In mathematics, the Bussgang theorem is a theorem of stochastic analysis. The theorem states that the cross-correlation between a Gaussian signal before and after it has passed through a nonlinear operation are equal to the signals auto-correlation 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

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Let   be a zero-mean stationary Gaussian random process and   where   is a nonlinear amplitude distortion.

If   is the autocorrelation function of  , then the cross-correlation function of   and   is

 

where   is a constant that depends only on  .

It can be further shown that

 

Derivation for One-bit Quantization

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It is a property of the two-dimensional normal distribution that the joint density of   and   depends only on their covariance and is given explicitly by the expression

 

where   and   are standard Gaussian random variables with correlation  .

Assume that  , the correlation between   and   is,

 .

Since

 ,

the correlation   may be simplified as

 .

The integral above is seen to depend only on the distortion characteristic   and is independent of  .

Remembering that  , we observe that for a given distortion characteristic  , the ratio   is  .

Therefore, the correlation can be rewritten in the form

 .

The above equation is the mathematical expression of the stated "Bussgang‘s theorem".

If  , or called one-bit quantization, then  .

[2][3][1][4]

Arcsine law

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If the two random variables are both distorted, i.e.,  , the correlation of   and   is

 .

When  , the expression becomes,

 

where  .

Noticing that

 ,

and  ,  ,

we can simplify the expression of   as

 

Also, it is convenient to introduce the polar coordinate  . It is thus found that

 .

Integration gives

 

This is called "Arcsine law", which was first found by J. H. Van Vleck in 1943 and republished in 1966.[2][3] The "Arcsine law" can also be proved in a simpler way by applying Price's Theorem.[4][5]

The function   can be approximated as   when   is small.

Price's Theorem

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Given two jointly normal random variables   and   with joint probability function

 ,

we form the mean

 

of some function   of  . If   as  , then

 .

Proof. The joint characteristic function of the random variables   and   is by definition the integral

 .

From the two-dimensional inversion formula of Fourier transform, it follows that

 .

Therefore, plugging the expression of   into  , and differentiating with respect to  , we obtain

 

After repeated integration by parts and using the condition at  , we obtain the Price's theorem.

 

[4][5]

Proof of Arcsine law by Price's Theorem

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If  , then   where   is the Dirac delta function.

Substituting into Price's Theorem, we obtain,

 .

When  ,  . Thus

 ,

which is Van Vleck's well-known result of "Arcsine law".

[2][3]

Application

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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.[citation needed]

References

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  1. ^ a b J.J. Bussgang,"Cross-correlation function of amplitude-distorted Gaussian signals", Res. Lab. Elec., Mas. Inst. Technol., Cambridge MA, Tech. Rep. 216, March 1952.
  2. ^ a b c Vleck, J. H. Van. "The Spectrum of Clipped Noise". Radio Research Laboratory Report of Harvard University (51).
  3. ^ a b c Vleck, J. H. Van; Middleton, D. (January 1966). "The spectrum of clipped noise". Proceedings of the IEEE. 54 (1): 2–19. doi:10.1109/PROC.1966.4567. ISSN 1558-2256.
  4. ^ a b c Price, R. (June 1958). "A useful theorem for nonlinear devices having Gaussian inputs". IRE Transactions on Information Theory. 4 (2): 69–72. doi:10.1109/TIT.1958.1057444. ISSN 2168-2712.
  5. ^ a b Papoulis, Athanasios (2002). Probability, Random Variables, and Stochastic Processes. McGraw-Hill. p. 396. ISBN 0-07-366011-6.

Further reading

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