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 be a zero-mean stationary Gaussian random process and where is a nonlinear amplitude distortion.
where is a constant that depends only on .
It can be further shown that
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.
- J.J. Bussgang,"Cross-correlation function of amplitude-distorted Gaussian signals", Res. Lab. Elec., Mas. Inst. Technol., Cambridge MA, Tech. Rep. 216, March 1952.
- 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