Output characteristic function for an analog crosscorrelator with bandpass inputs

An analysis is presented of an ideal two-channel cross-correlator in which each channel input consists of a deterministic signal combined additively with stationary Gaussian noise. It is assumed that all input quantities are bandlimited to some common passband,
0 < \omega_{0}- \Omega/2 \leq |\omega | \leq \omega_{0} + \Omega/2
, with angular bandwidth
\Omega > 0
. Moreover, the random noises
n_{1}(t)
and
n_{2}(t)
are assumed jointly normal with crosscorrelation function
E[n_{1}(t)\cdot n_{2}(t+ \tau)]
independent of
t
. After multiplication of the two composite inputs, the product process is passed through an ideal lowpass filter to produce the correlator output,
G(t)
. The main result of the paper is an explicit determination of the characteristic function of
G(t)
in closed form. This extends previous work by D. C. Cooper [1], who considered the same model with sinusoidal signals and a restricted form of dependency between the Gaussian noises in the two channels. The more general derivation given here makes use of canonical representations for the bandpass input quantities and exhibits the system output as a quadratic form in the (Gaussian) quadrature components. A recent result of Yu. S. Lezin [6] on the output probability density of an autocorrelator with bandpass inputs is shown to be a special case of the analysis.