Technical Report Austin, 1994.

abstract bibtex

abstract bibtex

A review of the Fourier, Mellin, and Hilbert transforms is provided to lay the foundation for developing a new transform denoted the K-transform: a composition of the Fourier and Mellin transforms. In this report, the transform is defined, an explicit expression as an integral operator is derived, and an asymptotic estimate for the transform kernel is obtained. Some properties of the K-transform are explored, and the application of this transform to the work of Altes on mammalian hearing is noted. Lastly, a unified setting for Fourier and wavelet analysis is explored. The connection of the Heisenberg group with the Gabor transform and Wigner-Ville distribution is shown, as is the connection of the affine group with the wavelet transform. These notions are then combined in a single setting, the affine-Heisenberg group. In this setting, the report is closed by introducing an affine version of the Wigner-Ville distribution in terms of the K-transform.

@techreport{ ruth_mellin_1994, address = {Austin}, title = {The Mellin transform in signal analysis}, abstract = {A review of the Fourier, Mellin, and Hilbert transforms is provided to lay the foundation for developing a new transform denoted the K-transform: a composition of the Fourier and Mellin transforms. In this report, the transform is defined, an explicit expression as an integral operator is derived, and an asymptotic estimate for the transform kernel is obtained. Some properties of the K-transform are explored, and the application of this transform to the work of Altes on mammalian hearing is noted. Lastly, a unified setting for Fourier and wavelet analysis is explored. The connection of the Heisenberg group with the Gabor transform and Wigner-Ville distribution is shown, as is the connection of the affine group with the wavelet transform. These notions are then combined in a single setting, the affine-Heisenberg group. In this setting, the report is closed by introducing an affine version of the Wigner-Ville distribution in terms of the K-transform.}, author = {Ruth, David M. and Gilbert, John E.}, year = {1994}, keywords = {{ANALYTIC} {FUNCTIONS}, {ASYMPTOTIC} {SERIES}, {CONVERGENCE}, {FOURIER} {TRANSFORMATION}, {GROUPS}({MATHEMATICS}), {HYPERGEOMETRIC} {FUNCTIONS}., {INTEGRAL} {EQUATIONS}, {INTEGRAL} {TRANSFORMS}, {KERNEL} {FUNCTIONS}, {MAPPING}({TRANSFORMATIONS}), {MATHEMATICAL} {MODELS}, {OPERATORS}({MATHEMATICS}), {RESOLUTION}, Signal processing, {TIME} {DOMAIN}, convolution, fourier analysis, frequency domain}, pages = {42--42} }

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