with α > 0, which has the desired symmetry, w(1/2 x) = w(x). Direct calculation followed by Poisson summation shows that ( )( ) det M w (x, ξ) = e α(2x 1/2+k)2 /2 e αm2 /2 e 2πimξ = k π 2α ( k m e α(2x 1/2+k)2 /2 )( m e 2π2 (ξ+m) 2 /α ).

8 170 4 Bases for time frequency analysis The upper and lower Riesz bounds then take the form [54] ( )( ) ( ) A = e α(1/2+k)2 /2 ( 1) m e αm2 /2 < e αk2 /2 = B k and the dual basis is generated by the dual window α 1/2 w(x) = 2π 1/2 m k γ ke α(2x 1/2+k)2 /2 ; γ l e α(2x 1/2+l)2 /2 k = When α = 2π, w itself is very nearly a shifted Gaussian. k e 2πikξ m e 2π2 (ξ+m) 2 /α dξ. 4.2 Local trigonometric bases Smooth localization The Wilson bases have good joint time and frequency-pair localization. The basis elements are approximately localized on frequency pairs of Heisenberg tiles of fixed length and width. In contrast, wavelet basis elements can be thought of as being concentrated on Heisenberg boxes of unit area but of constant relative bandwidth based on scale. By modifying the Wilson basis construction, it is possible to construct bases whose elements live under overlapping windows having compact supports of arbitrary lengths. This corresponds to tiling the time frequency plane with rectangle pairs whose sides have constant ratio over fixed time intervals, but whose ratios can change from one time interval to the next. The construction of these local trigonometric bases on R is attributed to Coifman and Meyer [89] (see also [10] and [240]), although discrete analogues were introduced first by Malvar [269] (see [189] for further insights). In the forthcoming construction only adjacent intervals have overlapping windows. The case of multiple but finitely overlapping windows was also considered by Herley et al. [187]. Remaining problems in higher dimensions will not addressed here. Let η(x) be a function that is non-negative, symmetric with respect to x = 0, supported in [ 1, 1] and having integral π/2. We can take η to be C if we wish. Set θ(x) = x η(t) dt. Then θ(x) π/4 is smooth, nondecreasing, and antisymmetric. One can also define θ ε (x) = θ(x/ε). It follows that sin(θ ε (x)) is also nondecreasing such that sin(θ ε (x)) = 0 if x < ε, sin(θ ε (x)) = 1 if x > ε and sin(θ ε (0)) = 1/ 2. Notice also that sin(θ ε ( x)) = sin(π/2 θ ε (x)) = cos(θ ε (x)). We will abbreviate sin(θ ε (x)) = s ε (x) and cos(θ ε (x)) = c ε (x). Suppose now that I = [x 0, x 1 ) is a nontrivial interval. Choose numbers α and β such that x 0 + α x 1 β. Define b I (x) = s α (x x 0 )c β (x x 1 ). The function b I (x) is called the bell over I because the graph of b I is, more or less, bell-shaped: it vanishes to the left of x 0 α, increases until it reaches the value one at x 0 + α, stays flat on (x 0 + α, x 1 β) then decreases steadily until it vanishes to the right of x 1 + β.

9 4.2 Local trigonometric bases 171 Normally one thinks of the localization of a function f to the interval [x 0, x 1 ) as the product of f with the characteristic function of [x 0, x 1 ). Such a cutoff clearly defines a projection onto a subspace of L 2 (R) and two such projections are orthogonal when their cutoff intervals are disjoint. We aim to build analogous projections with smooth, hence necessarily overlapping cutoffs. Orthogonality is no longer automatic and requires folding at endpoints. Definition Let I = [x 0, x 1 ) and let b I be as above. The localization S I f of f to I is defined as S I f(x) = b I (x)f(x) + b I (2x 0 x)f(2x 0 x) b I (2x 1 x)f(2x 1 x). One now defines an operator P I f(x) = b I (x)s I f(x). Symmetry properties of s α and c β show that S I f(x) is symmetric on (x 0 α, x 0 +α) with respect to its midpoint x 0 while it is antisymmetric with respect to x 1 on (x 1 β, x 1 +β). This guarantees that P I defines a projection. Proposition P I P I = P I. In fact, f = P I (f) if and only if f = b I S with S even with respect to x 0 on (x 0 α, x 0 + α) and odd with respect to x 1 on (x 1 β, x 1 + β). Proof. The first statement follows from the second. Suppose now that S has the indicated local symmetries. Then we claim that S I (b I S) = S on [x 0 α, x 1 + β) in which case b I S is certainly in the range of P I = b I S I. The trick is to analyze the regions on which the behavior of the bell varies. If x 0 α x x 0 + α then x 0 α 2x 0 x x 0 + α so b I (2x 1 x) = 0 and S I (b I S)(x) = b 2 I(x)S(x) + b 2 I(2x 0 x)s(2x 0 x) b 2 I(2x 1 x)s(2x 1 x) = s 2 α(x x 0 ) S(x) + c 2 α(x x 0 ) S(2x 0 x) = (s 2 α(x x 0 ) + c 2 α(x x 0 )) S(x) = S(x) where we have used the symmetry of S on (x 0 α, x 0 + α). If x 0 + α x x 0 β, then b I (x) = 1 and b I (2x 0 x) = b I (2x 1 x) = 0 so that S I (b I S)(x) = S(x). Finally, if x 1 β x x 1 + β, then x 1 β 2x 1 x x 1 + β and b I (2x 0 x) = 0 and S I (b I S)(x) = c 2 β(x x 1 ) S(x) c 2 β(x 1 x) S(2x 1 x) = S(x) where we have used the anti-symmetry of S on (x 1 β, x 1 + β). Thus for such a function S one has S I (b I S) = S and, hence, P I (b I S) = b I S and b I S is in the range of P I. This completes the proof. The aim here is to build a family of orthogonal projections that give rise to a basis for L 2 (R). Suppose that I = [x 0, x 1 ) and J = [x 1, x 2 ) are two adjacent intervals. Bells b I (x) and b J (x) for I and J are said to be compatible provided b I (x) = s α (x x 0 )c β (x x 1 ) while b J (x) = s β (x x 1 )c γ (x x 2 ) in which x 0 + α x 1 β < x 1 + β x 2 γ. The point is that the inner cutoff terms need to match up. Given such compatible bells one defines the bell b I J (x) = s α (x x 0 )c γ (x x 2 ).

10 172 4 Bases for time frequency analysis Proposition Suppose that I and J are contiguous intervals as above with compatible bells b I,b J and let b I J be defined as above. Then P I + P J = P I J while P I P J = P J P I = 0. Proof. The second statement follows from the first along with the idempotency of the projections, since then one has P I J = (P I + P J ) 2 = P I J + P I P J + P J P I which shows that P I P J = P J P I. On the other hand, P I P J = P 2 I P J = P I P J P I = P J P 2 I = P JP I, which provides a contradiction unless P I P J = 0. The first statement is proved directly from the definitions using observations similar to those used in proving Proposition and we leave it as an exercise. Armed with Proposition one can build a resolution of the identity by choosing a strictly increasing sequence {x k } R such that lim k ± x k = ± and assigning to each interval I k = [x k 1, x k ) the operator P k = P Ik = b Ik S Ik in such a way that contiguous intervals are equipped with compatible bells. It then follows from Proposition that M k= N P k f = P [x N 1,x M ]f f in L 2 (R) as N, M. Thus, to build an orthonormal basis for L 2 (R) of functions localized about the interval I k it suffices to build, for each k, an orthonormal basis e nk of functions on I k having the indicated symmetry properties. Then the functions b Ik e nk will be orthogonal in L 2 (R) and, being in the image of P k, will automatically be orthogonal to corresponding orthogonal families generated over P k for k k. Here is the idea. Proposition Suppose that {e n } forms an orthonormal basis for L 2 (I) where I = [x 0, x 1 ). Given α, β > 0 with α + β < x 1 x 0, let ẽ n be obtained by symmetric extension of e n on (x 0 α, x 0 +α) and by antisymmetric extension of e n on (x 1 β, x 1 +β). Then the functions b I ẽ n form an orthonormal family in L 2 (R). Proof. First notice that { x0 +α b I ẽ n, b I ẽ m = x 0 α x1 β x1 +β } + + b 2 I(x) ẽ n (x) ẽ m (x) dx. (4.8) x 0 +α x 1 β The symmetry properties of the extensions ẽ n and the change of variable x 2x 0 x gives x0+α x 0 α b 2 I(x) ẽ n (x) ẽ m (x) dx = = x0+α x 0 x0 +α x 0 [ b 2 I (2x 0 x) + b 2 I(x) ] e n (x) e m (x) dx e n (x) e m (x) dx

11 4.2 Local trigonometric bases 173 since b 2 I (2x 0 x) + b 2 I (x) = 1 on [x 0, x 0 + α). Similarly, since ẽ n ẽ m is locally symmetric about x 1, x1 +β x 1 β b 2 I(x) ẽ n (x) ẽ m (x) dx = x1 x 1 β e n (x) e m (x) dx. Adding now the terms in (4.8) and using the fact that b I (x) = 1 inside (x 0 + α, x 1 β), one finds that b I ẽ n, b I ẽ m = x 1 x 0 e n (x)e m (x) dx = δ nm as claimed. The particular structure of the local basis functions is not so crucial as is the local bell condition b 2 I (2x 0 x) + b 2 I (x) = 1. Corollary Suppose that for each k Z one has an orthonormal basis {e nk } for L 2 (I k ). Then {b Ik ẽ nk } n,k forms an orthonormal basis for L 2 (R). The corollary follows from the orthogonality of the projection operators P k along with the fact that P k (b Ik ẽ nk ) = b Ik ẽ nk by the symmetry properties of ẽ nk and the characterization of the range of P I from Proposition Example. The functions {cos ((2n + 1)πx/2)} n=0,1,... form an orthonormal basis for L 2 [0, 1]. Further, the basis functions are locally symmetric near x = 0 and locally antisymmetric near x = 1. By dilating and shifting one can obtain a corresponding basis for I = [x 0, x 1 ), namely ( ) 2 2n + 1 e n (x) = I cos π(x x 0 ) (n 0) 2 I which are locally symmetric near x = x 0 and locally antisymmetric near x = x 1. By choosing a corresponding basis for each interval I k one obtains an orthonormal basis for L 2 (R). Figure 4.1 illustrates several cosine packets. Remark. The particular symmetries chosen at the left and right endpoints of I are not crucial for building a projection over I. For example, by changing the to a + in the definition of S I one would obtain a function S I f that is locally symmetric at both endpoints of I; by changing the + to a one would obtain a function that is locally antisymmetric at both endpoints of I. The operator b I S I would be idempotent in either case. However, in order that projections corresponding to contiguous intervals be orthogonal, one must have opposite polarity of the two projections at any shared endpoint as is the case when symmetry is imposed at the left endpoint and antisymmetry at the right endpoint. By choosing other symmetries one can build local bases of sinusoids. The bases obtained in the corollary, starting from trigonometric functions {e nk }, are called local trigonometric bases Locally bandlimited functions The techniques just outlined are capable of generating large families of bases of L 2 (R). Intuitively, such bases are appropriate for analyzing signals having

12 174 4 Bases for time frequency analysis 8 Some Cosine Packets 7 (7,32, 1) 6 (6,16, 2) 5 (5, 8, 4) 4 (4, 4, 8) 3 (3, 2,16) 2 (2, 1,32) 1 (1, 0,64) Fig Plots of cosine packets of different scales and frequencies components of different onsets, durations, and basic waveforms. Best basis algorithms (e.g., [364]) seek to choose, among a given family of bases, one that optimizes a given function of the cost of computing approximate signal expansions. If, for the sake of argument, one wishes to expand f in local trigonometric functions, there is still the question of how to choose the segmentation points {x k } from which the basic bells are chosen. There are several issues here and we will consider just a few. First, often one does not know, a priori, what are good points at which to segment a given signal. One can attempt a type of recursive dyadic partitioning (see Section 2.4). However, the signal may have sharply localized features near points of the form k/2 L where k is odd and L large, meaning there is a high cost in searching for good break points. A second issue is that time frequency localization of bells b I (x) = s α (x x 0 )c β (x x 1 ) depends on the factors α, β that determine the steepness of the cutoff. One has α + β x 1 x 0 and, ideally, one wants α β (x 1 x 0 )/2 in order not to introduce artificial high frequencies into localized signal components. Good time frequency localization under adjacent intervals then requires that these intervals have lengths of roughly the same magnitude. Jawerth and Sweldens [216] suggest a type of multiscale folding to address this issue. A second issue is that of dealing with discrete data. Malvar s lapped orthogonal transforms (LOT s) are the discrete version of Coifman and Meyer s local trigonometric bases. When working with discrete data one is constrained by the Nyquist rate, whereas when working with LOTs one hopes that the signal is locally well approximated by local trigonometric terms of low degree. A local sampling theorem should tell us how long windows must be in order to contain well-defined frequency content of a sampled bandlimited signal. Such

13 4.2 Local trigonometric bases 175 a local sampling theorem was deduced by Bernardini and Vetterli [49]. The theorem really applies to the Wilson bases, that is, local cosine basis with uniform bells. Bernardini and Vetterli defined a subspace V j of L 2 (R) as V j = span{g jk : k Z} in which ( ( g jk = b(t k) cos π j + 1 ) ) t. 2 Here, b is a bell function based on [0, 1] as described in Section 4.2. From these V j one can build U (N) = span{g jk : 0 j < N, k Z}. Then U (N) may be thought of as a class of real signals that are locally bandlimited of order N. The space U (N) depends on the specific bell used in the definition of g jk, so the extent to which signals can be regarded as approximately locally bandlimited is open to interpretation. Given a continuous-time function f and a positive integer N, consider the sample sequence f (N) [n] = f(1/(2n) + n/n) (n Z). Similarly, we define b (N) [n] = b(1/(2n) + n/n) and g (N) jk [n] = g jk(1/(2n) + n/n). The sampling sequence {1/(2N) + n/n} n is chosen as it is in order to preserve local symmetry properties of signals with respect to integers. A space of discrete-time signals U (N) is obtained by sampling signals in U (N) : U (N) = {f (N) [n] : f U (N), n Z}. As is shown in [49], local symmetry properties inherited from {g jk } are enough to show that {g (N) jk }N 1 j=0 is an orthonormal basis for U (N). Consider now the kernel R(t, x) = k N 1 j=0 g jk (t)g jk (x) which, by the orthonormality of {g jk }, is the reproducing kernel for U (N). The following result can be viewed as an analogue of the classical sampling theorem. It applies to continuous-time signals that are locally bandlimited and was first proved by Bernardini and Vetterli [49]. Theorem If f U (N), then f admits the sampling representation f(t) = ( f (N) [n]r t, n ). N n As a basic component of the proof, one must relate the coefficients of the continuous representation to those of the discrete representation. Proof. If f U (N), then there are constants a jk such that f(t) = k N 1 j=0 a jk g jk (t). (4.9)

14 176 4 Bases for time frequency analysis Since f is continuous, sampling both sides of this equation at t = n/n yields f (N) [n] = f(1/(2n) + n/n) = k N 1 j=0 a jk g (N) jk [n]. Applying the discrete orthonormality of {g (N) jk } jk then gives a jk = n f (N) [n]g (N) jk [n]. (4.10) Substituting (4.10) into (4.9) gives the result. 4.3 Wavelet packet bases High- and low-pass filters Recall that scaling functions ϕ(x) are associated with scaling sequences {h k } such that (1.2), i.e., ϕ(2ξ) = H(ξ) ϕ(ξ) holds, where the low-pass scaling filter H(ξ) = k h ke 2πikξ. When ϕ is orthogonal to its shifts, by (1.3) the highpass filter G(ξ) = k g ke 2πikξ with g k = ( 1) k h 1 k satisfies the identity H(ξ) 2 + G(ξ) 2 1. As in Section one defines the discrete convolution decimation operator H acting on sequences by (Hc) k = 2 l h l 2k c l. The adjoint operator H, which acts by upsampling followed by convolution is given by (H c) k = 2 h k 2l c l. l Operators G and G are defined similarly by replacing the coefficients h k by g k = ( 1) k h 1 k, the Fourier coefficients of G. The QMF condition may be written H H + G G = I and GH = HG = 0 (4.11) which means that the dual high- and low-pass filter pairs give rise to a perfect reconstruction subband scheme. From the point of view of functions, on the other hand, the conditions (4.11) account for the direct sum decomposition V 0 = W 1 V 1 in terms of multiresolution spaces and this is the point of view that we will build on here. At the level of coefficients {c k } of f(x) = k c kϕ(x k) in V 0 this decomposition is accomplished through the mappings c (1) = Hc and d (1) = Gc. The operators H and G are also used to obtain V 1 = V 2 W 2 so that one has V 0 = V 2 W 2 W 1. Wavelet decompositions amount to iterations of this

15 4.3 Wavelet packet bases 177 procedure. The idea behind wavelet packets is simply that the wavelet spaces can also be decomposed. Given a scaling function ϕ that is orthogonal to its integer shifts, basic wavelet packets are defined recursively as follows. Set w 0 (x) = ϕ(x) = 2 h k ϕ(2x k); w 1 (x) = ψ(x) = 2 g k ϕ(2x k). Notice that if H : l 2 (Z) l 2 (Z) is the convolution decimation operator ( Hc) k = 2 l h l 2kc l with a similar definition for Ḡ, then w 0 (x) = 2 ( Hw 0 (2x )) 0, w 1 (x) = 2 (Ḡw 0(2x )) 0. Iterating the operators H and G on w 0 and w 1 gives rise to w 2n (x) = 2 h k w n (2x k) = 2 ( Hw n (2x )) 0, w 2n+1 (x) = 2 g k w n (2x k) = 2 (Ḡw n(2x )) 0. The QMF property of the filters H(ξ) and G(ξ) is of course shared by H( ξ) and G( ξ). Consequently, the discrete filters H and Ḡ satisfy H H+ Ḡ Ḡ = I. Hence, reconstruction of w n from w 2n and w 2n+1 is achieved by w n (2x) = 1 2 ( H w 2n (x )) 0 + (Ḡ w 2n+1 (x )) 0 = l h 2l w 2n (x + l) + ḡ 2l w 2n+1 (x + l). (4.12) The formulas express the fact that wavelet packets are being employed to decompose wavelet subspaces further into high- and low-frequency components. To express this view more precisely, define { Ω n = ak w n (x k) : {a k } l (Z)} 2. Then, setting δf(x) = 2f(2x), the formula (4.12) expresses the orthogonal decomposition δω n = Ω 2n Ω 2n+1. Since W 1 = δω 0 Ω 0 = Ω 1 and W n+1 = δw n, we have and by iterating, W 2 = δw 1 = δω 1 = Ω 2 Ω 3 W m = 2m 1 j=2 m 1 Ω j. It follows that the collection {w n (x k) : k Z and 2 m 1 n 2 m 1} forms an orthonormal basis for W m. Wavelet packet splittings can be viewed as tilings of the half-plane as follows. Recall that a wavelet can be thought of as being concentrated in

16 178 4 Bases for time frequency analysis the upper time frequency plane in the Heisenberg rectangle I ω where, if I = [k/2 j, (k + 1)/2 j ) then ω = [2 j, 2 j+1 ). Suppose now that k is even and consider the wavelet pair ψ I, ψ I where I = [(k + 1)/2 j, (k + 2)/2 j ). This pair occupies the bitile [k/2 j, (k + 2)/2 j ) [2 j, 2 j+1 ) which we think of as a pair of time sibling tiles. On the other hand, this same bitile can be expressed as the pair of frequency sibling tiles [l/2 j 1, (l + 1)/2 j 1 ) ( [2 2 j 1, 3 2 j 1 ) [3 2 j 1, 4 2 j 1 ) ) where k = 2l. These frequency siblings correspond to wavelet packets from spaces Ω j of lower order Subspaces and trees; splitting criteria Wavelet packet decompositions can be associated with dyadic trees and such a scheme is consistent with a general notion of wavelet packets. Notationally, set Γ = V 0 l 2 (Z) so that we have attached a concrete pair of orthogonal projection operators H H and G G associated to V 1 and W 1 which we now label Γ 0 and Γ 1, i.e., Γ 0 = V 1 = H H V 0, Γ 1 = W 1 = G G V 0 and Γ = Γ 0 Γ 1. Iterating the operators on these subspaces then gives Γ 0 = Γ 0,0 Γ 0,1 V 2 W 2 and Γ 1 = Γ 1,0 Γ 1,1 δ 2 Ω 2 δ 2 Ω 3. After m-iterations one has a family of 2 m subspaces indexed by binary sequences of length m or, even better, by dyadic subintervals of [0, 1) of length 2 m. We denote these as Γ I, understood to mean Γ ε1...ε m when the left endpoint of I has binary expansion 2 m m 1 k=0 ε k2 k. The following theorem is due to Coifman et al. [91]. Theorem Suppose that, except for a countable set, a dyadic interval I is expressed as the disjoint union I = j=1 I j. Then Γ I = j=1 Γ I j. To interpret Theorem 4.3.1, to each interval I one associates an orthonormal basis of Γ I. Splitting I into its left and right subintervals amounts to replacing an expansion of a signal in Γ I in terms of the standard basis for Γ I by the expansion in terms of the basis functions standard to the left and right subintervals. The point here is that, taken as a union, all of the different bases are vastly overcomplete but, by fixing a partition of I into dyadic subintervals, one chooses a specific basis from all of the ones available. In the concrete setting of wavelet packets, the entire family 2 m/2 w n (2 m x k), n N, m Z, k Z of wavelet packets is overcomplete. One would like a convenient way of labelling those subfamilies that form bases. If one identifies Γ l 2 (Z) with V m (rather than V 0 ) equipped with its standard basis {2 m/2 ϕ(2 m x k)} k= and denotes I( j, n) = [2 j n, 2 j (n + 1)) restricted to those n, j such that 2 m I( j, n) [0, 1) (denote these intervals by E m ) one has the following. Theorem If E N Z has the property that, except for a countable set, [0, ) is covered by a disjoint union of dyadic intervals I( j, n) where (j, n) E, then the wavelet packets

17 4.4 Information cells and tilings 179 { } 2 j/2 w n (2 j x k) : k Z, (j, n) E form an orthonormal basis of L 2 (R). 4.4 Information cells and tilings Theorem remains valid if, instead of using the same pair of operators H and G to split each Γ I into left and right subspaces, one simply has a consistent, but possibly interval-dependent, means of splitting Γ I into an orthogonal direct sum of two subspaces indexed by its left and right subintervals. Intuitively, wavelet packets split a frequency interval into its upper and lower parts, whereas local trigonometric bases serve to split an interval space spatially in terms of its left and right subintervals. One might wish to be able to interchange such splittings so as to optimize some function of them. Herley et al. [187] introduced a general approach to joint time frequency splittings of discrete signals, though lapped orthogonal transforms were at the technical heart of that work. Given a suitable means of interpolating between the Malvar time splitting and the wavelet packet frequency splitting, one could recursively construct signal decompositions associated with arbitrary tilings of the time frequency plane. We will not review the techniques of [187]. Rather, we will consider some subsequent ideas of Bernardini and Kovacevic [48] addressing the problem of designing appropriate filters for these arbitrary tilings. As usual there are two competing issues: the cost of computing an expansion (rate) and the cost of compressing (distortion). The problem of localizing a basic signal about a tile is fundamental here. As in [48], we consider discrete time vectors f[n], n = 0,..., M 1 thought of as critically sampled sequences of function values. To analyze f, choose an orthogonal family g k, k = 1,..., M of vectors in R M and extend them symmetrically to the left and antisymmetrically to the right, setting g k [n] = g k [ 1 n] when M/2 n < 0 and g k [n] = g k [2M 1 n] when M n < 3M/2 where we have assumed that M is even. The extension process is represented by left multiplication of the orthogonal matrix G having columns g k by a 2M M matrix E. The columns of EG remain orthogonal. Subsequent windowing is represented by multiplication of EG on the left by the 2M 2M matrix W = diag (w( M/2),..., w(3m/2 1)). Here we shall assume that the window sequence w(n) is chosen a priori to have some desirable properties as in the case of lapped orthogonal transforms. The columns of H = W EG then form a filter bank with extra properties determined by W and G. For example, in the standard lapped orthogonal transforms G is essentially a DFT matrix, hence is amenable to fast filter implementations while the columns h k of H admit some interpretation of being time frequency localized. We focus here on the problem of optimizing G for a fixed choice of W. Optimization here is expressed in terms of minimizing uncertainty.

18 180 4 Bases for time frequency analysis Localization of a sequence h(n), n Z about the tile I ω in T Z means that E I = n / I h(n) 2 and E ω = T\ω n h(n)e2πint 2 dt should each be small. In the finite case one wishes to impose corresponding localization constraints that E Ik and E ωk should be small jointly on the columns of H. The errors E Ik and E ωk can be expressed in terms of a pair of 2M 2M symmetric positive semidefinite uncertainty matrices C Ik which serves to restrict the sum to appropriate indices and C ωk which represents the analogous quantity E ωk (h k ) = h T k C ω k h k. The uncertainty cost associated with the basis h k can be expressed as C{h 1,..., h k } = = M h T k (C Ik + C ωk ) h k k=1 M gk T E T W T (C Ik + C ωk ) W E g k k=1 M gk T D k g k. (4.13) k=1 Thus, given W, one seeks a basis {g k } that minimizes the cost associated with localizing the basis elements about a corresponding collection of tiles in time frequency. The g k need not be Fourier vectors. Rather, they are the discrete analogues of the local basis functions e nk in Corollary Minimizing (4.13) is untenable insofar as the matrices D k have no preordained structure. A suboptimal approach is to minimize each term of (4.13) by finding an eigenvector a k of D k with minimal eigenvalue. The a k will not generally be orthogonal to one another, so it is desirable to find the cost of orthogonalizing the resultant set as well. This involves computing the singular value decomposition of the matrix A = [a 1 a M ], A = OΣQ, in the sense that OQ is orthogonal and its columns have the property of having the minimal distance, among all orthogonal bases, from the original vectors. Proposition The matrix OQ in the singular value decomposition A = OΣQ minimizes tr [(U A) T (U A)] among all orthogonal matrices U. Proof. A simple calculation gives tr [(OQ OΣQ) T (OQ OΣQ)] = M + tr (Σ 2 ) 2tr (Σ). Further, tr [(U OΣQ) T (U OΣQ)] = tr [(U T Q T ΣO T )(U OΣQ)] = tr [(I + Q T Σ 2 Q Q T ΣO T U U T OΣQ)] = M + tr (Q T Σ 2 Q) 2tr (Q T ΣO T U) = M + tr (Σ 2 ) 2tr (Q T ΣO T U), so the problem is to maximize tr (Q T ΣO T U) = tr (ΣO T UQ T ) over all choices of U. Equivalently, one needs to show that tr (ΣV ) is maximized over orthogonal V when V = I. But this is clear because

19 tr (ΣV ) = 4.5 The discrete Walsh model phase plane 181 M Σ nn V nn n=1 M Σ nn V nn tr (Σ) n=1 since V nn 1. Here one has also used the fact that Σ nn 0. This proves the proposition. Of course, minimizing uncertainty cost is but one issue to be confronted in choosing an optimal basis corresponding to a time frequency tiling. At this stage we turn to the problem of finding a best-adapted basis chosen from a fixed family of tiling bases, specifically in the setting of Walsh functions. 4.5 The discrete Walsh model phase plane In Thiele s thesis [347], the Walsh model phase plane was formalized as a model phase space for time frequency analysis. Walsh packets are supported in an interval. It is useful to think of them as being localized in a frequency bin as well but they are discontinuous, hence poorly localized in frequency. Thus a time frequency tile can be associated with a Walsh function only in a heuristic way. To each standard Walsh function on [0, 1) one can, in fact, attach a unique number n that specifies the number of times the function changes sign on [0, 1). This sign change frequency or sequency provides a rough notion of wavenumber. What one gains from this point of view is a pairwise disjoint (a.e.) decomposition of the plane, thought of loosely as time wavenumber space, defined by cells occupied by separate orthogonal Walsh functions. In short, the precise meaning of frequency is traded-off for precise orthogonal time wavenumber decompositions. The sequency order on Walsh functions is defined recursively on [0, 1) by W 0 (x) = χ [0,1) (x), W 2n (x) = W n (2x) + ( 1) n W n (2x 1), W 2n+1 (x) = W n (2x) ( 1) n W n (2x 1), so that W n has exactly n-sign changes in [0, 1) as is easily proved by induction. This sequency ordering was introduced by Walsh [357] cf., [179], [348]. The Walsh packet functions W nlj (x) = 2 j/2 W n (2 j x l) are then thought of as functions localized on the rectangles [l/2 j, (l + 1)/2 j ) [2 j n, 2 j (n + 1)) in the time frequency plane. Literally, W njl (x) is supported in [l/2 j, (l + 1)/2 j ). W 0 is the Haar scaling function having zero oscillations on [0, 1) while W 1 is the Haar wavelet which has one oscillation on [0, 1). The parameter j in W nlj (x) does not affect wavenumber, but rather wavelength. So, care must be taken when thinking of [2 j n, 2 j (n + 1)) as some type of

20 182 4 Bases for time frequency analysis Fourier support of W nlj. Nevertheless, interpreting [2 j n, 2 j (n + 1)) as a wave support allows one to associate a dyadic tile P = P nlj = [l/2 j, (l + 1)/2 j ) [2 j n, 2 j (n + 1)) with a unique Walsh packet W P = W nlj and, thereby the union R R + of these tiles can be called the Walsh phase plane. One calls I P = [l/2 j, (l + 1)/2 j ) the time interval of P and ω P = [2 j n, 2 j (n + 1)) the frequency interval of P. Figure 4.2 illustrates several Walsh packets. Figure 4.3 illustrates the nor- Fig Walsh packets of different shifts and sequencies malized tiles associated with a pair of Walsh packets. The phase planes for these packets are normalized so that a fixed number of congruent tiles will fit inside the square. Disjoint tiles give rise to orthogonal Walsh packets. Lemma Two Walsh packets having disjoint tiles are orthogonal. Proof. The statement is obvious when the tiles have disjoint time supports. Disjoint tiles with a common time interval of unit length are orthogonal because they are integer time shifts of orthogonal Walsh functions W n. In general, distinct Walsh packets sharing the same time interval are Walsh functions rescaled and shifted by the same factor, so orthogonality follows. If the time intervals I P and I P of W P and W P merely overlap then, since they are dyadic one is a dyadic subinterval of the other, say I P I P. But then W P χ IP is a multiple of a Walsh packet W P : this follows from the recursive definition of Walsh functions. On the other hand, the frequency intervals ω P and ω P must be disjoint since the tiles are. Since ω P = ω P it follows again from dyadic geometry that ω P and ω P are disjoint and so, as before, W P is orthogonal to W P and hence to W P.

21 4.5 The discrete Walsh model phase plane Fig Normalized Walsh packets and their associated tiles In the Walsh plane, Walsh packet tiles have unit area. For reasons that will become apparent, it will be useful to associate tree structures to certain subsets of tiles. Toward this end, it is also useful to work with admissible bitiles. These are pairs of tiles, or tiles of area two, in which both the time and frequency intervals are dyadic. Not all adjacent pairs of tiles form admissible bitiles. Bitiles can be split into tiles either by splitting in frequency into upper and lower sibling tiles or in time into left and right sibling tiles. The relevance of bitiles stems from combination rules for the different siblings. If B is a bitile with frequency siblings B + and B and time siblings B l and B r then [ ] WB = 1 [ ] [ ] 1 ( 1) n WBl W B ( 1) n+1. W Br That is, time or frequency sibling pairs of Walsh packets span one another Subspaces spanned by finite sets of tiles What follows is a sequence of lemmas concerning the geometry of tiles. The immediate goal is to show that the span of a finite collection of Walsh packets depends only on the region covered by their tiles. This will have consequences presently in terms of defining optimal bases for signal expansion and later, in Chapter 7 in proving boundedness of certain integral operators. Several geometric details will be left as exercises for the reader (see also [347]). The first observation offers a relationship between tiles and bitiles. Lemma Let B, C be distinct bitiles with upper and lower siblings B +, B and C +, C respectively. Then either B + C + = or B C =.

22 184 4 Bases for time frequency analysis It is simple to put a partial order on tiles based on containment of their time intervals: one writes P P if I P I P and ω P ω P. Given a set P of tiles one denotes by P min the set of minimal tiles in P and by P the set of all tiles contained in span (P) P P P. One has Lemma Given a finite set P of pairwise disjoint tiles, either P = or P contains a pair of frequency siblings. P min Suppose that P P min where P min is the set of tiles contained in the span of the minimal tiles in P. Choose a tile P with time interval I P of maximum length among tiles in P \ P min. The frequency sibling Q of this tile must then also belong to P. To see this, note that there is a tile P P in P and so P Q as well. Since P span (P), there must be a P P \ P min that intersects Q nontrivially. As the tiles in P are pairwise disjoint, P P = so Q P. But I P I P = I Q so one must have Q = P. Corollary If P is a pairwise disjoint set of tiles then span (P) = span ( P min ) = span ( P max ). The key observation here is that if span (P) contains a bitile then the frequency and time siblings can be substituted for one another in span (P). The same observation also yields the following. Corollary If P is a pairwise disjoint set of tiles then each R P belongs to a set of pairwise disjoint tiles having the same span as P. Bitiles can be given the same partial ordering by containment of time intervals as tiles have. One says that a collection B of bitiles is convex if B, B B and B B B implies B B. Lemma The union of a finite convex set of bitiles can be decomposed into a disjoint union of tiles. The lemma is proved by induction. The idea is to start with a minimal bitile B in the convex collection B and to use convexity to show that if P is one of the frequency siblings of B then P is either contained in or disjoint from the union of bitiles in the set B \ {B}. The thrust of the lemmas above is that there is a well-defined L 2 -projection attached to any union of pairwise disjoint tiles. Theorem If P and P are finite, pairwise disjoint sets of tiles having the same tile span then {W P : P P} and {W Q : Q P } form orthonormal families spanning the same subspace of L 2 (R). In particular, the union of a finite collection of pairwise disjoint tiles defines an orthogonal projection operator onto the corresponding subspace. The theorem follows from the fact that the transformation from P to P min only requires converting pairs of frequency siblings to time siblings. This is where bitiles come in. Each such conversion is just an orthogonal transformation. The result also uses Lemma

23 4.5.2 Tilings and the notion of best basis 4.5 The discrete Walsh model phase plane 185 As there is a large number of possible Walsh packet bases that can span the same disjoint union of tiles, it is important to have a criterion for choosing, among such bases, one that is best for performing a certain signal processing task, such as compressing a signal or image. One would also like an efficient algorithm for computing such an optimal signal representation. Definition Let S denote the set of all sequences defined on dyadic tiles. A functional H : S [0, ) is said to be additive if, whenever P is a pairwise disjoint set of tiles, one has H({a Q χ P (Q)}) = P P H({a Qδ QP }). Here χ P (Q) = 1 if Q P and is zero otherwise, while δ QP is the Kronecker delta for a pair of tiles. Examples include the entropy function H({a P }) = a 2 P log a2 P or H r({a P }) = a P r if r < 2. Fix an additive function H. If A is a subset of the Walsh plane that can be written as a pairwise disjoint union of tiles P P then one sets m A ({a P }) = inf{h({a Q χ P (Q)}) : P defines a pairwise disjoint cover of A}. The mapping A m A is subadditive. That is, if A 1 and A 2 are disjoint subregions spanned by pairwise disjoint collections of tiles, then m A1 A 2 m A1 + m A2. For dyadic rectangles one actually has the following. Lemma Let H be an additive functional and D a dyadic rectangle in the Walsh plane with area at least 2. Let D +, D, D l, D r denote the top, bottom, left and right subrectangles, respectively. Then m D = min{m D + + m D, m Dl + m Dr }. The important observation here is that if P is a tiling of D by pairwise disjoint tiles then, from interval length considerations, every tile is either contained entirely in one of D +, D or else every tile is contained entirely in one of D l, D r. Suppose now that P satisfies m D ({a P }) = H({a Q χ P (Q)}). Then P is the disjoint union of two pairwise disjoint tilings, either of D l and D r or of D + and D. The result now follows from additivity of H. The lemma gives a divide and conquer strategy for computing optimal Walsh decompositions, but the actual algorithm for doing so is obtained from the following bottom-up approach. We are thinking of analyzing discrete signals of length N = 2 J. To such a signal we associate a function that is piecewise constant on dyadic subintervals of [0, 1) of length 2 J. First compute the Walsh packet coefficients for each tile in S J = [0, 1) [0, 2 J ). There are O(J2 J ) of these. Next: compute H({a Q δ QP }) for each such tile. for l = 1,..., J do: For each dyadic rectangle of area 2 l, find a minimizing tiling built from subrectangle minimizers of size 2 l 1 by choosing the vertical or horizontal subrectangles. At the Jth iteration, a global minimizing tiling has been found for the sequence of Walsh packet coefficients. The algorithm is O(J2 J ).