The channels are independent but not identical, i.e., the fading coefficient of theithchannel hi ∼CN(0,σi2)

EE617Intro. to Wireless & Cellular Communications Tutorial 2 – February 3, 2012

1. A BPSK symbol (±a) is transmitted overLi.i.d Rayleigh fading channels. The fading coefficient of the i^th channel h_i ∼ CN(_0,σ²). Show that the average probability of error is given by

Pe = ¹ 2

"

1−µ

L−1 l

∑

=0

2l l

1−µ² 4

l#

whereµ =

q SNR

1+SNR andSNR = ^σ_N^2a2

0

2. A BPSK symbol (±a) is transmitted over L parallel Rayleigh fading channels. The channels are independent but not identical, i.e., the fading coefficient of thei^thchannel hi ∼_CN(0,σ_i²).

(a) What is the probability of error for a given channel realization?

(b) What is the diversity gain? Is it different for the case when all channels are i.i.d?

Argue using the deep fade event.

3. Consider the binary orthogonal signaling scheme (as discussed in class). One bit of information is transmitted during every block of two successive samples x[2m] and x[2m+₁]as follows: (x[2m]_,x[2m+₁]) = (a, 0)_or (x[2m]_,x[2m+₁]) = (_0,a)_{. This} scheme is used in 2 parallel i.i.d Rayleigh fading channels. The same bit is sent over both channels. The receiver does not know the channel realization in either channel.

Thus the channel model is given by

y₁[m] =h₁[m]x[m] +w₁[m] y₂[m] =h₂[m]x[m] +w₂[m] where

w₁,w2 ∼CN(0,N0) h₁,h₂ ∼CN(0, 1) Find the optimum decision rule at the receiver.