Bussgang theorem


In mathematics, the Bussgang theorem is a theorem of stochastic analysis. The theorem states that the crosscorrelation of a Gaussian signal before and after it has passed through a nonlinear operation are equal up to a constant. It was first published by Julian J. Bussgang in 1952 while he was at the Massachusetts Institute of Technology.

Statement

Let be a zero-mean stationary Gaussian random process and where is a nonlinear amplitude distortion.
If is the autocorrelation function of, then the cross-correlation function of and is
where is a constant that depends only on.
It can be further shown that

Application

This theorem implies that a simplified correlator can be designed. Instead of having to multiply two signals, the cross-correlation problem reduces to the gating of one signal with another.