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

604 PERFORMANCE ANALYSIS OF DIGITAL COMMUNICATION SYSTEMS
10.1-2 An alternative to the optimum filter is a suboptimum filter, where we assume a particular filter
form and adjust its parameters to maximize p. Such filters are inferior to the optimum filter but
may be simpler to design.
For a rectangular pulse p(t) ofheightA and width T b at the input (Fig. Pl0.1-2), determine
Pmax if, instead of the matched filter, a one-stage RC filter with H (cv) = 1 / (1 + jcvRc) is used.
Assume a white Gaussian noise of PSD N /2. Show that the optimum performance is achieved
when 1/RC = 1.26/T b .
Hint: Set dp 2 /dx = 0 (x = T b / RC).
Figure
P. 10.1-2
pt)
o
l o
(a) (b) (c)
10.2-1 In coherent detection of a binary PPM, a half-width pulse Po(t) of is transmitted with different
delays for binary digit "O" and " l" over O ::: t ::: T b . Note that
Po(t) = u(t) - u(t - T b /2)
The binary PPM transmission is to simply transmit
Po (t),
Po (t - T b /2),
if "O" is sent
if "1" is sent
The channel noise is AWGN with spectrum level of N /2.
(a) Determine the optimum receiver architecture for this binary system. Sketch the optimum
receiver filter respouse in the time domain.
(b) If P("O"] = 0.4 and P["l"] = 0.6, find the optimum threshold and the resulting receiver
bit error rate.
(c) The receiver was misinformed and believes that P["O"] = 0.5 = P["l"]. It hence designed
a receiver based on this information. Find the true probability of error when, in fact, the
actual prior probabilities are P["0"] = 0.4 and P["l "] = 0.6. Compare this result with the
result in part (b).
10.2-2 In the coherent detection of binary chirp modulations, the transmission over O ::: t ::: T b is
A cos(aot 2 + 0 o ).
Acos(a 1 t 2 + 0 1 ),
if "O" is sent
if "l" is sent
The channel noise is AWGN with spectrum N /2. The binary digits are equally likely.
(a) Design the optimum receiver.
(b) Find the probability of bit error for the optimum receiver in part (a).
10.2-3 In coherent schemes, a small pilot is added for synchronization. Because the pilot does not
carry information, it causes degradation in Ph. Consider a coherent PS K that uses the following
two pulses of duration T h :
p(t) = A J I - m 2 cos W e t+ Am sin W e t
q(t) = -A J 1 - m 2 cos W e t+ Am sin (v e t