Z-Transform

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Richard Frayne - One of the best experts on this subject based on the ideXlab platform.

  • a general description of linear time frequency transforms and formulation of a fast invertible transform that samples the continuous s transform spectrum nonredundantly
    IEEE Transactions on Signal Processing, 2010
    Co-Authors: Robert A Brown, M L Lauzon, Richard Frayne
    Abstract:

    Examining the frequency content of signals is critical in many applications, from neuroscience to astronomy. Many techniques have been proposed to accomplish this. One of these, the S-transform, provides simultaneous time and frequency information similar to the wavelet transform, but uses sinusoidal basis functions to produce frequency and globally referenced phase measurements. It has shown promise in many medical imaging applications but has high computational requirements. This paper presents a general transform that describes Fourier-family transforms, including the Fourier, short-time Fourier, and S- transforms. A discrete, nonredundant formulation of this transform, as well as algorithms for calculating the forward and inverse transforms are also developed. These utilize efficient sampling of the time-frequency plane and have the same computational complexity as the fast Fourier transform. When configured appropriately, this new algorithm samples the continuous S-transform spectrum efficiently and nonredundantly, allowing signals to be transformed in milliseconds rather than days, as compared to the original S-transform algorithm. The new and efficient algorithms make practical many existing signal and image processing techniques, both in biomedical and other applications.

  • a general description of linear time frequency transforms and formulation of a fast invertible transform that samples the continuous s transform spectrum nonredundantly
    IEEE Transactions on Signal Processing, 2010
    Co-Authors: Robert A Brown, M L Lauzon, Richard Frayne
    Abstract:

    Examining the frequency content of signals is critical in many applications, from neuroscience to astronomy. Many techniques have been proposed to accomplish this. One of these, the S-transform, provides simultaneous time and frequency information similar to the wavelet transform, but uses sinusoidal basis functions to produce frequency and globally referenced phase measurements. It has shown promise in many medical imaging applications but has high computational requirements. This paper presents a general transform that describes Fourier-family transforms, including the Fourier, short-time Fourier, and S- transforms. A discrete, nonredundant formulation of this transform, as well as algorithms for calculating the forward and inverse transforms are also developed. These utilize efficient sampling of the time-frequency plane and have the same computational complexity as the fast Fourier transform. When configured appropriately, this new algorithm samples the continuous S-transform spectrum efficiently and nonredundantly, allowing signals to be transformed in milliseconds rather than days, as compared to the original S-transform algorithm. The new and efficient algorithms make practical many existing signal and image processing techniques, both in biomedical and other applications.

Robert A Brown - One of the best experts on this subject based on the ideXlab platform.

  • a general description of linear time frequency transforms and formulation of a fast invertible transform that samples the continuous s transform spectrum nonredundantly
    IEEE Transactions on Signal Processing, 2010
    Co-Authors: Robert A Brown, M L Lauzon, Richard Frayne
    Abstract:

    Examining the frequency content of signals is critical in many applications, from neuroscience to astronomy. Many techniques have been proposed to accomplish this. One of these, the S-transform, provides simultaneous time and frequency information similar to the wavelet transform, but uses sinusoidal basis functions to produce frequency and globally referenced phase measurements. It has shown promise in many medical imaging applications but has high computational requirements. This paper presents a general transform that describes Fourier-family transforms, including the Fourier, short-time Fourier, and S- transforms. A discrete, nonredundant formulation of this transform, as well as algorithms for calculating the forward and inverse transforms are also developed. These utilize efficient sampling of the time-frequency plane and have the same computational complexity as the fast Fourier transform. When configured appropriately, this new algorithm samples the continuous S-transform spectrum efficiently and nonredundantly, allowing signals to be transformed in milliseconds rather than days, as compared to the original S-transform algorithm. The new and efficient algorithms make practical many existing signal and image processing techniques, both in biomedical and other applications.

  • a general description of linear time frequency transforms and formulation of a fast invertible transform that samples the continuous s transform spectrum nonredundantly
    IEEE Transactions on Signal Processing, 2010
    Co-Authors: Robert A Brown, M L Lauzon, Richard Frayne
    Abstract:

    Examining the frequency content of signals is critical in many applications, from neuroscience to astronomy. Many techniques have been proposed to accomplish this. One of these, the S-transform, provides simultaneous time and frequency information similar to the wavelet transform, but uses sinusoidal basis functions to produce frequency and globally referenced phase measurements. It has shown promise in many medical imaging applications but has high computational requirements. This paper presents a general transform that describes Fourier-family transforms, including the Fourier, short-time Fourier, and S- transforms. A discrete, nonredundant formulation of this transform, as well as algorithms for calculating the forward and inverse transforms are also developed. These utilize efficient sampling of the time-frequency plane and have the same computational complexity as the fast Fourier transform. When configured appropriately, this new algorithm samples the continuous S-transform spectrum efficiently and nonredundantly, allowing signals to be transformed in milliseconds rather than days, as compared to the original S-transform algorithm. The new and efficient algorithms make practical many existing signal and image processing techniques, both in biomedical and other applications.

M L Lauzon - One of the best experts on this subject based on the ideXlab platform.

  • a general description of linear time frequency transforms and formulation of a fast invertible transform that samples the continuous s transform spectrum nonredundantly
    IEEE Transactions on Signal Processing, 2010
    Co-Authors: Robert A Brown, M L Lauzon, Richard Frayne
    Abstract:

    Examining the frequency content of signals is critical in many applications, from neuroscience to astronomy. Many techniques have been proposed to accomplish this. One of these, the S-transform, provides simultaneous time and frequency information similar to the wavelet transform, but uses sinusoidal basis functions to produce frequency and globally referenced phase measurements. It has shown promise in many medical imaging applications but has high computational requirements. This paper presents a general transform that describes Fourier-family transforms, including the Fourier, short-time Fourier, and S- transforms. A discrete, nonredundant formulation of this transform, as well as algorithms for calculating the forward and inverse transforms are also developed. These utilize efficient sampling of the time-frequency plane and have the same computational complexity as the fast Fourier transform. When configured appropriately, this new algorithm samples the continuous S-transform spectrum efficiently and nonredundantly, allowing signals to be transformed in milliseconds rather than days, as compared to the original S-transform algorithm. The new and efficient algorithms make practical many existing signal and image processing techniques, both in biomedical and other applications.

  • a general description of linear time frequency transforms and formulation of a fast invertible transform that samples the continuous s transform spectrum nonredundantly
    IEEE Transactions on Signal Processing, 2010
    Co-Authors: Robert A Brown, M L Lauzon, Richard Frayne
    Abstract:

    Examining the frequency content of signals is critical in many applications, from neuroscience to astronomy. Many techniques have been proposed to accomplish this. One of these, the S-transform, provides simultaneous time and frequency information similar to the wavelet transform, but uses sinusoidal basis functions to produce frequency and globally referenced phase measurements. It has shown promise in many medical imaging applications but has high computational requirements. This paper presents a general transform that describes Fourier-family transforms, including the Fourier, short-time Fourier, and S- transforms. A discrete, nonredundant formulation of this transform, as well as algorithms for calculating the forward and inverse transforms are also developed. These utilize efficient sampling of the time-frequency plane and have the same computational complexity as the fast Fourier transform. When configured appropriately, this new algorithm samples the continuous S-transform spectrum efficiently and nonredundantly, allowing signals to be transformed in milliseconds rather than days, as compared to the original S-transform algorithm. The new and efficient algorithms make practical many existing signal and image processing techniques, both in biomedical and other applications.

Michael Unser - One of the best experts on this subject based on the ideXlab platform.

  • ICASSP - A family of discrete Fourier transforms with pseudo-cyclic convolution properties
    ICASSP '87. IEEE International Conference on Acoustics Speech and Signal Processing, 1
    Co-Authors: Michael Unser
    Abstract:

    An extended family of Discrete Fourier transforms is introduced. These transforms, which may be implemented by using FFTs, allow the computation of pseudo-cyclic convolutions by multiplication in the transform domain. The choice of a suitable transform (DFT 1/4 ) or the combined use of two complementary transforms allows a fast and efficient computation of aperiodic convolutions of waveforms of duration N by using N-point transforms that require no zero padding. Finally, all members of this family are shown to be equivalent asymptotically to the Karhunen-Loeve transform of an arbitrary wide sense stationary process.

Adrien Hardy - One of the best experts on this subject based on the ideXlab platform.

  • Time-frequency transforms of white noises and Gaussian analytic functions
    arXiv: Probability, 2018
    Co-Authors: Rémi Bardenet, Adrien Hardy
    Abstract:

    A family of Gaussian analytic functions (GAFs) has recently been linked to the Gabor transform of white Gaussian noise [Bardenet et al., 2017]. This answered pioneering work by Flandrin [2015], who observed that the zeros of the Gabor transform of white noise had a very regular distribution and proposed filtering algorithms based on the zeros of a spectrogram. The mathematical link with GAFs provides a wealth of probabilistic results to inform the design of such signal processing procedures. In this paper, we study in a systematic way the link between GAFs and a class of time-frequency transforms of Gaussian white noises on Hilbert spaces of signals. Our main observation is a conceptual correspondence between pairs (transform, GAF) and generating functions for classical orthogonal polynomials. This correspondence covers some classical time-frequency transforms, such as the Gabor transform and the Daubechies-Paul analytic wavelet transform. It also unveils new windowed discrete Fourier transforms, which map white noises to fundamental GAFs. All these transforms may thus be of interest to the research program `filtering with zeros'. We also identify the GAF whose zeros are the extrema of the Gabor transform of the white noise and derive their first intensity. Moreover, we discuss important subtleties in defining a white noise and its transform on infinite dimensional Hilbert spaces. Finally, we provide quantitative estimates concerning the finite-dimensional approximations of these white noises, which is of practical interest when it comes to implementing signal processing algorithms based on GAFs.

  • Time-frequency transforms of white noises and Gaussian analytic functions
    Applied and Computational Harmonic Analysis, 1
    Co-Authors: Rémi Bardenet, Adrien Hardy
    Abstract:

    Abstract A family of Gaussian analytic functions (GAFs) has recently been linked to the Gabor transform of Gaussian white noises [4] . This answered pioneering work by Flandrin [10] , who observed that the zeros of the Gabor transform of white noise had a regular distribution and proposed filtering algorithms based on the zeros of a spectrogram. In this paper, we study in a systematic way the link between GAFs and a class of time-frequency transforms of Gaussian white noises. Our main observation is a correspondence between pairs (transform, GAF) and generating functions for classical orthogonal polynomials. This covers some classical time-frequency transforms, such as the Gabor transform and the Daubechies-Paul wavelet transform. It also unveils new windowed discrete Fourier transforms, which map white noises to fundamental GAFs. Moreover, we discuss subtleties in defining a white noise and its transform on infinite dimensional Hilbert spaces and its finite dimensional approximations.