The output signal-to-noise ratio of correlation detectors

Expressions are derived for the output signal-to-noise ratio of a correlation detector when the two input functions to be correlated differ only by the presence of an arbitrary linear filter in each path, and the addition of noise to each. It is assumed that the signal and noises are Gaussian with arbitrary power density spectra, and the integration is performed by a filter of arbitrary transfer function. Two types of correlation detectors are distinguished, the low-pass detector in which the integrator is a low-pass filter, and the band-pass detector in which one of the two input functions is deliberately displaced in frequency by
\Delta
and the integrator is therefore a band-pass filter tuned to
\Delta
. Output signal-to-noise ratio expressions for the two types are almost identical.