Assume that
${N}_{}$ is a white Gaussian process with zero mean and spectral height
$\frac{{N}_{0}}{2}$ .

If
$b$ is "0" then
${X}_{}=A{p}_{T}()$ and if
$b$ is "1" then
${X}_{}=-A{p}_{T}()$ where
${p}_{T}()=\begin{cases}1 & \text{if $0\le \le T$}\\ 0 & \text{otherwise}\end{cases}$ . Suppose
$(b=1)=(b=0)=1/2$ .

Find the probability density function
${Z}_{T}$ when bit "0" is transmitted and also when bit "1" is transmitted.
Refer to these two densities as
$f({Z}_{T}{H}_{0}, z)$ and
$f({Z}_{T}{H}_{1}, z)$ ,
where
${H}_{0}$ denotes the hypothesis that bit "0" is transmitted and
${H}_{1}$ denotes the hypothesis that bit "1" is transmitted.

Consider the ratio of the above two densities;
i.e. ,

and its natural log
$\ln (z)$ .
A reasonable scheme to decide which bit was actually transmittedis to compare
$\ln (z)$ to a fixed threshold
$$ .
(
$(z)$ is often referred to as the likelihood function and
$\ln (z)$ as the log likelihood function). Given threshold
$$ is used
to decide
$\hat{b}=0$ when
$\ln (z)\ge $ then find
$(\hat{b}\neq b)$ (note that we will say
$\hat{b}=1$ when
$\ln (z)< $ ).

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