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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.

Statement of the theorem

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

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

$R_{XY}(\tau )=C{R}_X(\tau ),$

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

It can be further shown that

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

Application

This theorem implies that a simplified correlator can be designed.Template:Clarify Instead of having to multiply two signals, the cross-correlation problem reduces to the gatingTemplate:Clarify of one signal with another.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

References

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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