A probabilistic analysis of time delay extraction by the cepstrum in stationary Gaussian noise

Often, the cepstrum has been used when estimating the time delay r between elements of a composite signal embedded in noise,
n(t)
. Here a probabilistic analysis is conducted dealing with the effect of stationary Gaussian noise on the characteristics of such a nonlinear processor. The input noise,
n(t)
is reflected at the cepstrum's output as: 1) modulation
M_{1}(\omega)
of sinusoidal information carrier with subsequent reduction in the height of echo peak at
\tau
; and 2) generation of background noises
\hat{phi}(\omega)
and
\hat{psi}(\omega)
with tendency to obscure that echo peak. The resulting probabilities of these terms are derived. The expected mean and standard deviation of reduction in the peak level at
\tau
due to noise are analytically described. Similar statistical measures on the background noise are also obtained. The results point out the dependence of the-statistical measures upon the pointwise variation of input signal to noise spectra. Thus the total signal power to total noise power
(S/N)
alone is an insufficient measure for setting the cepstrum performance, and relative bandwidths of signal and noise are also needed. The detection probability
P_{D}
of a peak corresponding to
\tau
is calculated for exponential spectra with equal bandwidths in signal and noise and with echo strength
a = 0.4; P_{D}
is found to be close to one for an
SIN
greater than -4 dB; deterioration in detection is found to be rapid with thresholding near an
S/N \sim -6
db where
P_{D}
falls to 0.3.