Further derivations of statistical measures in the cepstrum (Corresp.)

Abstract

Expanded statistical measures in the cepstrum are obtained when estimating the time delay\taubetween the wavelets of a composite signal embedded in noise. Previous considerations have been limited to physical instances where the modulusaand the ratio of signal to noise spectra\Phi_{y}/ \Phi_{n}yield values forx = (1 +a^{2}) \Phi_{y}/ \Phi_{n}and/or\gamma = 2|a| / (1 + a^{2})much less than one. The physical parameters,a, \Phi_{y}, and\Phi_{n}are allowed to vary throughout the spectrum, thus introducing new elements to the analysis. The required statistical measures are derived for the complete series of cross-terms, for a higher order perturbation in the modulation term, and for its logarithmically generated harmonics. Under these open conditions, the explicit behavior of the additional statistical measures are delineated as a function of the modulusaand the ratio of signal to noise spectra. The effect of the interfering harmonics with the modulation term is assessed for signal and noise with exponential autocorrelation functions.