Decomposition of time-frequency distributions using scaled-window spectrograms

William J. Williams; Jeffrey C. O Neill

doi:10.1117/12.211424

7 June 1995 Decomposition of time-frequency distributions using scaled-window spectrograms

William J. Williams, Jeffrey C. O Neill

Proceedings Volume 2563, Advanced Signal Processing Algorithms; (1995) https://doi.org/10.1117/12.211424
Event: SPIE's 1995 International Symposium on Optical Science, Engineering, and Instrumentation, 1995, San Diego, CA, United States

Abstract

The decomposition of time-frequency representations (TFRs) in terms of weighted spectrograms has been recently proposed by several authors. Spectrogram decomposition concepts allow inner products to be used in the computations rather than the more cumbersome outer products usually associated with TFR computations. Kernels are decomposed in terms of eigenvectors in such a manner that a TFR may be represented by a truncated spectrogram series according to the strength of the eigenvalues associated with these eigenvectors. Many TFRs can be represented by relatively few spectrograms due to the small contributions of the remaining spectrograms with eigenvalues below some threshold value. The windows of the spectrograms forming the spectrogram series are the eigenvectors of the decomposition of the kernel of the particular representation. In the present paper it is shown how full TFR representations can be obtained by using a carefully chosen set of scaled cross spectrogram windows, thus avoiding the inherent approximations of the eigenvector approach. Much redundancy can be taken advantage of to permit computation of a small number of short-time Fourier transforms (STFTs). It is not practical to compute the Wigner distribution via the spectrogram decomposition approach due to the fact that the singular values of the decomposition are plus or minus one, precluding truncation of the spectrogram series. The new approach, on the other hand, can represent a Wigner distribution and other TFRs with a small number of STFTs. These STFTs can be used to compute a number of spectrograms and cross-spectrograms which, when appropriately weighted and summed, yield a given TFR, depending on the kernel used in the decomposition.

Citation Download Citation

William J. Williams and Jeffrey C. O Neill "Decomposition of time-frequency distributions using scaled-window spectrograms", Proc. SPIE 2563, Advanced Signal Processing Algorithms, (7 June 1995); https://doi.org/10.1117/12.211424

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