Zak Transform

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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
    2015
    Co-Authors: Mary Jane Bastiaans, Faculteit Elektrotechniek
    Abstract:

    Gabor’s expansion of a discrete-time 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 anal-ysis 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 finite-support, leads to the con-cept 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 trans-form as matrix-vector products. Using these matrix-vector products, a relationship between the analysis win-dow and the synthesis window is derived. It is shown how this relationship enables us to determine the opti-mum 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, 1999
    Co-Authors: A.j. Van Leest, Mary Jane Bastiaans
    Abstract:

    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 non-separable) lattice.

  • gabor s signal expansion on a quincunx lattice and the modified Zak Transform
    International Conference on Acoustics Speech and Signal Processing, 1999
    Co-Authors: A.j. Van Leest, Mary Jane Bastiaans
    Abstract:

    Gabor's expansion of a signal on a quincunx lattice with oversampling by a rational factor is presented for continuous-time 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), 1999
    Co-Authors: A.j. Van Leest, Mary Jane Bastiaans
    Abstract:

    Gabor's expansion of a signal on a quincunx lattice with oversampling by a rational factor is presented for continuous-time 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 continuous-time and discrete-time signals
    Signal and Image Representation in Combined Spaces, 1998
    Co-Authors: Mary Jane Bastiaans
    Abstract:

    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 sliding-window spectrum is shown. It is shown how Gabor's expansion coefficients can be found as samples of the sliding-window 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 bi-orthonormal 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 continuous-time as well as the discrete-time case are con­sidered, and, by sampling the continuous frequency variable that still occurs in the discrete-time 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 well-known fast Fourier Transform algorithm. Not only Gabor's critical sampling is considered, but also, for continuous-time 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, one-dimensional signal by coherent-optical means.

Min-hung 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, 1996
    Co-Authors: Soo-chang Pei, Min-hung Yeh
    Abstract:

    Abstract The relationship between finite discrete Zak Transform and finite Gabor expansion are well discussed in this paper. In this paper, we present two DFT-based algorithms for computing Gabor coefficients. One is based upon the time-split Zak Transform, the other is based upon the frequency-split Zak Transform. These two methods are time and frequency dual pairs. With the help of Zak Transform, the closed-form 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 2-D case and compute 2-D Gabor expansion coefficients through 2-D discrete Zak Transform and 4-D DFT. Four methods can be applied in the 2-D case. They are time-time-split, time-frequency-split, frequency-time-split and frequency-frequency-split.

  • time and frequency split Zak Transform for finite gabor expansion
    International Symposium on Circuits and Systems, 1995
    Co-Authors: Soo-chang Pei, Min-hung Yeh
    Abstract:

    The relationship between the finite discrete Zak Transform and the finite Gabor expansion are discussed in this paper. We present two DFT-based algorithms for computing Gabor coefficients. One is based upon the time-split Zak Transform, the other upon the frequency-split 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 2-D case and compute 2-D Gabor expansion coefficients through the 2-D discrete Zak Transform and 4-D DFT. Four methods can be applied in the 2-D case. They are time-time-split, time-frequency-split, frequency-time-split and frequency-frequency-split.

  • ISCAS - Time and frequency split Zak Transform for finite Gabor expansion
    Proceedings of ISCAS'95 - International Symposium on Circuits and Systems, 1
    Co-Authors: Soo-chang Pei, Min-hung Yeh
    Abstract:

    The relationship between the finite discrete Zak Transform and the finite Gabor expansion are discussed in this paper. We present two DFT-based algorithms for computing Gabor coefficients. One is based upon the time-split Zak Transform, the other upon the frequency-split 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 2-D case and compute 2-D Gabor expansion coefficients through the 2-D discrete Zak Transform and 4-D DFT. Four methods can be applied in the 2-D case. They are time-time-split, time-frequency-split, frequency-time-split and frequency-frequency-split.

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, 2010
    Co-Authors: Andrzej K. Brodzik
    Abstract:

    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 time-domain 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, 2002
    Co-Authors: Andrzej K. Brodzik
    Abstract:

    A new formulation of the Gerchberg-Papoulis (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 complex-valued 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 four-fold reduction in the computational complexity of the complex Zak space approach.

  • Gerchberg-Papoulis algorithm and the finite Zak Transform
    Wavelet Applications in Signal and Image Processing VIII, 2000
    Co-Authors: Andrzej K. Brodzik, Richard Tolimieri
    Abstract:

    We propose a new, time-frequency formulation of the Gerchberg-Papoulis 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 time-frequency 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 Gerchberg-Papoulis procedure and in an increased flexibility of the algorithm.

  • Extrapolation of band-limited signals and the finite Zak Transform
    Signal Processing, 2000
    Co-Authors: Andrzej K. Brodzik, Richard Tolimieri
    Abstract:

    Abstract We propose a new formulation of the Gerchberg–Papoulis algorithm for extrapolation of band-limited 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 narrow-band and certain wide-band 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.

  • Weyl-Heisenberg Systems and the Finite Zak Transform
    Signal and Image Representation in Combined Spaces, 1998
    Co-Authors: Andrzej K. Brodzik, Izidor Gertner, Richard Tolimieri
    Abstract:

    Abstract. Previously, a theoretical foundation for designing algorithms for computing Weyl-Heisenberg (W-H) coefficients at critical sampling was established by applying the finite Zak Transform. This theory established clear and easily computable conditions for the existence of W-H expansion and for stability of computations. The main computational task in the resulting algorithm was a 2-D finite Fourier Transform. In this work we extend the applicability of the approach to rationally over-sampled W-H systems by developing a deeper understanding of the relationship established by the finite Zak trans-form between linear algebra properties of W-H systems and function theory in Zak space. This relationship will impact on questions of existence, parameterization, and computation of W-H 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, parallel-processing environment. Timing results show that real-time computation of W-H expansions is realizable.

A.j. Van Leest - One of the best experts on this subject based on the ideXlab platform.

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 Society-second Series, 2018
    Co-Authors: Dominik Jüstel
    Abstract:

    The Zak Transform on $\mathbb{R}^d$ is an important tool in condensed matter physics, signal processing, time-frequency 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, 2017
    Co-Authors: Dominik Jüstel
    Abstract:

    The Zak Transform on $\mathbb{R}^d$ is an important tool in condensed matter physics, signal processing, time-frequency 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.