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 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
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.
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
^
abJ.J. Bussgang,"Cross-correlation function of amplitude-distorted Gaussian signals", Res. Lab. Elec., Mas. Inst. Technol., Cambridge MA, Tech. Rep. 216, March 1952.
^
abcVleck, J. H. Van. "The Spectrum of Clipped Noise". Radio Research Laboratory Report of Harvard University (51).
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 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
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.
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
^
abJ.J. Bussgang,"Cross-correlation function of amplitude-distorted Gaussian signals", Res. Lab. Elec., Mas. Inst. Technol., Cambridge MA, Tech. Rep. 216, March 1952.
^
abcVleck, J. H. Van. "The Spectrum of Clipped Noise". Radio Research Laboratory Report of Harvard University (51).