Truncated sampling for the Fourier-Mellin transform with applications to wideband WVD computation

Jeffrey Allen

doi:10.1117/12.23485

1 November 1990 Truncated sampling for the Fourier-Mellin transform with applications to wideband WVD computation

Jeffrey Allen

Proceedings Volume 1348, Advanced Signal Processing Algorithms, Architectures, and Implementations; (1990) https://doi.org/10.1117/12.23485
Event: 34th Annual International Technical Symposium on Optical and Optoelectronic Applied Science and Engineering, 1990, San Diego, CA, United States

Abstract

The Fourier-Mellin transform (FMT) of an input function is defined as and is the magnitude squared of the Mellin transform of the magnitude squared of the Fourier transform of the input function [1]. As such the FMT is unchanged by translations and dilations of the input function. While the FMT has found applications in optical pattern recognition [3] [5] ship classification by sonar and radar [15] and image processing [10] only cursory attention has been paid to the truncation error incurred by using a finite number of samples of the input function. This paper establishes truncation bounds for computing the FMT for band-limited functions from a finite number of samples of the input function. These bounds naturally suggest an implementation of the FMT by the method of direct expansions [4] [14]. This approach readily generalizes to a direct expansion for the Wigner-Ville distribution [13] and the Q distribution [2]. 1 Principal Notation u(x) fff00 e_2tu(t)dt Fourier transform of u M(u s) fD X_i2r8() Mellin transform of u . FM(u s) M(lI(x)I2 s)________ Fourier-Mellin transform of u Q(U V f002rt U(wft)_V(w/fr) Q distribution of U and V W(U V t w) fe_i2ntY U(w + y/2) V(w y/2) dy Wigner-Ville distribution of U and V

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Jeffrey Allen "Truncated sampling for the Fourier-Mellin transform with applications to wideband WVD computation", Proc. SPIE 1348, Advanced Signal Processing Algorithms, Architectures, and Implementations, (1 November 1990); https://doi.org/10.1117/12.23485

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