The USC Andrew and Erna Viterbi School of Engineering USC Signal and Image Processing Institute USC Ming Hsieh Department of Electrical Engineering University of Southern California

Technical Report USC-SIPI-177

“Comparison between Correlation-Based and Cumulant-Based Approaches to the Harmonic Retrieval and Related Problems”

by Dae C. Shin and Jerry M. Mendel

April 1991

In signal processing, we frequently encounter the problem of estimating the number of harmonics, frequencies, and amplitudes in a sum of sinusoids. The observed signals are usually corrupted by spatially and/or temporally colored noise with unknown power spectral density. It has been shown by Swami and Mendel that a cumulant-based approach to this problem is very effective. In this report, we compare the use of biased and unbiased, segmented and unsegmented estimators for both correlation and 1-D diagonal slice of the fourth-order cumulant function. We suggest using accumulated singular values to determine the number of harmonics. We compare correlation-based and cumulant-based methods for determining the number of harmonics when the amplitude of one harmonic decreases and when the frequency of one harmonic approaches the other for the case of two sinusoids measured in colored Gaussian noise. We also compare the performance of the Pisarenko, MUSIC, and minimum-norm algorithms for frequency estimation, and the performance of least square (LS), total least square (TLS), and constrained total least square (CTLS) for amplitude estimation using either correlations or cumulants. Our studies: (1) provide further support for using cumulants over correlations when measurement noise is colored and Gaussian; (2) demonstrate that one should use unbiased unsegmented correlation or cumulant estimators; (3) indicate that high-resolution results are best obtained using cumulant-based MUSIC or minimum-norm algorithms; and (4) show that LS estimates of amplitudes suffice.

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