ECE 8700 Communication System Engineering, Spring 2009
Homework Set # 3
Suggested Problems from the Text
2.38,46,47,52 (random processes);
3.1-6 (PAM, PSK, QAM)
Homework # 3 (Due Wed., Feb. 11 before class): (Do all. Submit problems 4, 5, 7,
8.)
1. Binary Communications: Consider receiving a binary symbol in additive Gaussian
noise. Let the two transmitted symbols be denotes as Otand 1t. The received real-
valued random variable, from which a decision is to be made, is denoted as R. Condi-
tioned on the transmitted symbol, it has Gaussian PDF’s
pR(r/0t) = 1
√2π0.09 e−r2/0.18 (1)
pR(r/1t) = 1
√2π0.09 e−(r−0.8)2/0.18 (2)
Assume that P(0t) = P(1t) = 0.5. Let Orand 1rrepresent the received symbols (i.e.
the symbols decided on at the receiver).
(a) Using a detection threshold (on R) of value T= 0.4, determine the probability of
making a bit error, P(e).
(b) Using a detection threshold (on R) of value T= 0.5, determine P(1r/1t), P(0r/0r),
P(0r) and P(e).
2. Consider Gaussian random vector X= [X1, X2, X3]Twith mean vector
mx= [1,2,3]Tand covariance matrix
Cx=
σ11 0σ13
0σ22 0
σ31 0σ33
.(3)
Consider a new random vector
Y=
100
020
101
X(4)
and random variable Z= [1,1,1] Y. Determine the expression for PDF of Z(this
will be in terms of the σij ).
3. Weighted Sum of Multiple Random Variables: Consider four statistically independent
random variables ri;i= 1,2,3,4 with PDF’s
p(ri) = 1
q2πσ2
i
e−(ri−si)2/2σ2
i(5)
1
