The heuristics of cepstrum analysis of a stationary complex echoed Gaussian signal in stationary Gaussian noise

We consider cepstrum analysis of a model consisting of a detector receiving a Gaussian single complex echoed signal in Gaussian noise of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x(t) + β[cos θ x(t- τ) + sin θ xH(t- τ)] + n(t),</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">β</tex> is the echo amplitude, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">θ</tex> an arbitrary constant, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">xH(t - τ)</tex> is the Hilbert transform of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x(t - τ)</tex> . We assume no a priori knowledge of the spectra of the signal and noise except that they are smooth. The cepstrum is obtained as follows. We compute the log power spectral density estimate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Â(f)</tex> of the received echoed signal and noise. The estimate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Â(f)</tex> equals the "true" log spectral density <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Â(f)</tex> plus the sampling fluctuation of the log spectral density estimates. The echo produces a ripple of frequency <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">τ</tex> in the log spectral density. <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Â(f)</tex> is filtered to remove the slowly varying spectrum components and the result is analyzed by power spectrum estimation procedures to yield the cepstrum in which there is a peak due to the echo. To assess performance, we compare the cepstrum due to the log spectrum sampling fluctuation with the cepstrum peak due to the echo. The echo detectability for a given value of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">τ</tex> can then be found by assuming the cepstrum estimates due to the log spectrum sampling fluctuations alone to he distributed as a scaled chi-square, and when an echo is present the distribution is assumed to be a scaled noncentral chi-square. We compare a special case of these results to an adaptation of a maximum likelihood procedure discussed by Whittle, for which <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">θ = 0</tex> , and find that the cepstrum is only <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1.8</tex> dB worse for the echo amplitude and data length assumed. This degradation is not surprising since the cepstrum is equally effective for a complex echo in which <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">θ</tex> is not known a priori. Results of computer experiments using irregular spectra are presented which indicate the superiority of the cepstrum over autocovariance.