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, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n(t)</tex> . 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, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n(t)</tex> is reflected at the cepstrum's output as: 1) modulation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M₁(ω)</tex> of sinusoidal information carrier with subsequent reduction in the height of echo peak at <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">τ</tex> ; and 2) generation of background noises <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">phî(ω)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">psî(ω)</tex> 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">τ</tex> 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(S/N)</tex> alone is an insufficient measure for setting the cepstrum performance, and relative bandwidths of signal and noise are also needed. The detection probability <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PD</tex> of a peak corresponding to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">τ</tex> is calculated for exponential spectra with equal bandwidths in signal and noise and with echo strength <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a = 0.4; PD</tex> is found to be close to one for an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SIN</tex> greater than -4 dB; deterioration in detection is found to be rapid with thresholding near an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S/N ~ -6</tex> db where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PD</tex> falls to 0.3.