Indian Institute of Technology Bhubaneswar
School of Electrical Sciences
Wireless and Mobile Communications
(EC6L020)
Assignment 1
1. Using the result y 2 ≥ (y − x)2 + x2, prove the following inequality for the
Gaussian-Q function.


 x2 
Q(x) < exp − 
2
2. We have a multi-input single-output (MISO) communication system with
M transmit antennas. The received symbol r can be represented as
r = hT 1M ×1 s + nT 1M ×1,
where h = [h1 , h2 , . . . , hM ]T , n = [n1, n2, . . . , nM ]T , and the vector 1M ×1 =
[1, 1, . . . , 1]T1×M . Furthermore, hi ∼ CN (0, σh2 ) and ni ∼ CN (0, σn2 ) for
i = 1, . . . , M. It is suggested that full diversity order can be achieved for
such a communication system.
(a) Is this statement true if the channel gains and the noise in the various
channels are all independent of each other?
(b) Is this statement true if E [hi hj ] = ǫ for i, j = 1, . . . , M, i 6= j and the
noise are independent in each of the channels and independent of the
channel gains?
3. In order to improve the performance of cellular systems, multiple base
stations can receive the signal transmitted from a given mobile unit and
combine these multiple signals either by selecting the strongest one or
summing the signals together, perhaps with some optimized weights.
This typically increases SNR and reduces the effects of shadowing and
fading. Combining of signals received from multiple base stations is
called macrodiversity, and here we explore the benefits of this technique.
Consider a mobile at the midpoint between two base stations in a cellular
network. The received signals from the base stations are given by
Pr,1 = W + Z1 , Pr,2 = W + Z2 ,
Please go on to the next page. . .
EC6L020
Wireless and Mobile Communications
Page 2 of 2
where Z1 , Z2 are N (0, σn2 ) random variables. We define outage with
macrodiversity to be the event that α1 Pr,1 +α2 Pr,2 falls below a threshold
T with α1 + α2 = 1.
(a) If Z1 and Z2 are independent, find the values of α1 and α2 such that
the received SNR is maximized.
(b) If Z1 and Z2 are independent, and α1 = α2 = 1, find the outage
probability Pout of the system .
(c) If Z1 and Z2 are dependent with E [Z1 Z2] = ρσn2 , find the values of
α1 and α2 such that the received SNR is maximized.
(d) If Z1 and Z2 are dependent with E [Z1 Z2 ] = ρσn2 , and α1 = α2 = 1,
find the outage probability Pout of the system .
4. A Gaussian random vector X = [X1 , X2]T is distributed as N µ, K ,
where




−4
2
−1
 .
µ=  , K=
3
−1 2
We wish to obtain from X a zero-mean white random vector Y = [Y1, Y2]T
with co-variance matrix I 2 such that Y = A X + b. Find A and b using
(a) eigendecomposition
(b) causal whitening
5. Derive the probability of error of a SIMO system employing EGC technique
at the receiver for the wireless communication system as discussed in the
class.
6. Obtain the received symbol for a 2 × 3 MIMO wireless communication
system using MRC and O-STBC techniques such that the system has
full diversity.
End of assignment