Connexions

You are here: Home » Content » Homework 5 of Elec 430
Content Actions
Lenses

What is a lens?

Lenses

A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual Connexions member, a community, or a respected organization.

This content is ...
Affiliated with (?)
This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.

  • This module is included inLens: Rice University OpenCourseWare
    By: OpenCourseWare ConsortiumAs a part of collection:"Digital Communication Systems"

    Click the "Rice University OCW" link to see all content affiliated with them.

    Rice University OCW
Tags

(?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

Homework 5 of Elec 430

Module by: Behnaam Aazhang

Summary: (Blank Abstract)

Problem 1

Consider a ternary communication system where the source produces three possible symbols: 0, 1, 2.
a) Assign three modulation signals s 1 t s 1 t , s 2 t s 2 t , and s 3 t s 3 t defined on t∈0T t 0 T to these symbols, 0, 1, and 2, respectively. Make sure that these signals are not orthogonal and assume that the symbols have an equal probability of being generated.
b) Consider an orthonormal basis ψ 1 t ψ 1 t , ψ 2 t ψ 2 t , ..., ψ N t ψ N t to represent these three signals. Obviously NN could be either 1, 2, or 3.
HW5Fig1.png
Figure 1
Now consider two different receivers to decide which one of the symbols were transmitted when r t = s m t+ N t r t s m t N t is received where m=123 m 1 2 3 and N t N t is a zero mean white Gaussian process with S N f= N 0 2 S N f N 0 2 for all ff. What is f r | s m ( t ) f r | s m ( t ) and what is f Y | s m ( t ) f Y | s m ( t ) ?
HW5Fig2.png
Figure 2
Find the probability that m ^ ≠m m ^ m for both receivers. P e =Pr m ^ ≠m P e m ^ m .

Problem 2

Proakis and Salehi problems 7.18, 7.26, and 7.32

Problem 3

Suppose our modulation signals are s 1 t s 1 t and s 2 t s 2 t where s 1 t=ⅇ-t2 s 1 t t 2 for all tt and s 2 t=- s 1 t s 2 t s 1 t . The channel noise is AWGN with zero mean and spectral height N 0 2 N 0 2 . The signals are transmitted equally likely.
HW5Fig3.png
Figure 3
Find the impulse response of the optimum filter. Find the signal component of the output of the matched filter at t=T t T where s 1 t s 1 t is transmitted; i.e., u 1 t u 1 t . Find the probability of error Pr m ^ ≠m m ^ m .
In this part, assume that the power spectral density of the noise is not flat and in fact is
S N f=12πf2+α2 S N f 1 2 f 2 α 2 (1)
for all ff, where αα is real and positive. Can you show that the optimum filter in this case is a cascade of two filters, one to whiten the noise and one to match to the signal at the output of the whitening filter?
HW5Fig4.png
Figure 4
c) Find an expression for the probability of error.

Comments, questions, feedback, criticisms?

Send feedback