Generalized scale transforms

Robert M. Nickel; William J. Williams

doi:10.1117/12.367663

2 November 1999 Generalized scale transforms

Proceedings Volume 3807, Advanced Signal Processing Algorithms, Architectures, and Implementations IX; (1999) https://doi.org/10.1117/12.367663
Event: SPIE's International Symposium on Optical Science, Engineering, and Instrumentation, 1999, Denver, CO, United States

Abstract

We are presenting a new class of transforms which facilitates the processing of signals that are nonlinearly stretched or compressed in time. We refer to nonlinear stretching and compression as warping. While the magnitude of the Fourier transform is invariant under time shift operations, and the magnitude of the scale transform is invariant under (linear) scaling operations, the new class of transforms is magnitude invariant under warping operations. The new class contains the Fourier transform and the scale transform as special cases. Important theorems, like the convolution theorem for Fourier transforms, are generalized into theorems that apply to arbitrary members of the transform class. Cohen's class of time-frequency distributions is generalized to joint representations in time and arbitrary warping variables. Special attention is paid to a modification of the new class of transforms that maps an arbitrary time-frequency contour into an impulse in the transforms that maps an arbitrary time-frequency contour into an impulse in the transform domain. A chirp transform is derived as an example.

Citation Download Citation

Robert M. Nickel and William J. Williams "Generalized scale transforms", Proc. SPIE 3807, Advanced Signal Processing Algorithms, Architectures, and Implementations IX, (2 November 1999); https://doi.org/10.1117/12.367663

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