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Mary Jane Bastiaans  One of the best experts on this subject based on the ideXlab platform.

On the discrete Gabor Transform and the discrete Zak Transform,” accepted for publication in Signal Process
2015CoAuthors: Mary Jane Bastiaans, Faculteit ElektrotechniekAbstract:Gabor’s expansion of a discretetime signal into a set of shifted and modulated versions of an elementary signal or synthesis window is introduced, along with the inverse operation, i.e. the Gabor Transform, which uses an analysis window that is related to the synthesis window and with the help of which Gabor’s expansion coefficients can be determined. The restriction to a signal and an analysis window that both have finitesupport, leads to the concept of a discrete Gabor expansion and a discrete Gabor Transform. After introduction of the discrete Fourier Transform and the discrete Zak Transform, it is possible to express the discrete Gabor expansion and the discrete Gabor transform as matrixvector products. Using these matrixvector products, a relationship between the analysis window and the synthesis window is derived. It is shown how this relationship enables us to determine the optimum synthesis window in the sense that it has minimum L2 norm, and it is shown that this optimum synthesis window resembles best the analysis window

on the non separable discrete gabor signal expansion and the Zak Transform
Information Sciences Signal Processing and their Applications, 1999CoAuthors: A.j. Van Leest, Mary Jane BastiaansAbstract:The discrete Gabor signal expansion on a lattice that is obtained by linear combinations of two independent vectors, and its relation with the discrete Zak Transform are presented. It is shown how the Zak Transform can be helpful in determining Gabor's signal expansion coefficients and how it can be used in finding the dual window that corresponds to a given window for this (generally nonseparable) lattice.

gabor s signal expansion on a quincunx lattice and the modified Zak Transform
International Conference on Acoustics Speech and Signal Processing, 1999CoAuthors: A.j. Van Leest, Mary Jane BastiaansAbstract:Gabor's expansion of a signal on a quincunx lattice with oversampling by a rational factor is presented for continuoustime signals. It is shown how a modified Zak Transform instead of the ordinary Zak Transform can be helpful in determining Gabor's signal expansion coefficients and how it can be used in finding the dual window. Furthermore, some examples of dual windows for the quincunx case are given and compared with dual windows for the rectangular case.

ICASSP  Gabor's signal expansion on a quincunx lattice and the modified Zak Transform
1999 IEEE International Conference on Acoustics Speech and Signal Processing. Proceedings. ICASSP99 (Cat. No.99CH36258), 1999CoAuthors: A.j. Van Leest, Mary Jane BastiaansAbstract:Gabor's expansion of a signal on a quincunx lattice with oversampling by a rational factor is presented for continuoustime signals. It is shown how a modified Zak Transform instead of the ordinary Zak Transform can be helpful in determining Gabor's signal expansion coefficients and how it can be used in finding the dual window. Furthermore, some examples of dual windows for the quincunx case are given and compared with dual windows for the rectangular case.

Gabor's expansion and the Zak Transform for continuoustime and discretetime signals
Signal and Image Representation in Combined Spaces, 1998CoAuthors: Mary Jane BastiaansAbstract:Abstract. Gabor's expansion of a signal into a discrete set of shifted and modulated versions of an elementary signal is introduced and its relation to sampling of the slidingwindow spectrum is shown. It is shown how Gabor's expansion coefficients can be found as samples of the slidingwindow spectrum, where, at least in the case of critical sampling, the window function is related to the elementary signal in such a way that the set of shifted and modulated elementary signals is biorthonormal to the corresponding set of window functions. The Zak Transform is introduced and its intimate relationship to Gabor's signal expansion is demonstrated. It is shown how the Zak Transform can be helpful in determining the window function that corresponds to a given elementary signal and how it can be used to find Gabor's expansion coefficients. The continuoustime as well as the discretetime case are considered, and, by sampling the continuous frequency variable that still occurs in the discretetime case, the discrete Zak Transform and the discrete Gabor Transform are introduced. It is shown how the discrete Transforms enable us to determine Gabor's expansion coefficients via a fast computer algorithm, analogous to the wellknown fast Fourier Transform algorithm. Not only Gabor's critical sampling is considered, but also, for continuoustime signals, the case of oversampling by an integer factor. It is shown again how, in this case, the Zak Transform can be helpful in determining a (no longer unique) window function corresponding to a given elementary signal. An arrangement is described which is able to generate Gabor's expansion coefficients of a rastered, onedimensional signal by coherentoptical means.
Minhung Yeh  One of the best experts on this subject based on the ideXlab platform.

Time and frequency split Zak Transform for finite Gabor expansion
Signal Processing, 1996CoAuthors: Soochang Pei, Minhung YehAbstract:Abstract The relationship between finite discrete Zak Transform and finite Gabor expansion are well discussed in this paper. In this paper, we present two DFTbased algorithms for computing Gabor coefficients. One is based upon the timesplit Zak Transform, the other is based upon the frequencysplit Zak Transform. These two methods are time and frequency dual pairs. With the help of Zak Transform, the closedform solutions for analysis basis can also be derived while the oversampling ratio is an integer. Moreover, we extend the relationship between finite discrete Zak Transform and Gabor expansion to the 2D case and compute 2D Gabor expansion coefficients through 2D discrete Zak Transform and 4D DFT. Four methods can be applied in the 2D case. They are timetimesplit, timefrequencysplit, frequencytimesplit and frequencyfrequencysplit.

time and frequency split Zak Transform for finite gabor expansion
International Symposium on Circuits and Systems, 1995CoAuthors: Soochang Pei, Minhung YehAbstract:The relationship between the finite discrete Zak Transform and the finite Gabor expansion are discussed in this paper. We present two DFTbased algorithms for computing Gabor coefficients. One is based upon the timesplit Zak Transform, the other upon the frequencysplit Zak Transform. These two methods are time and frequency dual pairs. Furthermore, we extend the relationship between the finite discrete Zak Transform and the Gabor expansion to the 2D case and compute 2D Gabor expansion coefficients through the 2D discrete Zak Transform and 4D DFT. Four methods can be applied in the 2D case. They are timetimesplit, timefrequencysplit, frequencytimesplit and frequencyfrequencysplit.

ISCAS  Time and frequency split Zak Transform for finite Gabor expansion
Proceedings of ISCAS'95  International Symposium on Circuits and Systems, 1CoAuthors: Soochang Pei, Minhung YehAbstract:The relationship between the finite discrete Zak Transform and the finite Gabor expansion are discussed in this paper. We present two DFTbased algorithms for computing Gabor coefficients. One is based upon the timesplit Zak Transform, the other upon the frequencysplit Zak Transform. These two methods are time and frequency dual pairs. Furthermore, we extend the relationship between the finite discrete Zak Transform and the Gabor expansion to the 2D case and compute 2D Gabor expansion coefficients through the 2D discrete Zak Transform and 4D DFT. Four methods can be applied in the 2D case. They are timetimesplit, timefrequencysplit, frequencytimesplit and frequencyfrequencysplit.
Andrzej K. Brodzik  One of the best experts on this subject based on the ideXlab platform.

Construction of sparse representations of perfect polyphase sequences in Zak space with applications to radar and communications
EURASIP Journal on Advances in Signal Processing, 2010CoAuthors: Andrzej K. BrodzikAbstract:Sparse representations of sequences facilitate signal processing tasks in many radar, sonar, communications, and information hiding applications. Previously, conditions for the construction of a compactly supported finite Zak Transform of the linear FM chirp were investigated. It was shown that the discrete Fourier Transform of a chirp is, essentially, a chirp, with support similar to the support of the timedomain signal. In contrast, the Zak space analysis produces a highly compactified chirp, with support restricted to an algebraic line. Further investigation leads to relaxation of the original restriction to chirps, permitting construction of a wide range of polyphase sequence families with ideal correlation properties. This paper contains an elementary introduction to the Zak Transform methods, a survey of recent results in Zak space sequence design and analysis, and a discussion of the main open problems in this area.

Signal extrapolation in the real Zak space
IEEE Transactions on Signal Processing, 2002CoAuthors: Andrzej K. BrodzikAbstract:A new formulation of the GerchbergPapoulis (1974, 1975) algorithm for extrapolation of bandlimited signals was introduced. The new formulation was obtained by translating the fundamental operations of the GP procedure, the truncation, and the Fourier Transform into the language of the finite Zak (1967) Transform. However, the Zak Transform formulation of the GP algorithm assumes complexvalued signals, whereas the GP procedure is usually applied to real signals. We present a new and more efficient algorithm that acts directly on a real signal via the real Zak Transform (RZT) relation between a signal and its Hartley Transform, leading, in effect, to approximately a fourfold reduction in the computational complexity of the complex Zak space approach.

GerchbergPapoulis algorithm and the finite Zak Transform
Wavelet Applications in Signal and Image Processing VIII, 2000CoAuthors: Andrzej K. Brodzik, Richard TolimieriAbstract:We propose a new, timefrequency formulation of the GerchbergPapoulis algorithm for extrapolation of band limited signals. The new formulation is obtained by translating the constituent operations of the Gerchberg Papoulis procedure, the truncation and the Fourier Transform, into the language of the finite Zak Transform, a timefrequency tool intimately related to the Fourier Transform. We will show that the use of the Zak Transform results in a significant reduction of the computational complexity of the GerchbergPapoulis procedure and in an increased flexibility of the algorithm.

Extrapolation of bandlimited signals and the finite Zak Transform
Signal Processing, 2000CoAuthors: Andrzej K. Brodzik, Richard TolimieriAbstract:Abstract We propose a new formulation of the Gerchberg–Papoulis algorithm for extrapolation of bandlimited signals. The new formulation is obtained by translating the fundamental operations of the Gerchberg–Papoulis procedure, the truncation and the Fourier Transform, into the language of the finite Zak Transform. The use of the Zak Transform results in a significant reduction in the number of multiplications required by the Gerchberg–Papoulis algorithm in the case of narrowband and certain wideband signals, and in an increased flexibility of the algorithm. Tensor product formalism will be used throughout this work as this formalism provides powerful tools for analyzing and coding constructions and algorithms.

WeylHeisenberg Systems and the Finite Zak Transform
Signal and Image Representation in Combined Spaces, 1998CoAuthors: Andrzej K. Brodzik, Izidor Gertner, Richard TolimieriAbstract:Abstract. Previously, a theoretical foundation for designing algorithms for computing WeylHeisenberg (WH) coefficients at critical sampling was established by applying the finite Zak Transform. This theory established clear and easily computable conditions for the existence of WH expansion and for stability of computations. The main computational task in the resulting algorithm was a 2D finite Fourier Transform. In this work we extend the applicability of the approach to rationally oversampled WH systems by developing a deeper understanding of the relationship established by the finite Zak transform between linear algebra properties of WH systems and function theory in Zak space. This relationship will impact on questions of existence, parameterization, and computation of WH expansions. Implementation results on single RISC processor of i860 and the PARAGON parallel multiprocessor system are given. The algorithms described in this paper possess highly parallel structure and are especially suited in a distributed memory, parallelprocessing environment. Timing results show that realtime computation of WH expansions is realizable.
A.j. Van Leest  One of the best experts on this subject based on the ideXlab platform.

on the non separable discrete gabor signal expansion and the Zak Transform
Information Sciences Signal Processing and their Applications, 1999CoAuthors: A.j. Van Leest, Mary Jane BastiaansAbstract:The discrete Gabor signal expansion on a lattice that is obtained by linear combinations of two independent vectors, and its relation with the discrete Zak Transform are presented. It is shown how the Zak Transform can be helpful in determining Gabor's signal expansion coefficients and how it can be used in finding the dual window that corresponds to a given window for this (generally nonseparable) lattice.

gabor s signal expansion on a quincunx lattice and the modified Zak Transform
International Conference on Acoustics Speech and Signal Processing, 1999CoAuthors: A.j. Van Leest, Mary Jane BastiaansAbstract:Gabor's expansion of a signal on a quincunx lattice with oversampling by a rational factor is presented for continuoustime signals. It is shown how a modified Zak Transform instead of the ordinary Zak Transform can be helpful in determining Gabor's signal expansion coefficients and how it can be used in finding the dual window. Furthermore, some examples of dual windows for the quincunx case are given and compared with dual windows for the rectangular case.

ICASSP  Gabor's signal expansion on a quincunx lattice and the modified Zak Transform
1999 IEEE International Conference on Acoustics Speech and Signal Processing. Proceedings. ICASSP99 (Cat. No.99CH36258), 1999CoAuthors: A.j. Van Leest, Mary Jane BastiaansAbstract:Gabor's expansion of a signal on a quincunx lattice with oversampling by a rational factor is presented for continuoustime signals. It is shown how a modified Zak Transform instead of the ordinary Zak Transform can be helpful in determining Gabor's signal expansion coefficients and how it can be used in finding the dual window. Furthermore, some examples of dual windows for the quincunx case are given and compared with dual windows for the rectangular case.

ISSPA  On the nonseparable discrete Gabor signal expansion and the Zak Transform
ISSPA '99. Proceedings of the Fifth International Symposium on Signal Processing and its Applications (IEEE Cat. No.99EX359), 1CoAuthors: A.j. Van Leest, Mary Jane BastiaansAbstract:The discrete Gabor signal expansion on a lattice that is obtained by linear combinations of two independent vectors, and its relation with the discrete Zak Transform are presented. It is shown how the Zak Transform can be helpful in determining Gabor's signal expansion coefficients and how it can be used in finding the dual window that corresponds to a given window for this (generally nonseparable) lattice.

The discrete Gabor Transform and the discrete Zak Transform on a quincunx lattice
Proceedings of the IEEESP International Symposium on TimeFrequency and TimeScale Analysis (Cat. No.98TH8380), 1CoAuthors: A.j. Van LeestAbstract:The discrete Gabor expansion on a quincunx lattice and its relation with the discrete Zak Transform is presented. It is shown how the Zak Transform can be helpful in determining Gabor's signal expansion coefficients and how it can be used in finding the dual window functions that correspond to a given elementary signal for this quincunx lattice. Furthermore, some examples are given and compared with Gabor's signal expansion on a rectangular lattice.
Dominik Jüstel  One of the best experts on this subject based on the ideXlab platform.

the Zak Transform on strongly proper g spaces and its applications
Journal of The London Mathematical Societysecond Series, 2018CoAuthors: Dominik JüstelAbstract:The Zak Transform on $\mathbb{R}^d$ is an important tool in condensed matter physics, signal processing, timefrequency analysis, and harmonic analysis in general. This article introduces a generalization of the Zak Transform to a class of locally compact $G$spaces, where $G$ is either a locally compact abelian or a second countable unimodular type I group. This framework unifies previously proposed generalizations of the Zak Transform. It is shown that the Zak Transform has invariance properties analog to the classic case and is a Hilbert space isomorphism between the space of $L^2$functions and a direct integral of Hilbert spaces that is explicitly determined via a Weil formula for $G$spaces and a Poisson summation formula for compact subgroups. Some applications in physics are outlined.

The Zak Transform on strongly proper G‐spaces and its applications
Journal of the London Mathematical Society, 2017CoAuthors: Dominik JüstelAbstract:The Zak Transform on $\mathbb{R}^d$ is an important tool in condensed matter physics, signal processing, timefrequency analysis, and harmonic analysis in general. This article introduces a generalization of the Zak Transform to a class of locally compact $G$spaces, where $G$ is either a locally compact abelian or a second countable unimodular type I group. This framework unifies previously proposed generalizations of the Zak Transform. It is shown that the Zak Transform has invariance properties analog to the classic case and is a Hilbert space isomorphism between the space of $L^2$functions and a direct integral of Hilbert spaces that is explicitly determined via a Weil formula for $G$spaces and a Poisson summation formula for compact subgroups. Some applications in physics are outlined.