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

[4] [5]
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".
[2] [3]