# 1.5 Homework 6

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

## Problem 1

Consider the following modulation system

${s}_{0}(t)=A{P}_{T}(t)-1$
and
${s}_{1}(t)=-(A{P}_{T}(t))-1$
for $0\le t\le T$ where ${P}_{T}(t)=\begin{cases}1 & \text{if 0\le t\le T}\\ 0 & \text{otherwise}\end{cases}$

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 ${a}_{n}\in \{2A, A, -A, -(2A)\}$ . The transmitted signal is

${X}_{t}=\sum_{n=1}^{4} {a}_{n}s(t-nT)$
where $s(t)$ is a rectangular pulse of duration $T$ and height of 1. Assume that we have a channel with impulse response $g(t)$ which is a rectangular pulse of duration $T$ and height 1, with white Gaussian noise with ${S}_{N}(f)=\frac{{N}_{0}}{2}$ for all $f$ .

• 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

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Damian
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I think
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