Bussgang theorem

In mathematics, the Bussgang's 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 of the theorem

Let $\left\{X(t)\right\}={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {x^{2}}{2\sigma ^{2}}}}$ (i.e. a zero-mean stationary Gaussian random process)

and $\left\{Y(t)\right\}=g(X(t))$ where $g(\cdot )$ is a nonlinear amplitude distortion.

If $R_{X}(\tau )$ is the autocorrelation of $\left\{X(t)\right\}$), then the cross correlation of $\left\{X(t)\right\}$ and $\left\{Y(t)\right\}$ is

$R_{XY}(\tau )=CR_{X}(\tau )$, where $C$ is a constant and depends only on $g(\cdot )$.

It can be further shown that

$C={\frac {1}{\sigma ^{3}{\sqrt {2\pi }}}}\int \limits _{-\infty }^{\infty }ug(u)e^{-{\frac {u^{2}}{2\sigma ^{2}}}}du$

Application

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