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Homework 6 of Elec 430

Module by: Behnaam Aazhang. E-mail the author

Summary: (Blank Abstract)

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 s 0 t A P T t 1
(1)
and
s 1 ⁢t=(−(A⁢ P T ⁢t))−1 s 1 t A P T t 1
(2)
for 0≤t≤T 0 t T where P T ⁢t={1  if  0≤t≤T0  otherwise   P T t 1 0 t T 0

Figure 1
Figure 1 (Fig1.png)

The channel is ideal with Gaussian noise which is μ N ⁢t=1 μ N t 1 for all t t, wide sense stationary with R N ⁢τ=b2⁢e−|τ| R N τ b 2 τ for all τ∈R τ . Consider the following receiver structure

Figure 2
(a) (b)
r t = s m ⁢t+ N t r t s m t N t
Figure 2(b) (Fig2.png)
  • a) Find the optimum value of the threshold for the system (e.g., γγ that minimizes the P e P e ). Assume that π 0 = π 1 π 0 π 1
  • b) Find the error probability when this threshold is used.

Problem 2

Consider a PAM system where symbols a 1 a 1 , a 2 a 2 , a 3 a 3 , a 4 a 4 are transmitted where a n ∈2⁢AA−A−(2⁢A) a n 2 A A A 2 A . The transmitted signal is

X t =∑n=14 a n ⁢s⁢t−n⁢T X t n 1 4 a n s t n T
(3)
where s⁢t s t is a rectangular pulse of duration TT and height of 1. Assume that we have a channel with impulse response g⁢t g t which is a rectangular pulse of duration TT and height 1, with white Gaussian noise with S N ⁢f= N 0 2 S N f N 0 2 for all f f.

  • a) Draw a typical sample path (realization) of X t X t and of the received signal r t r t (do not forget to add a bit of noise!)
  • b) Assume that the receiver knows g⁢t g t . Design a matched filter for this transmission system.
  • c) Draw a typical sample path of Y t Y t , the output of the matched filter (do not forget to add a bit of noise!)
  • d) Find an expression (or draw) u⁢n⁢T u n T where u⁢t=s*g* h opt ⁢t u t s g h opt t .

Problem 3

Proakis and Salehi, problem 7.35

Problem 4

Proakis and Salehi, problem 7.39

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