A DISCRETE ZAK TRANSFORM. Christopher Heil. The MITRE Corporation McLean, Virginia Technical Report MTR-89W00128.

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1 A DISCRETE ZAK TRANSFORM Christopher Heil The MITRE Corporation McLean, Virginia Technical Report MTR-89W00128 August 1989 Abstract. A discrete version of the Zak transfor is defined and used to analyze discrete Weyl Heisenberg fraes, which are nonorthogonal systes in the space of square-suable sequences that, although not necessarily bases, provide representations of square-suable sequences as sus of the frae eleents. While the general theory is essentially siilar to the continuous case, ajor differences occur when specific Weyl Heisenberg fraes are evaluated. In particular, it is shown that while Weyl Heisenberg fraes in the continuous case are bases only if they are generated by functions that are not sooth or have poor decay, it is possible in the discrete case to construct Weyl Heisenberg fraes that are bases and are generated by sequences with good decay. The sapled Gaussian provides an exaple of such a signal. 1. Introduction The recent paper [HW1] surveyed the current literature on fraes in L 2 (R). Fraes are generalizations of orthonoral bases and L 2 (R) is the space of finite-energy signals on the real line. The intent of the frae constructions described in [HW1] is to provide a atheatical foundation for siultaneous tie and frequency analysis in signal processing. However, before these fraes can be ipleented as algoriths on a digital coputer, the theory ust be extended to the type of signals actually used in digital signal processing, naely, discrete or sapled signals. D. Walnut, in [W2], began this work by describing frae constructions for l 2 (Z), the space of square-suable sequences indexed by Z, the set of integers. Each such sequence can be thought of as a sapled finite-energy signal. This paper continues in the sae spirit by extending another aspect of the continuoussignal theory to discrete signals, naely, we define a discrete version of the so-called Zak transfor. We show that the definition of this transfor and the general theores obtained by applying it to Weyl Heisenberg fraes are essentially the sae as in the continuous case, but that very different results can occur when specific Weyl Heisenberg fraes are evaluated. In particular, discrete Weyl Heisenberg fraes appear to be ore tractable than their continuous counterparts, indicating that they have good potential for digital signal processing analysis. This work was supported by the MITRE Sponsored Research Progra. 1

2 2 CHRISTOPHER HEIL To give a ore precise eaning to the above rearks, let us first recall the definition of orthonoral basis. If H is a Hilbert space (e.g., L 2 (R) or l 2 (Z)) with inner product, and nor, then a set of vectors {e n } in H is an orthonoral basis if { 1, if = n; 1. e, e n = 0, if n; 2. x, e n 2 = x 2 for all x H. A fundaental property of orthonoral bases is that x = x, e n e n for every x H, so that inforation about x is stored in the set of scalars { x, e n }. Because the orthonorality condition (1) is a stringent and sensitive condition, we define fraes, which have no orthonorality condition and are therefore often easier to generate. A set of vectors {x n } in H is a frae if there exist constants A, B > 0 such that A x 2 n x, x n 2 B x 2 for all x H. As exposited in [HW1], the frae operator Sx = x, x n x n is then welldefined, though not necessarily the identity, as is the case for orthonoral bases. However, S is an isoorphis (i.e., is bijective, continuous, and has a continuous inverse), and every x H can be written x = x, S 1 x n x n, so x is again characterized by its frae transfor { x, S 1 x n }. The paper [HW1] discussed the proble of finding certain fraes for L 2 (R). In particular, the fraes of interest are those in which the frae eleents are easily generated fro a single fixed function through a cobination of the eleentary operations of translation, odulation, and dilation. For exaple, a Weyl Heisenberg syste for L 2 (R) is a set of the for {G n },n Z, where G n (t) = e 2πi(t na)/b G(t na) for soe fixed function G L 2 (R) and fixed a, b > 0. An affine syste is a set of the for {H n },n Z, where H n (t) = a n/2 H(a n t b) for soe fixed H L 2 (R) and fixed a > 1, b > 0. Because of the difficulty involved in tie-copressing sapled signals, discrete versions of affine fraes are fairly coplex (see [W2]). This akes discrete Weyl Heisenberg fraes correspondingly ore attractive, and so we will not ention affine fraes again in this paper. The discrete analogue of Weyl Heisenberg systes is a set of the for {g n } n Z,=0,...,a 1, where g n (k) = e 2πi(k na)/b g(k na) for soe fixed g l 2 (Z) and fixed integers a, b > 0. For quantu echanical reasons, J. Zak in the 1960s proved results about the properties of the Weyl Heisenberg syste {G n } when G was the Gaussian signal G(t) = e πt2 and when a = b = 1. In particular, he showed that the set {G n } was coplete (in fact, overcoplete by one eleent) in the sense that its linear span was dense in L 2 (R) (see [Z]). However, I. Daubechies, A. Grossann, and Y. Meyer later showed that the Gaussian does not generate a Weyl Heisenberg frae (see [DGM]). This is unfortunate since, by the

3 A DISCRETE ZAK TRANSFORM 3 uncertainty principle, the Gaussian is the function in L 2 (R) which is best localized both in tie and frequency, in the sense of iniizing the value tg(t) 2 γĝ(γ) 2 G 2. (UP) Thus it is the signal which should provide best results when attepting siultaneous tie and frequency signal analysis. Recently, it has been shown that no function G L 2 (R) for which (UP) is even finite can generate a Weyl Heisenberg frae with a = b (see [BHW] for history and review of proofs, also [Bal; Bat; D; DJ; L]). It can be shown that Weyl Heisenberg fraes are bases (i.e., the frae representations are unique) only when a = b. Thus the only Weyl Heisenberg fraes that will have this desirable feature are those for which (UP) is infinite, which iplies that G is either not sooth or does not decay quickly. This akes Weyl Heisenberg fraes in L 2 (R) soewhat unattractive. However, we show in this paper that the situation for discrete Weyl Heisenberg fraes is quite different. For exaple, we show that the discrete Weyl Heisenberg syste generated by the sapled Gaussian signal, g(k) = e πk2 for k Z, does generate a frae when a = b, and that this frae is a basis for l 2 (Z). 2. Notation R is the real line thought of as the tie axis, and ˆR is its dual group, the real line thought of as the frequency axis. Z is the set of integers, and C the coplex nubers. Z + is the set of positive integers, and R + = (0, ) the set of positive real nubers. Given a Z +, we set Z a = {0, 1,..., a 1}. We will attept to distinguish between sequences and functions by using lowercase letters for the forer and uppercase for the latter. A sequence is a apping f defined on the integers, while a function is a apping F defined on the real line. l 2 (Z) is the Hilbert space of all square-suable sequences, with nor and inner product ( ) 1/2 f 2 = f(k) 2 and f, g = f(k) g(k). k Z k Z The Banach spaces l p (Z) for 1 p < are defined analogously, and l (Z) is the space of all bounded sequences. L 2 (R) is the Hilbert space of all square-integrable functions, with nor and inner product F 2 = ( R F (t) 2 dt) 1/2 and F, G = R F (t) G(t) dt.

4 4 CHRISTOPHER HEIL The Banach spaces L p (R) for 1 p < are defined analogously, and L (R) is the space of all essentially bounded functions. Given a function F we define the translation of F by a R as Modulation is defined as T a F (t) = F (t a). E b F (t) = e 2πibt F (t) for b R. Translation and odulation are also defined on sequences, as long as a Z and b R. We also use the sybol E b by itself to refer to the exponential function E b (t) = e 2πibt. The two-diensional exponentials are E (a,b) (t, ω) = e 2πiat e 2πibω. Given a Hilbert space H and sets of vectors {x n } and {y n }, we define the following ters. 1. {x n } and {y n } are biorthonoral if x, y n = δ n = { 1, if = n; 0, if n. 2. {x n } is coplete if its linear span is dense in H, or equivalently, if the only x H with x, x n = 0 for all n is x = {x n } is inial if x / span{x n } n for any (where span stands for the closed linear span). 4. {x n } is a basis if for each x H there exist unique scalars {c n } such that x = cn x n. The basis is unconditional if every rearrangeent of these series converge. It is bounded if 0 < inf x n sup x n <. 3. Fraes We review soe of the properties of fraes. More details can be found in [H2; HW1; D; DS; Y]. Definition 3.1 [DS]. A set of vectors {x n } in a Hilbert space H is a frae if there exist nubers A, B > 0 such that for every x H we have A x 2 x, x n 2 B x 2. n The nubers A, B are the frae bounds. The frae is tight if A = B. The frae is exact if it ceases to be a frae whenever any single eleent is deleted fro the sequence. Given two operators S, T : H H we write S T if Sx, x T x, x for all x H. We denote by I the identity ap on H, i.e., Ix = x for all x H.

5 A DISCRETE ZAK TRANSFORM 5 Theore 3.2 [DS]. Given a sequence {x n } in a Hilbert space H, the following two stateents are equivalent: 1. {x n } is a frae with bounds A, B. 2. Sx = x, x n x n is a bounded linear operator with AI S BI, called the frae operator for {x n }. Corollary 3.3 [DS]. 1. S is invertible and B 1 I S 1 A 1 I. 2. {S 1 x n } is a frae with bounds B 1, A 1, called the dual frae. 3. Every x H can be written x = x, S 1 x n x n = x, x n S 1 x n. Theore 3.4 [DS]. The reoval of a vector fro a frae leaves either a frae or an incoplete set. In particular, given fixed, x, S 1 x 1 {x n } n is a frae; x, S 1 x = 1 {x n } n is incoplete. Theore 3.5 [Y]. 1. Inexact fraes are not bases, i.e., the representations in Corollary 2.3 part 3 are not unique. 2. A sequence {x n } in a Hilbert space H is an exact frae for H if and only if it is a bounded unconditional basis. This occurs if and only if there exists an orthonoral basis {e n } and an isoorphis U: H H such that x n = Ue n for all n. 4. The Zak Transfor Two good articles on the properties of the continuous version of the Zak transfor are [J1; J2]. Definition The (continuous) Zak transfor of a function F is (forally) ZF (t, ω) = j Z F (t + ja) e 2πijω for (t, ω) R ˆR, and where a R + is a fixed paraeter.

6 6 CHRISTOPHER HEIL 2. The (discrete) Zak transfor of a sequence f is (forally) zf(k, ω) = f(k + ja) e 2πijω j Z for (k, ω) Z ˆR, and where a Z + is a fixed paraeter. The series defining ZF or zf can converge in various senses. For exaple, if F is a continuous function with copact support, then ZF clearly converges pointwise since it reduces everywhere to a finite su. First note that ZF and zf are quasiperiodic in the following sense: 1. For all (t, ω) R ˆR we have ZF (t + a, ω) = e 2πiω ZF (t, ω); ZF (t, ω + 1) = ZF (t, ω). 2. For all (k, ω) Z ˆR we have zf(k + a, ω) = e 2πiω zf(k, ω); zf(k, ω + 1) = zf(k, ω). Therefore, ZF is copletely deterined by its values on the fundaental rectangle Q = [0, a) [0, 1), while zf is copletely deterined by its values on the set q = Z a [0, 1) = {0,..., a 1} [0, 1). Note that q is the subset of Q consisting of vertical lines with integer abscissa coordinates. Moreover, if F is a continuous function, and f is the integer sapled values of F, i.e., f(k) = F (k) for k Z, then ZF (k, ω) = zf(k, ω) for all (k, ω) q (in fact, for all (k, ω) Z ˆR). Definition 4.2. We let L 2 (Q) be the Hilbert space of all appings Φ: Q C for which Φ 2 = ( a Φ(t, ω) 2 dt dω) 1/2 <,

7 A DISCRETE ZAK TRANSFORM 7 with the usual inner product. Siilarly, l 2 (q) is the Hilbert space of appings ϕ: q C for which ( 1/2 1 ϕ 2 = ϕ(k, ω) dω) 2 <, k Z a again with the expected inner product. 0 The proof of the following theore for Z can be found in [HW1]. The proof for the discrete case is virtually identical, so will be oitted. Theore Z is a unitary ap of L 2 (R) onto L 2 (Q), with the series defining Zf converging in the nor of L 2 (Q). 2. z is a unitary ap of l 2 (Z) onto l 2 (q), with the series defining zf converging in the nor of l 2 (q). 5. Analysis of Weyl Heisenberg Fraes with the Zak Transfor Definition Given a fixed function G L 2 (R) and given a, b R +, the (continuous) Weyl Heisenberg syste generated by (G, a, b) is {G n },n Z, where G n (t) = T na E b G(t) = e2πi(t na)/b G(t na). We call G the other wavelet and a, b the syste paraeters. 2. Given a fixed sequence g l 2 (Z) and given a, b Z +, the (discrete) Weyl Heisenberg syste generated by (g, a, b) is {g n } Zb,n Z, where g n (k) = T na E b g(k) = e2πi(k na)/b g(k na). We call g the other sequence and a, b the syste paraeters. We are ost interested, of course, in Weyl Heisenberg fraes. We refer to [HW1] for exaples and conditions for the existence of continuous Weyl Heisenberg fraes, and to [W2] for discrete W H fraes. Since Z and z are unitary aps, we can use the to transfor probles fro the tie doain into a Zak transfor doain. More precisely, the unitarity iplies that a discrete W H syste {g n } in l 2 (Z) is coplete, a frae, an exact frae, an orthonoral basis, etc., if and only if the sae is true of the syste {zg n } in l 2 (q), with analogous rearks holding for continuous systes. This transforation turns out to be particularly useful in the case a = b, for zg n then has the following siple for.

8 8 CHRISTOPHER HEIL Theore 5.2. Given g l 2 (Z) and a Z +. If b = a then zg n = E ( a,n) zg. Proof. We copute: zg n (k, ω) = j = j g n (k + ja) e 2πijω e 2πi(k+ja na)/a g(k + ja na) e 2πijω = e 2πik/a j = e 2πik/a j g(k + ja na) e 2πijω g(k + ja) e 2πi(j+n)ω = e 2πik/a e 2πinω j g(k + ja) e 2πijω = E ( a,n) (k, ω) zg(k, ω). An analogous result holds for the continuous case. The iportance of Theore 5.2 is that the set of functions {a 1/2 E ( a,n) } Za,n Z fors an orthonoral basis for l 2 (q). Thus the question of when {g n } can for a frae for l 2 (Z) when a = b is transfored by z into the question of when is possible to take an orthonoral basis for l 2 (q), ultiply each eleent of the basis by the single function zg, and still obtain a frae for l 2 (q). This iposes severe restrictions on the function zg, which are given precisely in the following theore. Theore 5.3. Given a fixed g L 2 (R) and a Z +. If we define b = a then the following stateents hold. 1. {g n } is coplete in l 2 (Z) if and only if zg 0 a.e. 2. {g n } is inial and coplete in l 2 (Z) if and only if 1 zg l2 (q). 3. {g n } is a frae for l 2 (Z) with frae bounds A, B if and only if In this case the frae is exact. 0 < a 1 A zg 2 a 1 B < a.e. 4. {g n } is an orthonoral basis for l 2 (Z) if and only if zg 2 = a 1 a.e.

9 A DISCRETE ZAK TRANSFORM 9 Proof. 1. Assue {g n } was coplete in l 2 (Z). Then {zg n } is coplete in l 2 (q) by the unitarity of z. Define ϕ on q by ϕ(k, ω) = { 1, if zg(k, ω) = 0; 0, if zg(k, ω) 0. Then zg n, ϕ = E ( a,n) zg, ϕ = E ( a,n), zg ϕ = E ( a,n), 0 = 0 for all, n. But {zg n } is coplete, so this iplies ϕ = 0 a.e., which iplies zg = 0 a.e. Now assue zg 0 a.e., and suppose ϕ l 2 (q) is such that zg n, ϕ = 0 for all, n. Then 0 = zg n, ϕ = E ( a,n) zg, ϕ = E ( a,n), zg ϕ. Since zg ϕ l 1 (q) and {E ( a,n) } is coplete in l 1 (q), it follows that ϕ = 0 a.e., so {zg n } is coplete is l 2 (q). Since z is unitary, this iplies that {g n } is coplete in l 2 (Z). 2. Assue {g n } is inial and coplete in l 2 (Z), so the sae is true of {zg n } in l 2 (q). By part 1, we also know that zg 0 a.e. Now, the iniality iplies (see [S]) that there is a set of functions {ϕ n } in l 2 (q) which are biorthonoral to {zg n }, i.e., zg n, ϕ n = δ δ nn = { 1, if = and n = n ; 0, otherwise. Now zg n, ϕ n = E ( a,n), zg ϕ n, and zg ϕ n l 1 (q) while E ( a,n) l (q). Thus the set {zg ϕ n } l 1 (q) is biorthonoral to the set {E ( a,n) } l (q). But there is a unique sequence in l 1 (q) which is biorthonoral to {E ( a,n) } l (q), naely {a 1 E ( a,n) }. Therefore, zg ϕ n = a 1 E ( a,n), so E ( a,n) /zg = a 1 ϕ n for all, n. In particular, 1/zg l 2 (q). Conversely, assue 1/zg l 2 (q). Then zg 0 a.e., so {g n } is coplete by part 1. Let g = a 1 Z 1 (1/zg) l 2 (Z). Then g n, g n = zg n, z g n = E ( a,n) zg, E ( a,n ) z g = a 1 E ( a,n), E ( a,n ) = δ δ nn.

10 10 CHRISTOPHER HEIL Thus {g n } and { g n } are biorthonoral in l 2 (Z). The existence of a biorthonoral set iplies iediately that {g n } is inial (see [S]). 3. Siilar to the continuous Zak arguent presented in [HW1]. 4. Follows easily fro 3. An analogous continuous version of Theore 5.3 can be found in [H2] and [HW1]. We are especially interested in parts 3 and 4 of Theore 5.3. Part 3 iplies that, in the a = b case, any W H frae will be exact, hence a basis for l 2 (Z). This is desirable, since in this case we have unique decopositions of l 2 (Z), i.e., the frae representations in Corollary 2.3 part 3 are unique. In both the continuous and discrete cases, it can be shown that if the ratio a/b is greater than 1 then it is ipossible to for W H fraes, while if it is less than 1, any W H frae will be inexact, i.e., the frae representations will not be unique. This is why we have concentrated our attention on the a = b case. Our goal now is to deterine what sorts of functions will or will not satisfy the conditions of Theore 5.3. To do this, we ust deterine when the Zak transfor of a sequence or function will have zeroes. In the continuous case, it can be shown that if G L 2 (R) is such that ZG is continuous on the plane R ˆR, then ZG has a zero, and so G cannot generate a W H frae for the case a = b. It is shown in [H2] that if G is continuous and satisfies the ild decay condition sup k Z t [k,k+1] g(t) <, then ZG will be continuous. Thus only unpleasant functions will generate W H fraes for L 2 (R). In particular, the Gaussian function G(t) = e πt2 will not generate a W H frae when a = b. The situation is ore coplicated in the discrete case, for here zg is defined only on disconnected vertical lines. Therefore, there is no concept of continuity in the k variable, a fact which will allow us to construct discrete W H fraes fro sequences which do have good decay. In the sequel, when we speak of the continuity of zg, we will ean only that zg is continuous in the ω variable on each vertical line. Theore 5.4. If g l 1 (Z) then zg is well-defined; in fact, the series defining zg(k, ) converges uniforly for each k Z to a continuous function. Proof. Given any fixed k Z, we have the following for each ω ˆR: zg(k, ω) = g(k + ja) e 2πijω k Z

11 A DISCRETE ZAK TRANSFORM 11 k Z g(k + ja) g(k) k Z <. Therefore zg(k, ) converges uniforly, and so the liit ust be continuous since each ter g(k + ja) e 2πijω is continuous. Although not needed, we ention the following additional facts about the Zak transfor of l 1 -sequences. The proof is siilar to the continuous version in [H2]. Theore 5.5. Given 1 p < 2, the Zak transfor is a linear, continuous, injective apping of l p (Z) into l p (q) with z = 1. Moreover, Range(z) is dense in l p (q), but z is not surjective, and z 1 : Range(z) l p (Z) is not continuous. Theore 5.6. The Zak transfor cannot be defined as a ap of l q (Z) into any l r (q) when q > 2. We now exaine the properties of sequences whose Zak transfor is continuous. Theore 5.7. Assue g l 2 (Z) is such that zg is continuous. 1. If g is odd (i.e., g( k) = g(k) for k Z), then zg(0, 0) = zg(0, 1/2) = 0. If in addition a is even, then zg(a/2, 0) = If g is even (i.e., g( k) = g(k) for k Z), then zg(a/2, 1/2) = 0. Proof. We prove only 2 since 1 is siilar. Since g is even we can copute: zg( k, ω) = k Z = k Z g( k + ja) e 2πijω g( k ja) e 2πijω = g(k + ja) e 2πijω k Z = zg(k, ω). Since a is even, a/2 is an integer. The quasiperiodicity of zg therefore iplies that zg(a/2, 1/2) = zg( a/2, 1/2) = e 2πi( 1/2) zg(a/2, 1/2) = e πi zg(a/2, 1/2)

12 12 CHRISTOPHER HEIL = zg(a/2, 1/2). Therefore zg(a/2, 1/2) = 0. As a result of this theore, no sequence g l 1 (Z) which is odd can generate a W H frae for any integer a Z + if we take b = a. Also, no even sequence g l 1 (Z) can generate a W H frae for any even values of a Z + (with b = a). In particular, the sapled Gaussian sequence g(k) = e πk2 will not generate a frae for any even a Z +. This still leaves the question of odd values of a, which we address in the next section. The Guassian function and sequence are: 6. The Gaussian G(t) = e rt2, t R; g(k) = e rk2, k Z; where r > 0 is fixed. Since G is well-behaved, we know that ZG will be continuous and ust therefore have at least one zero in the fundaental rectangle Q. Direct calculation gives ZG(t, ω) = j G(t + ja) e 2πijω where θ 3 is the third Jacobi theta function = e rt2 e 2rtja e rj2 a 2 e 2πijω j ( ) ( ) = e rt2 e ra 2 j 2 ω e 4πij 2 rta 2πi j = e rt2 θ 3 ( ω 2 + irta 2π, e ra2 ), θ 3 (w, q) = q j2 cos(4πjw) = j=1 j= q j2 e 4πijw. (see [R] for an exposition of the properties of the four theta functions). Now, the zeroes of θ 3 (, e ra2 ) occur precisely at the points ra2 i 4π nra2 i 2π, for, n Z. These correspond to the values t = a 2 + na and ω = 1 2 a single zero in Q. +. Hence ZG has

13 A DISCRETE ZAK TRANSFORM 13 Now, zg is given by zg(k, ω) = ZG(k, ω) for (k, ω) q. If a is even then the point (a/2, 1/2) lies in q, so zg has a zero. Since ZG is continuous and zg is its restriction, this iplies that zg is not bounded below, hence cannot generate a frae for the case a = b. However, if a is odd then the point (a/2, 1/2) does not lie in q, so zg has no zeroes in q. As zg is continuous on q, this iplies that it is bounded above and below on q, and therefore does generate a W H frae for l 2 (Z) if b = a. Moreover, by Theore 5.3, this frae is exact, so is a bounded unconditional basis for l 2 (Z) by Theore 2.5. References [Bal] R. Balian, Un principe d incertitude fort en théorie du signal ou en écanique quantique, C. R. Acad. Sci. Paris 292 (1981), [Bat] G. Battle, Heisenberg proof of the Balian-Low theore, Lett. Math. Phys. 15 (1988), [B1] J. Benedetto, Gabor representations and wavelets, Coutative Haronic Analysis, 1987 (D. Colella, eds.), Contep. Math. 91, Aerican Matheatical Society, Providence, RI, 1989, pp [B2] J. Benedetto, 1990, Uncertainty principle inequalities and spectru estiation, Proceedings of the NATO Advanced Study Institute on Fourier Analysis and its Applications, 1989 (J. Byrnes, eds.), Kluwer Acadeic Publishers, The Netherlands, [BHW] J. Benedetto, C. Heil, and D. Walnut, Rearks on the proof of the Balian Low theore, preprint. [D] I. Daubechies, The wavelet transfor, tie-frequency localization and signal analysis, IEEE Trans. Infor. Theory (to appear). [DGM] I. Daubechies, A. Grossann, and Y. Meyer, 1986, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986), [DJ] I. Daubechies and A.J.E.M. Janssen, to appear, Two theores on lattice expansionsn, IEEE Trans. Infor. Theory 39 (1993), 3 6. [G] D. Gabor, Theory of counications, J. Inst. Elec. Eng. (London) 93 (1946), [H1] C. Heil, Wavelets and fraes, Proceedings of the Suer Signal Processing Conference, 1988, The Institute for Matheatics and Its Applications, Springer-Verlag, New York, [H2], Wiener aalga spaces in generalized haronic analysis and wavelet theory, Ph.D. thesis, University of Maryland, College Park, MD, [HW1] C. Heil and D. Walnut, Continuous and discrete wavelet transfors, SIAM Review 31 (1989), [HW2], Gabor and wavelet expansions, Proceedings of the NATO Advanced Study Institute on Fourier Analysis and its Applications, 1989 (J. Byrnes, eds.), Kluwer Acadeic Publishers, The Netherlands, [J1] A.J.E.M. Janssen, Bargann transfor, Zak transfor, and coherent states, J. Math. Phys. 23 (1982), [J2], The Zak transfor: a signal transfor for sapled tie-continuous signals, Philips J. Res. 43 (1988), [L] F. Low, 1985, Coplete sets of wave packets, A Passion for Physics Essays in Honor of Geoffrey Chew (C. DeTar, et al., eds.), World Scientific, Singapore, 1985, pp [R] E. Rainville, Special Functions, MacMillan, New York, [S] I. Singer, Bases in Banach Spaces, I, Springer-Verlag, New York, [W1] D. Walnut, Weyl Heisenberg wavelet expansions: existence and stability in weighted spaces (1989), University of Maryland, College Park, MD. [W2], Wavelets in Discrete Doains, The MITRE Corporation, Technical Report MP-89W00040 (1989). [Y] R. Young, An Introduction to Nonharonic Fourier Series, Acadeic Press, New York, [Z] J. Zak, Finite translations in solid state physics, Phys. Rev. Lett. 19 (1967),