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

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

Statement of the theorem

Application

References

Further reading