1.5 Homework 6

Homework set 6 of ELEC 430, Rice University, Department of Electrical and Computer Engineering

Problem 1

Consider the following modulation system

The channel is ideal with Gaussian noise which is ${}_N(t)=1$ for all $t$ , wide sense stationary with $R_N()=b^{2}e^{-\left|\right|}$ for all $\in \mathbb{R}$ . Consider the following receiver structure

• a) Find the optimum value of the threshold for the system ( e.g. , $$ that minimizes the $P_e$ ). Assume that ${}_0={}_1$
• b) Find the error probability when this threshold is used.

Problem 2

Consider a PAM system where symbols $a_1$ , $a_2$ , $a_3$ , $a_4$ are transmitted where $a_{\mathrm{n}}\in \{2A, A, -A, -(2A)\}$ . The transmitted signal is

• a) Draw a typical sample path (realization) of $X_t$ and of the received signal $r_t$ (do not forget to add a bit of noise!)
• b) Assume that the receiver knows $g(t)$ . Design a matched filter for this transmission system.
• c) Draw a typical sample path of $Y_t$ , the output of the matched filter (do not forget to add a bit ofnoise!)
• d) Find an expression (or draw) $u(nT)$ where $u(t)=(s, g, h^{\mathrm{opt}}(t))$ .

Problem 3

Proakis and Salehi, problem 7.35

Problem 4

Proakis and Salehi, problem 7.39