1.4 Exercises on systems and density

Consider the following system

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. ,
  $(z)=\frac{f(Z_T{H}_0, z)}{f(Z_T{H}_1, z)}$
  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 $\ln (z)<$ ).
• Find a $$ that minimizes $(\hat{b}\neq b)$ .

Proakis and Salehi , problems 7.7, 7.17, and 7.19

Proakis and Salehi , problem 7.20, 7.28, and 7.23