On the Statistics of Spectral Amplitudes After Variance Reduction by Temporal Cepstrum Smoothing and Cepstral Nulling

In this paper, we derive the signal power bias that arises when spectral amplitudes are smoothed by reducing their variance in the cepstral domain (often referred to as cepstral smoothing) and develop a power bias compensation method. We show that if chi-distributed spectral amplitudes are smoothed in the cepstral domain, the resulting smoothed spectral amplitudes are also approximately chi-distributed but with more degrees of freedom and less signal power. The key finding for the proposed power bias compensation method is that the degrees of freedom of chi-distributed spectral amplitudes are directly related to their average cepstral variance. Furthermore, this work gives new insights into the statistics of the cepstral coefficients derived from chi-distributed spectral amplitudes using tapered spectral analysis windows. We derive explicit expressions for the variance and covariance of correlated chi-distributed spectral amplitudes and the resulting cepstral coefficients, parameterized by the degrees of freedom. The results in this work allow for a cepstral smoothing of spectral quantities without affecting their signal power. As we assume the parameterized chi-distribution for the spectral amplitudes, the results hold for Gaussian, super-Gaussian, and sub-Gaussian distributed complex spectral coefficients. The proposed bias compensation method is computationally inexpensive and shown to work very well for white and colored signals, as well as for rectangular and tapered spectral analysis windows.