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 ${\displaystyle \left\{X(t)\right\}}$ be a zero-mean stationary Gaussian random process and ${\displaystyle \left\{Y(t)\right\}=g(X(t))}$ where ${\displaystyle g(\cdot )}$ is a nonlinear amplitude distortion.

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

${\displaystyle R_{XY}(\tau )=CR_{X}(\tau ),}$

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

It can be further shown that

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

Application

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

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.

Further reading

• E.W. Bai; V. Cerone; D. Regruto (2007) "Separable inputs for the identification of block-oriented nonlinear systems", Proceedings of the 2007 American Control Conference (New York City, July 11–13, 2007) 1548–1553