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

Expanded statistical measures in the cepstrum are obtained when estimating the time delay
\tau
between the wavelets of a composite signal embedded in noise. Previous considerations have been limited to physical instances where the modulus
a
and the ratio of signal to noise spectra
\Phi_{y}/ \Phi_{n}
yield values for
x = (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 modulus
a
and 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.