B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

608 PERFORMANCE ANALYSIS OF DIGITAL COMMUNICATION SYSTEMS
(b) If x(t) is the input to a linear time-varying system whose output is
r: y (t) = h(t, r)x(r) dr
show what kind of output process this generates, and determine the mean and the
autocorrelation function of the linear system output y(t).
10.5-4 Determine the output PSD of the linear system in part (a) of Prob. 10.5-3.
10.5-5 Determine the output PSD of the linear system in part (b) of Prob. 10.5-3.
10.6-1 Consider the preprocessing of Fig. 10.17. The channel noise nw (t) is white Gaussian.
(a) Find the signal energy of r(t) and q (t) over the finite time interval [0, TM ].
(b) Prove that although r(t) and q(t) are not equal, both contain all the useful signal content.
(c) Show that the joint probability density function of (q1 , q2, . .. , (1N ), under the condition
that sk (t) is transmitted, can be written as
10.6-2 Consider an additive white noise channel. After signal projection, the received N x 1 signal
vector is given by
q = S; +n
when message m; is transmitted. The noise vector n has joint probability density function
N
n I (-ln;I).
-exp --
. r (2r)
1=1
(a) Find the (MAP) detector that can minimize the probability of detection error.
(b) Follow the derivations of optimum detector for A WGN noise to derive the optimum receiver
structure for this non-Gaussian white noise channel.
(c) Show how the decision regions are different between Gaussian and non-Gaussian noises
in a two-dimensional (N = 2) signal space.
10.6-3 A binary source emits data at a rate of 400,000 bit/s. Multiamplitude shift keying (PAM) with
M = 2, 16, and 32 is considered. In each case, determine the signal power required at the
receiver input and the minimum transmission bandwidth required if S 11 (cv) = 1 o- 8 and the bit
error rate P b , is required to be less than I o- 6 .
10.6-4 Repeat Prob. 10.6-3 for M-ary PSK.
10.6-5 A source emits M equiprobablemessages, which are assigned signals s 1 , s2, ... , SM . as shown
in Fig. PI 0.6-5. Determine the optimum receiver and the corresponding error probability P eM
for an AWGN channel as a function of E b /N.