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
x(t) + \beta[\cos \theta x(t- \tau) + \sin \theta x_{H}(t- \tau)] + n(t),
where
\beta
is the echo amplitude,
\theta
an arbitrary constant, and
x_{H}(t - \tau)
is the Hilbert transform of
x(t - \tau)
. 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
Â(f)
of the received echoed signal and noise. The estimate
Â(f)
equals the "true" log spectral density
Â(f)
plus the sampling fluctuation of the log spectral density estimates. The echo produces a ripple of frequency
\tau
in the log spectral density.
Â(f)
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
\tau
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
\theta = 0
, and find that the cepstrum is only
1.8
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
\theta
is not known a priori. Results of computer experiments using irregular spectra are presented which indicate the superiority of the cepstrum over autocovariance.