Hyperbolic Secants Yield Gabor Frames

Transcription

1 Applied and Computational Harmonic Analysis 1, ( doi:1.16/acha.1.376, available online at on Hyperbolic Secants Yield Gabor Frames A. J. E. M. Janssen Philips Research Laboratories WY-81, 5656 AA Eindhoven, The Netherlands a.j.e.m.janssen@philips.com and Thomas Strohmer 1 Department of Mathematics, University of California, Davis, California strohmer@math.ucdavis.edu Communicated by Charles K. Chui Received April 19, 1; revised October 31, 1 We show that (g,a,bis a Gabor frame when a>, b>, ab < 1, and g (t = ( 1 πγ1/ (cosh πγt 1 is a hyperbolic secant with scaling parameter γ>. This is accomplished by expressing the Zak transform of g in terms of the Zak transform of the Gaussian g 1 (t = (γ 1/4 exp( πγt, together with an appropriate use of the Ron Shen criterion for being a Gabor frame. As a side result it follows that the windows, generating tight Gabor frames, that are canonically associated to g and g 1 are the same at critical density a = b = 1. Also, we display the singular dual function corresponding to the hyperbolic secant at critical density. Elsevier Science (USA Key Words: Gabor frame; Zak transform; hyperbolic secant; theta functions. 1. INTRODUCTION AND RESULTS Let a>, b>, and let g L (R. A Gabor system (g,a,b consists of all time-andfrequency shifted functions g na,mb with integer n, m, where for x,y R we denote g x,y (t = e πiyt g(t x, t R. (1 We say that (g,a,b is a Gabor frame when there are A>, B< (lower and upper frame bounds, respectively such that for all f L (R we have A f f,g na,mb B f. ( n,m 1 T.S. acknowledges support from NSF Grant / $35. Elsevier Science (USA All rights reserved.

2 6 JANSSEN AND STROHMER Here and, denote the standard norm and inner product of L (R. We refer to [ 4] for generalities about frames for a Hilbert space and for both basic and in-depth information about Gabor frames, frame operators, dual frames, tight frames, etc. For a very recent and comprehensive treatment of Gabor frames, we refer to [5, Chap. 5 9, 11 13]; many of the more advanced results of modern Gabor theory are covered there in a unified manner. It is well known that a triple (g,a,b cannot be a Gabor frame when ab > 1; see for instance [5, Corollary 7.5.1]. Also such a system cannot be a Gabor frame when ab = 1and g L (R, tg(t L (R (extended Balian Low theorem; see for instance [4, Chap. ]. In this paper we demonstrate that (g,a,bis a Gabor frame when ab < 1and ( πγ 1/ 1 g,γ (t =, t R, (3 coshπγt with γ> (normalization such that g,γ =1. The result is obtained by (i relating the Zak transform (Zg(t, ν = g(t le πilν, t,ν R, (4 of g = g,γ to the Zak transform of g = g 1,γ,whereg 1,γ is the normalized Gaussian g 1,γ (t = (γ 1/4 e πγt, t R, (5 (ii using the observation that (g 1,γ,a,b is a Gabor frame and (iii an propriate use of the Ron Shen time domain criterion [9] for a Gabor system (g,a,bto be a Gabor frame. Explicitly, we show that there is a positive constant E such that (Zg,γ (t, ν = Here ϑ 4 (z; q is the theta function; see [1, Chap. 1], ϑ 4 (z; q= E(Zg 1,γ (t, ν ϑ 4 (πν; e πγ ϑ 4 (πt; e π/γ, t,ν R. (6 ( 1 n q n e inz, z C. (7 This ϑ 4 is positive and bounded for real arguments z and parameter q (, 1. The rest of the paper is organized as follows. In Section we present the proof or our main result. In Section 3 we prove formula (6 by using elementary properties of theta functions and basic complex analysis. These same methods, combined with formula (6, allow us to compute explicitly the singular dual function g,γ d (at critical density a = b = 1 canonically associated to g,γ according to g d,γ (t = dν (Zg,γ, t R. (8 (t, ν This is briefly presented in Section 4. We also show in Section 4 that the tight Gabor frame generating windows g t 1,γ, gt,γ, canonically associated to g 1,γ, g,γ and given at critical

3 HYPERBOLIC SECANTS YIELD GABOR FRAMES 61 density a = b = 1by g t (Zg(t, ν (t = dν, t R, (9 (Zg(t, ν for g = g 1,γ,g,γ, are the same. In [Janssen, 1, submitted for publication] a comprehensive study is made of the process, embodied by formula (9, of passing from windows g with few zeros in the Zak transform domain to tight Gabor frame generating windows, g t at critical density. One of the observations in [Janssen, 1, submitted for publication] is that the operation in (9 seems to diminish distances between positive, even and unimodal windows enormously. The fact that g1,γ t = gt,γ is an absurdly accurate illustration of this phenomenon.. PROOF OF THE MAIN RESULT In this section we present a proof for the result that (g,γ,a,b is a Gabor frame when a>, b>, ab < 1, and g,γ is given by (3. According to the Ron Shen criterion in the time domain [9] we have that (g,a,bis a Gabor frame with frame bounds A>, B< if and only if AI 1 b M g(tmg (t BI, a.e. t br. (1 Here I is the identity operator of l (Z and M g (t is the linear operator of l (Z (in the notation of [4, Sect. 1.3.], whose matrix with respect to the standard basis of l (Z is given by M g (t = ( g(t na l/b, a.e. t R (11 l Z, n Z (row index l, column index n. Since g,γ is rapidly decaying, the finite frame upper bound condition is easily seen to be satisfied, and we therefore concentrate on the positive lower frame bound condition. We may restrict here to the case where a<1, b = 1, since (g,a,b is a Gabor frame if and only if (D c g,a/c,bc is a Gabor frame. Here D c is the dilation operator (D c f (t c 1/ f(ct, t R, defined for f L (R when c>. Since D c g,γ = g,γ c, we only need to replace γ>byγb>whenb 1. Hence we shall show that there is an A> such that c l g(t na l A c, c= (c l l Z l (Z, t R, (1 with g = g,γ. Taking c l (Z it follows from Parseval s theorem for Fourier series and the definition of the Zak transform in (4 that for any n Z c l g(t na l = (Zg(t na, νc (ν dν, (13

4 6 JANSSEN AND STROHMER where C(ν is defined by C(ν = Now assuming the result (6 (with E> we have that c l g,γ (t na l = = c l eπilν, a.e. ν R. (14 (Zg,γ (t na, νc (ν dν E ϑ 4 (π(t na; e π/γ Define the one-periodic function D(ν by D(ν = d l eπilν := C (ν (Zg 1,γ (t na, ν ϑ 4 (πν; e πγ dν. (15 C(ν ϑ 4 (πν; e πγ, ν R. (16 It is well known that (g 1,γ,a,1 is a Gabor frame; see for instance [5, Theorem 7.5.3]. Accordingly (see (1, there is an A 1,γ > such that Letting d l g 1,γ (t na l A 1,γ d, d l (Z, t R. (17 m δ := min z R for δ>, we then see from (13 (18 that c l g,γ (t na l m 1/γ E = m 1/γ E 1 ϑ4 (z; e πδ > (18 (Zg 1,γ (t na, νd (ν dν d l g 1,γ (t na l m 1/γ E A 1/γ d. (19 Finally, with D and C related as in (16 we have from Parseval s theorem for Fourier series that d = D(ν 1 dν = ϑ 4 (πν; e πγ C(ν dν m γ C(ν dν = m γ c. (

5 HYPERBOLIC SECANTS YIELD GABOR FRAMES 63 Hence as required. c l g,γ (t na l m γ m 1/γ E A 1,γ c, (1 3. EXPRESSING Zg,γ IN TERMS OF Zg 1,γ In this section we express Zg,γ, with g,γ given in (3, in terms of Zg 1,γ, with g 1,γ given in (5. For this we need some basic facts of the theory of theta functions as they can be found in [1, Chap. 1]. The four theta functions are for q (, 1 given by ϑ 1 (z; q= 1 ( 1 n q (n+1/ e (n+1iz, ( i ϑ (z; q= ϑ 3 (z; q= ϑ 4 (z; q= where z C. One thus easily gets from (4 and (5 that q (n+1/ e (n+1iz, (3 q n e niz, (4 ( 1 n q n e niz, (5 (Zg 1,γ (t, ν = (γ 1/4 e πγt ϑ 3 ( π(ν iγt; e πγ, t,ν R. (6 THEOREM 3.1. We have for t,ν R (Zg,γ (t, ν = 1/ π 1/ γϑ 1 (; e πγ e ϑ 3(π(ν iγt; e πγ πγt ϑ 4 (πν; e πγ ϑ 4 (πt; e π/γ ( γ 3/4 = π 1/ ϑ 1 (; (Zg 1,γ (t, ν e πγ ϑ 4 (πν; e πγ ϑ 4 (πt; e π/γ. (7 Proof. For fixed t R we compute the Fourier coefficients b n (t of the one-periodic function ϑ 3 (π(ν iγt; e πγ ϑ 4 (πν; e πγ = b n (te πinν, ν R, (8 by the method that can be found in [1, Sect..6]. For brevity we shall suppress the expression e πγ in ϑ 3 (π(ν iγt; e πγ, etc., in the remainder of this proof. We thus have / b n (t = 1/ and we consider for n Z c n (t = ϑ 3 (π(ν iγt e πinν dν, n Z, (9 ϑ 4 (πν C ϑ 3 (π(ν iγt e πinν dν, (3 ϑ 4 (πν

6 64 JANSSEN AND STROHMER where C is the edge of the rectangle with corner points 1, 1, 1 + iγ, 1 + iγ, taken with positive orientation. It follows from [1, Sect. 1.1] that the function ϑ 4 (πν has a first-order zero at ν = 1 iγ and no zeros elsewhere on or within C. Therefore, by Cauchy s theorem, c n (t = πi Res ν=(1/iγ [ ϑ3 (π(ν iγt ϑ 4 (πν e πinν ] = iϑ 3(πi( 1 tγeπγn ϑ 4 ( 1 πiγ. (31 By [1, Example, first identity, on p. 464] we have (τ = iγ, q = exp( πγ, z = ϑ 4 ( 1 πiγ = ie (1/4πγ ϑ 1 (. (3 Next, by [1, p. 475, first formula] (z = πi( 1 tγ, τ = 1/τ = i/γ, q = exp(πiτ = exp( π/γ ( ( ( 1 1 ( ( 1 ϑ 3 (πi t γ = γ 1/ exp πγ ϑ t 3 π ; t e π/γ, (33 and by [1, Example, fourth identity, on p. 464] (z = πt, q instead of q ( 1 ϑ 3 (π ; t e π/γ = ϑ 4 ( πt; e π/γ = ϑ 4 (πt; e π/γ, (34 wherewealsohaveusedthatϑ 4 is an even function. It thus follows from (31 (34 that c n (t = ϑ 1 (γ 1/ e (1/4πγ e πγ(1/ t ϑ 4 (πt; e π/γ e πγn. (35 On the other hand, we have by 1-periodicity of the integrand in (3 (in ν that the two integrals along the vertical edges of C cancel one another. Hence c n (t = / 1/ ϑ 3 (π(ν iγt e πinν dν ϑ 4 (πν = b n (t e πγn / 1/ /+iγ 1/+iγ ϑ 3 (π(ν iγt e πinν dν ϑ 4 (πν ϑ 3 (π(ν iγt + πiγ e πinν dν. (36 ϑ 4 (πν + πiγ Furthermore, by the table in [1, Example 3 on p. 465] we have and It thus follows that ϑ 3 ( π(ν iγt + πiγ = e πγ e iπ(ν iγt ϑ 3 ( π(ν iγt, (37 ϑ 4 (πν + πiγ= e πγ e iπν ϑ 4 (πν. (38 c n (t = b n (t + e πγ(t n / 1/ ϑ 3 (π(ν iγt e πinν dν ϑ 4 (πν = b n (t(1 + e πγ(t n. (39

7 HYPERBOLIC SECANTS YIELD GABOR FRAMES 65 By (35 we then find that b n (t = γ 1/ We thus conclude (see (8 that That is, e πγn ϑ 1 ( e (1/4πγ e πγ(1/ t ϑ 4 (πt; e π/γ 1 + e πγ(t n = γ 1/ ϑ 1 ( e (1/4πγ e πγ(1/ t +πγt ϑ 4 (πt; e π/γ cosh πγ(t n = γ 1/ ϑ 1 ( eπγt ϑ 4(πt; e π/γ cosh πγ(t n. (4 ϑ 3 (π(ν iγt ϑ 4 (πν = γ 1/ ϑ 1 ( eπγt ϑ 4 (πt; e π/γ ( πγ 1/ e πinν (Zg,γ (t, ν = cosh πγ(t n e πinν cosh πγ(t n. (41 = 1/ π 1/ γϑ 1 (e πγt ϑ 3 (π(ν iγt ϑ 4 (πνϑ 4 (πt; e π/γ, (4 and this is the first line identity in (7. The second line identity in (7 follows from (6. This completes the proof. Note. Wehavethat ϑ 1 (; e πγ, ϑ 4 (πν; e πγ, ϑ 4 (πt; e π/γ > (43 for ν R, t R. Indeed, we have by [1, Sect. 1.41] that ϑ 1 (; e πγ = ϑ (; e πγ ϑ 3 (; e πγ ϑ 4 (; e πγ. (44 One sees directly from (3 that ϑ (; q >. Also, from [1, p. 476, the formula just before Example 1], one sees that ( ϑ 3 (z; q= ϑ 4 z + 1 π; q >, z R. (45 Remark. Theorem 3.1 implies that (Zg (t, ν/(zg 1 (t, ν can be factored into a function of t and a function of v. This factorization is crucial in the proof of the main result in Section. It seems possible to extend the approach in Section to other pairs of windows g 1, g,where(g 1,a,b is a frame and Zg /Zg 1 (nearly factorizes. We do not pursue this extension in this paper. 4. CANONICAL DUAL WINDOW AND TIGHT WINDOW AT CRITICAL DENSITY Theorem 3.1 allows us to calculate the Zak transform of the canonical dual g d,γ of g,γ for rational values of ab < 1 using Zibulski Zeevi matrices representing the frame

8 66 JANSSEN AND STROHMER operator in the Zak transform domain: see for instance [8, Sect. 1.5], [5, Sect. 8.3], and, of course [11]. The dual window can then be obtained as g d,γ (t = (Zg d,γ (t, ν dν, t R. (46 For a = b = 1 the Zibulski Zeevi matrices are just scalars, viz. (Zg,γ (t, ν and we have formally (Zg,γ d (t, ν = 1 (Zg,γ, t,ν R, (47 (t, ν and g,γ d (t = dν (Zg,γ, t R. (48 (t, ν The identities here hold only formally since 1/Zg,γ L loc (R, and, as said, (g,γ, 1, 1 is not a frame. However, nothing prevents us from computing the right-hand side of (48 for t R not of the form n + 1, n Z.Fortheset we have that (Zg,γ (t, ν is a well-behaved zero-free function of ν R. When we do this calculation for the case that γ = 1, we find, using the same methods as those in the proof of Theorem 3.1, that for t ( 1, 1, n Z, g d,1 (t + n = 1/ ( 1 n ϑ 4 (πteπt +πnt π 1/ (ϑ 1 ( cosh π(t + n. (49 Here all theta functions are with parameter q = e π. The same procedure for the Gaussian g 1,1 yields for t R not of the form n + 1/, n Z, g1,1 d 1/4 (t = ϑ 1 (eπt n 1/ t ( 1 n e π(n 1/ ; (5 see for instance [1, 6, Sect..14, and 7, Sect. 4.4]. Although the two functions in (49 (5 seem quite different, one can show that both functions are bounded but not in L p (R when 1 p<. Moreover, there is for both functions exponential decay as t away from the set of half-integers. In a similar fashion one can consider the canonical tight windows g t as sociated with g = g 1,γ,g,γ at critical density a = b = 1. We have that these windows are given in the Zak transform domain through the formula Zg t = Zg/ Zg, whence g t (Zg(t, ν (t = dν, t R. (51 (Zg(t, ν In this case the definitions are more than just formal since Zg/ Zg L (R, while one can show (see [Janssen, 1, submitted for publication] for details that in L sense g t = lim a 1 S 1/ a g, (5 where for g = g 1,γ,g,γ we have denoted the frame operator corresponding to the Gabor frame (g,a,a by S a. Now as a surprising consequence of Theorem 3.1 we have that

9 HYPERBOLIC SECANTS YIELD GABOR FRAMES 67 g1,γ t = gt,γ since Zg 1,γ /Zg,γ is positive everywhere. In [Janssen, 1, submitted for publication] it has been observed that the mapping g g t in (51 diminishes distances between even, positive, and unimodal windows g enormously when scaling parameters are set correctly. The fact that g1,γ t = gt,γ is an extremely accurate demonstration of this phenomenon. The systems (g t, 1, 1 form orthonormal bases for L (R; see, for instance, [5, Corollary 7.5.]. Interestingly, in the case that g γ = g 1,γ or g,γ, we have that, in L sense, lim γ gt γ = sinc π, lim γ gt γ = χ ( 1/,1/. (53 Hencewehaveafamilygγ t, γ>ofreasonably behaved tight frame generating windows at critical density a = b = 1 that interpolates between the Haar window χ ( 1/,1/ and its Fourier transform sinc π as γ varies between and. See [Janssen, 1, submitted for publication] for details. REFERENCES 1. M. J. Bastiaans, Gabor s expansion of a signal into Gaussian elementary signals, IEEE Proc. 68 (198, I. Daubechies, The wavelet transform, time frequency localization and signal analysis, IEEE Trans. Inform. Theory 35, No. 5 (199, I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Reg. Conf. Series in Applied Math., SIAM, Philadelphia, H. G. Feichtinger and T. Strohmer, Eds., Gabor Analysis and Algorithms: Theory and Applications, Birkhäuser, Boston, K. Gröchenig, Foundations of Time Frequency Analysis, Birkhäuser, Boston, A. J. E. M. Janssen, Weighted Wigner distributions vanishing on lattices, J. Math. Anal. Appl. 8, No. 1 (1981, A. J. E. M. Janssen, Bargmann transform, Zak transform, and coherent states, J. Math. Phys. 3, No. 5 (198, A. J. E. M. Janssen, The duality condition for Weyl Heisenberg frames, in Gabor Analysis and Algorithms: Theory and Applications (H. G. Feichtinger and T. Strohmer, Eds., Chap. 1, pp , Birkhäuser, Boston, A. Ron and Z. Shen, Weyl Heisenberg frames and Riesz bases in L (R d, Duke Math. J. 89, No. (1997, E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed., Cambridge Univ. Press, New York, M. Zibulski and Y. Y. Zeevi, Oversampling in the Gabor scheme, IEEE Trans. SP 41, No. 8 (1993,