Problem 1. The received signal energy E is uniformly distributed in between [0 10] . Let the mean of E is 5.

a) Consider a particular communication system that the threshold is 1.5. If the signal energy falls below the threshold, the system is in the outage state. What is the outage probability P?

b) If there are two diversity branches (E1 and E2) and pure selection is the diversity selection algorithm what is the outage probability if E1= E2? (assume the outage probability of one branch is P, you don't have to know the right answer of part a)

c) If there are two diversity branches (E1 and E2) and pure selection is the diversity selection algorithm what is the outage probability if E1 =10-E2? (assume the outage probability of one branch is P, you don't have to know the right answer of part a)

Problem 2.

A white noise x(t) appears at the input of a transmitter (the power is turned on). Let the transmitter frequency response be H_r(ω). the channel frequency response be H_r(ω) and the receiver frequency response be We). What is the autocorrelation of the receiver output?

Problem 3

The received signal

r = hd + n

where h is the channel response. d is the data and n is the noise. If the channel response h is known to the receiver. use the MMSE approach to estimate data d. Show all detailed steps of derivation.

Problem 4

A channel (see the channel model below) is estimated using a training sequence u(k) as the input of the channel. Here. u(k) for all k's are independent random variables with Prob(u(k)=1)=0.5 and Prob(u(k)=1)=0.5. The correlations of the output v(t) of the channel at any time k with the input training sequence [ u(k) u(k-1) u(k-2).....] are

[ 3 2 1.2 2.1 0.3 0.1 0 0 0 .....]. i.e..

E{v(k)*[u(k) u(k-1) u(k-2)]} = [3 2 1.2 2.1 0.3 0.1 0 0 ...]

What is the number of taps and what are the tap weights of the channel?

Problem 5

Consider a 3-tap channel as an example (Φ = 3). If the noise is ignored. the input and output relation can be written as

h₁u₁ = v₁;

h₂u₂ + h₂u₁ = v₂

h₁u₃ +h₂u₂ +h₃u₁ =v₃

h₁u₄ +h₂u₃ +h₃u₂ = v₄

...

h₁u_k +h₂u_k-1 +h₃u_k-2 = v_k

Find e₁, e₂, f₁ and f₂ such that the following equalizer will restore the input data.