# 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.[1]

## Statement of the theorem

Let $\left\{X(t)\right\}$ be 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 function of $\left\{ X(t) \right\}$, then the cross-correlation function of $\left\{ X(t) \right\}$ and $\left\{ Y(t) \right\}$ is

$R_{XY}(\tau) = CR_X(\tau),$

where $C$ is a constant that depends only on $g(\cdot)$.

It can be further shown that

$C = \frac{1}{\sigma^3\sqrt{2\pi}}\int_{-\infty}^\infty ug(u)e^{-\frac{u^2}{2\sigma^2}} \, du.$

## Application

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

## References

1. ^ J.J. Bussgang,"Cross-correlation function of amplitude-distorted Gaussian signals", Res. Lab. Elec., Mas. Inst. Technol., Cambridge MA, Tech. Rep. 216, March 1952.