Assume that
is a white Gaussian process with zero mean and spectral height
.
If
is "0" then
and if
is "1" then
where
. Suppose
.
Find the probability density function
when bit "0" is transmitted and also when bit "1" is transmitted.
Refer to these two densities as
and
,
where
denotes the hypothesis that bit "0" is transmitted and
denotes the hypothesis that bit "1" is transmitted.
Consider the ratio of the above two densities;
i.e. ,
and its natural log
.
A reasonable scheme to decide which bit was actually transmittedis to compare
to a fixed threshold
.
(
is often referred to as the likelihood function and
as the log likelihood function). Given threshold
is used
to decide
when
then find
(note that we will say
when
).