Bussgang theorem


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.

Statement

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

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.

Arcsine law

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. The "Arcsine law" can also be proved in a simpler way by applying Price's Theorem.
The function can be approximated as when is small.

Price's Theorem

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.

Proof of Arcsine law by Price's Theorem

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".

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.