The Experts below are selected from a list of 93 Experts worldwide ranked by ideXlab platform
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
Advances in Gabor Analysis, 2003CoAuthors: A.j.e.m. JanssenAbstract: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 nonnegative 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, nonnegative windowsg that are supported by and strictly decreasing on [0, ∞). The second class consists of all even, nonnegative, 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 lowbudget tie.

On rationally oversampled WeylHeisenberg frames
Signal Processing, 1995CoAuthors: A.j.e.m. JanssenAbstract:Abstract We relate the matrix elements of the linear systems, arising in the ZibulskiZeevi method for computing dual functions for rationally oversampled WeylHeisenberg frames, to the WexlerRaz 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 TolimieriOrr condition A is satisfied. The latter result is used to show that a g generating a rationally oversampled WeylHeisenberg frame and satisfying condition A has a minimal dual function that satisfies condition A as well.

Zak Transforms With Few Zeros and the Tie
2023CoAuthors: A.j.e.m. JanssenAbstract: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 nonnegative 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, nonnegative 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, 2001CoAuthors: A J BrodzikAbstract:We develop algorithms for computing blockrecursive Zak Transforms and WeylHeisenberg expansions, which achieve p/logL and (logM+p)/(logN+logL+1) multiplicative complexity reduction, respectively, over direct computations, where p'=pM, and Np' 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 timeevolving signal. These two algorithm classes exhibit typical tradeoffs 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, 1997CoAuthors: Helmut Bölcskei, Franz HlawatschAbstract:We consider three different versions of the Zak (1967) transform (ZT) for discretetime signals, namely, the discretetime ZT, the polyphase transform, and a cyclic discrete ZT. In particular, we show that the extension of the discretetime ZT to the complex zplane results in the polyphase transform, an important and wellknown 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 discretetime ZT and the cyclic discrete ZT are important for discretetime Gabor (1946) expansion (WeylHeisenberg frame) theory since they diagonalize the WeylHeisenberg 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
1997CoAuthors: Helmut Bölcskei, Student Member, Franz HlawatschAbstract:Abstract — We consider three different versions of the Zak transform (ZT) for discretetime signals, namely, the discretetime ZT, the polyphase transform, and a cyclic discrete ZT. In particular, we show that the extension of the discretetime ZT to the complex �plane results in the polyphase transform, an important and wellknown 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 discretetime ZT and the cyclic discrete ZT are important for discretetime 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
2014CoAuthors: Kloos TobiasAbstract: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 quasiperiodicity. 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 Bsplines.Comment: The results were presented on the "International Conference on Modern TimeFrequency Analysis" in Strobl, Austria, June 2  6th, 201

Zak Transforms and Gabor frames of totally positive functions and exponential Bsplines
'Elsevier BV', 2013CoAuthors: Kloos Tobias, Stöckler JoachimAbstract:We study totally positive (TP) functions of finite type and exponential Bsplines 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 quasiperiodicity. Our proof is based on the variationdiminishing property of shifts of exponential Bsplines. For the exponential Bspline B_m of order m, we determine a large set of lattice parameters a,b>0 such that the Gabor family of timefrequency 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, 2015CoAuthors: Tobias KloosAbstract: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 quasiperiodicity. 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 Bsplines.

full length article Zak Transforms and gabor frames of totally positive functions and exponential b splines
Journal of Approximation Theory, 2014CoAuthors: Tobias Kloos, Joachim StocklerAbstract:We study totally positive (TP) functions of finite type and exponential Bsplines 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 quasiperiodicity. Our proof is based on the variationdiminishing property of shifts of exponential Bsplines. For the exponential Bspline 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 timefrequency 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 twosided exponentials g(x)=@l2e^^@l^^x^ we find lower framebounds A, which show the asymptotically linear decay A~(1@a@b) as the density @a@b of the timefrequency lattice tends to the critical density @a@b=1.