مرکز منطقه ای اطلاع رساني علوم و فناوري - A probabilistic analysis of time delay extraction by the cepstrum in stationary Gaussian noise

Abstract :

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.