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

Problems 605
where Am sin wet is the pilot. Show that when the channel noise is white Gaussian,
Hint: Use Eq. (I 0.25b).
10.2-4 For polar binary communication systems, each error in the decision has some cost. Suppose
that when m = 1 is transmitted and we read it as m = 0 at the receiver, a quantitative penalty,
or cost, C10 is assigned to such an error, and, similarly, a cost Co 1 is assigned when m = 0 is
transmitted and we read it as m = 1. For the polar case where P m (0) = P m (1) = 0.5, show
that for white Gaussian channel noise, the optimum threshold that minimizes the overall cost
is not O but is a 0 , given by
Hint: See Hint for Prob. 8.2-11.
a 0 = N ln C0 1
4 C10
10.2-5 For a polar binary system with unequal message probabilities, show that the optimum decision
threshold a 0 is given by
N Pm (0)Co1
a 0 =-ln ----
4 Pm (l)C10
where Co1 and C10 are the cost of the errors as explained in Prob. 10.2-4, and P m (0) and Pm (l)
are the probabilities of transmitting O and 1, respectively.
Hint: See Hint for Prob. 8.2-11.
10.2-6 For 4-ary communication, messages are chosen from any one of four message symbols, m1 =
00, m2 = 01, m3 = 10, and m4 = 11. which are transmitted by pulses ±p(t), 0, and ±3p(t).
respectively. A filter matched to p(t) is used at the receiver. Denote the energy of p(t) as Ep .
The channel noise is AWGN with spectrnm N /2.
(a) If r is the matched filter output at t m , plotpr(rlmJ (00, 01, 10, and 11) for the four message
symbols, assuming that all message symbols are equally likely.
(b) To minimize the probability of detection error in part (a), determine the optimum decision
thresholds and the corresponding error probability P e as a function of the average symbol
energy to noise ratio.
10.2-7 Binary data is transmitted by using a pulse p(t) for 0 and a pulse yp(t) for 1. Let y > 1. Show
that the optimum receiver for this case consists of a filter matched to p(t) plus a detection
threshold as shown in Fig. Pl0.2-7. Determine the error probability P b of this receiver as a
function of Eb/ N if O and 1 are equiprobable.
Figure
P. 10.2-7
Decision:
0 if r < threshold
1 if r > threshold