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Homework 5 of Elec 430

Module by: Behnaam Aazhang. E-mail the author

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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.

Figure 1
Figure 1 (HW5Fig1.png)

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 ) ?

Figure 2
Figure 2 (HW5Fig2.png)

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.

Figure 3
Figure 3 (HW5Fig3.png)

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?

Figure 4
Figure 4 (HW5Fig4.png)

c) Find an expression for the probability of error.

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