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.
be a zero-mean stationary Gaussian random process and
is a nonlinear amplitude distortion.
It can be further shown that It is a property of the two-dimensional normal distribution that the joint density of
depends only on their covariance and is given explicitly by the expression where
are standard Gaussian random variables with correlation
may be simplified as The integral above is seen to depend only on the distortion characteristic
, we observe that for a given distortion characteristic
Therefore, the correlation can be rewritten in the form
.The above equation is the mathematical expression of the stated "Bussgang‘s theorem".
, or called one-bit quantization, then
If the two random variables are both distorted, i.e.,
Also, it is convenient to introduce the polar coordinate
( 1 − ρ sin 2 θ )
1 − ρ sin 2 θ
ρ − tan θ
，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.
Given two jointly normal random variables
with joint probability function
The joint characteristic function of the random variables
.From the two-dimensional inversion formula of Fourier transform, it follows that
After repeated integration by parts and using the condition at
, we obtain the Price's theorem.
is the Dirac delta function.
Substituting into Price's Theorem, we obtain,
,which is Van Vleck's well-known result of "Arcsine law".
[2][3] 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.