## Zak Transforms

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### A.j.e.m. Janssen - One of the best experts on this subject based on the ideXlab platform.

• ##### Zak Transforms with Few Zeros and the Tie
Co-Authors: A.j.e.m. Janssen
Abstract:

We consider the difficult problem of deciding whether a triple(g,a,b),with window g ∈ L 2(ℝ) and time shift parameter a and frequency shift parameterbis a Gabor frame from two different points of view. We first identify two classes of non-negative windows g ∈ L 2(ℝ) such that their Zak Transforms have no and just one zero per unit square, respectively. The first class consists of all integrable, non-negative windowsg that are supported by and strictly decreasing on [0, ∞). The second class consists of all even, non-negative, continuous, integrable windowsg that satisfy on [0, ∞) a condition slightly stronger than strict convexity (superconvexity). Accordingly, the members of these two classes generate Gabor frames for integer oversampling factor (ab)-1 ≥ 1 and ≥ 2, respectively. When we weaken the condition of superconvexity into strict convexity, the Zak TransformsZg may have as many zeros as one wants, but in all cases (g, a, b) is still a Gabor frame when(ab) -1 is an integer ≥ 2. As a second issue we consider the question for which a, b > 0 the triple (g, a, b) is a Gabor frame, where gis the characteristic function of an interval [0, ∞) with c0 > 0 fixed. It turns out that the answer to the latter question is quite complicated, where irrationality or rationality of abgives rise to quite different situations. A pictorial display, in which the various cases are indicated in the positive (a, b)-quadrant, shows a remarkable resemblance to the design of a low-budget tie.

• ##### On rationally oversampled Weyl-Heisenberg frames
Signal Processing, 1995
Co-Authors: A.j.e.m. Janssen
Abstract:

Abstract We relate the matrix elements of the linear systems, arising in the Zibulski-Zeevi method for computing dual functions for rationally oversampled Weyl-Heisenberg frames, to the Wexler-Raz method for computing dual functions. We give a necessary and sufficient condition for two functions g , γ having a frame upper bound to be dual in terms of their Zak Transforms, we characterize the minimal dual function °γ and we present a necessary and sufficient condition, in terms of the Zak transform, for a function g so that the Tolimieri-Orr condition A is satisfied. The latter result is used to show that a g generating a rationally oversampled Weyl-Heisenberg frame and satisfying condition A has a minimal dual function that satisfies condition A as well.

• ##### Zak Transforms With Few Zeros and the Tie
2023
Co-Authors: A.j.e.m. Janssen
Abstract:

We consider the dicult problem of deciding whether a triple (g; a; b), with window g 2 L 2 (R) and time shift parameter a and frequency shift parameter b, is a Gabor frame from two dierent points of view. We rst identify two classes of non-negative windows g 2 L 2 (R) such that their Zak Transforms have no and just one zero per unit square, respectively. The rst class consists of all integrable, non-negative windows g that are supported by [0; 1)

### A J Brodzik - One of the best experts on this subject based on the ideXlab platform.

• ##### recursive Zak Transforms and weyl heisenberg expansions
International Conference on Acoustics Speech and Signal Processing, 2001
Co-Authors: A J Brodzik
Abstract:

We develop algorithms for computing block-recursive Zak Transforms and Weyl-Heisenberg expansions, which achieve p/logL and (logM+p)/(logN+logL+1) multiplicative complexity reduction, respectively, over direct computations, where p'=pM, and N-p' is the number of overlapping samples in subsequent signal segments. For each transform we offer a choice of two algorithms based on two different implementations of the Zak transform of the time-evolving signal. These two algorithm classes exhibit typical trade-offs between computational complexity and memory requirements.

### Franz Hlawatsch - One of the best experts on this subject based on the ideXlab platform.

• ##### Discrete Zak Transforms, polyphase Transforms, and applications
IEEE Transactions on Signal Processing, 1997
Co-Authors: Helmut Bölcskei, Franz Hlawatsch
Abstract:

We consider three different versions of the Zak (1967) transform (ZT) for discrete-time signals, namely, the discrete-time ZT, the polyphase transform, and a cyclic discrete ZT. In particular, we show that the extension of the discrete-time ZT to the complex z-plane results in the polyphase transform, an important and well-known concept in multirate signal processing and filter bank theory. We discuss fundamental properties, relations, and transform pairs of the three discrete ZT versions, and we summarize applications of these Transforms. In particular, the discrete-time ZT and the cyclic discrete ZT are important for discrete-time Gabor (1946) expansion (Weyl-Heisenberg frame) theory since they diagonalize the Weyl-Heisenberg frame operator for critical sampling and integer oversampling. The polyphase representation plays a fundamental role in the theory of filter banks, especially DFT filter banks. Simulation results are presented to demonstrate the application of the discrete ZT to the efficient calculation of dual Gabor windows, tight Gabor windows, and frame bounds.

• ##### Discrete Zak Transforms, polyphase Transforms, and applications
1997
Co-Authors: Helmut Bölcskei, Student Member, Franz Hlawatsch
Abstract:

Abstract — We consider three different versions of the Zak transform (ZT) for discrete-time signals, namely, the discretetime ZT, the polyphase transform, and a cyclic discrete ZT. In particular, we show that the extension of the discrete-time ZT to the complex �-plane results in the polyphase transform, an important and well-known concept in multirate signal processing and filter bank theory. We discuss fundamental properties, relations, and transform pairs of the three discrete ZT versions, and we summarize applications of these Transforms. In particular, the discrete-time ZT and the cyclic discrete ZT are important for discrete-time Gabor expansion (Weyl–Heisenberg frame) theory since they diagonalize the Weyl–Heisenberg frame operator for critical sampling and integer oversampling. The polyphase representation plays a fundamental role in the theory of filter banks, especiall

### Kloos Tobias - One of the best experts on this subject based on the ideXlab platform.

• ##### Zeros of the Zak transform of totally positive functions
2014
Co-Authors: Kloos Tobias
Abstract:

We study the Zak transform of totally positive (TP) functions. We use the convergence of the Zak transform of TP functions of finite type to prove that the Zak Transforms of all TP functions without Gaussian factor in the Fourier transform have only one zero in their fundamental domain of quasi-periodicity. Our proof is based on complex analysis, especially the Theorem of Hurwitz and some real analytic arguments, where we use the connection of TP functions of finite type and exponential B-splines.Comment: The results were presented on the "International Conference on Modern Time-Frequency Analysis" in Strobl, Austria, June 2 - 6th, 201

• ##### Zak Transforms and Gabor frames of totally positive functions and exponential B-splines
'Elsevier BV', 2013
Co-Authors: Kloos Tobias, Stöckler Joachim
Abstract:

We study totally positive (TP) functions of finite type and exponential B-splines as window functions for Gabor frames. We establish the connection of the Zak transform of these two classes of functions and prove that the Zak Transforms have only one zero in their fundamental domain of quasi-periodicity. Our proof is based on the variation-diminishing property of shifts of exponential B-splines. For the exponential B-spline B_m of order m, we determine a large set of lattice parameters a,b>0 such that the Gabor family of time-frequency shifts is a frame for L^2(R). By the connection of its Zak transform to the Zak transform of TP functions of finite type, our result provides an alternative proof that TP functions of finite type provide Gabor frames for all lattice parameters with ab

### Tobias Kloos - One of the best experts on this subject based on the ideXlab platform.

• ##### Zeros of the Zak Transform of Totally Positive Functions
Journal of Fourier Analysis and Applications, 2015
Co-Authors: Tobias Kloos
Abstract:

We study the Zak transform of totally positive (TP) functions. We use the convergence of the Zak transform of TP functions of finite type to prove that the Zak Transforms of all TP functions without Gaussian factor in the Fourier transform have only one zero in their fundamental domain of quasi-periodicity. Our proof is based on complex analysis, especially the Theorem of Hurwitz and some real analytic arguments, where we use the connection of TP functions of finite type and exponential B-splines.

• ##### full length article Zak Transforms and gabor frames of totally positive functions and exponential b splines
Journal of Approximation Theory, 2014
Co-Authors: Tobias Kloos, Joachim Stockler
Abstract:

We study totally positive (TP) functions of finite type and exponential B-splines as window functions for Gabor frames. We establish the connection of the Zak transform of these two classes of functions and prove that the Zak Transforms have only one zero in their fundamental domain of quasi-periodicity. Our proof is based on the variation-diminishing property of shifts of exponential B-splines. For the exponential B-spline B"m of order m, we determine a set of lattice parameters @a,@b>0 such that the Gabor family G(B"m,@a,@b) of time-frequency shifts e^2^@p^i^l^@bB"m(@?-k@a), k,l@?Z, is a frame for L^2(R). By the connection of its Zak transform to the Zak transform of TP functions of finite type, our result provides an alternative proof that TP functions of finite type provide Gabor frames for all lattice parameters with @a@b<1. For even two-sided exponentials g(x)=@l2e^-^@l^|^x^| we find lower frame-bounds A, which show the asymptotically linear decay A~(1-@a@b) as the density @a@b of the time-frequency lattice tends to the critical density @a@b=1.