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In [[mathematics]], the '''Bussgang theorem''' is a [[theorem]] of [[Stochastic process|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]].<ref>J.J. Bussgang,"Cross-correlation function of amplitude-distorted Gaussian signals", Res. Lab. Elec., Mas. Inst. Technol., Cambridge MA, Tech. Rep. 216, March 1952.</ref>

==Statement of the theorem==

Let <math> \left\{X(t)\right\} </math> be a zero-mean stationary [[Gaussian process|Gaussian random process]] and <math> \left \{ Y(t) \right\} = g(X(t)) </math> where <math> g(\cdot) </math> is a nonlinear amplitude distortion.

If <math> R_X(\tau) </math> is the [[autocorrelation function]] of <math> \left\{ X(t) \right\}</math>, then the [[cross-correlation function]] of <math> \left\{ X(t) \right\}</math> and <math> \left\{ Y(t) \right\}</math> is

:<math> R_{XY}(\tau) = CR_X(\tau), </math>
where <math>C</math> is a constant that depends only on <math> g(\cdot) </math>.

It can be further shown that

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

==Application==
This theorem implies that a simplified correlator can be designed.{{clarify|reason=compared to what?|date=December 2010}} Instead of having to multiply two signals, the cross-correlation problem reduces to the gating{{clarify|reason=undefined|date=December 2010}} of one signal with another.{{citation needed|date=December 2010}}

==References==
{{reflist}}

==Further reading==
* E.W. Bai; V. Cerone; D. Regruto (2007) [http://diegoregruto.com/P14.pdf "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

{{DEFAULTSORT:Bussgang Theorem}}
[[Category:Probability theorems]]
[[Category:Stochastic processes]]

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