Samiksha Jaiswal (Editor)

Bussgang theorem

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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.

Contents

Statement of the theorem

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

If R X ( τ ) is the autocorrelation function of { X ( t ) } , then the cross-correlation function of { X ( t ) } and { Y ( t ) } is

R X Y ( τ ) = C R X ( τ ) ,

where C is a constant that depends only on g ( ⋅ ) .

It can be further shown that

C = 1 σ 3 2 π ∫ − ∞ ∞ u g ( u ) e − u 2 2 σ 2 d u .

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

Bussgang theorem Wikipedia
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