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

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P. 10.6·5
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Problems 609
10.6-6 A source emits eight equiprobable messages, which are assigned QAM signals s 1 , s2 • ... , sg,
as shown in Fig. Pl 0.6-6.
(a) Find the optimum receiver for an AWGN channel.
(b) Determine the decision regions and the error probability P eM of the optimum receiver as
a function of E b .
Figure
P. 10.6·6
10.6-7 Prove that for E b /N » 1 and M »
for MPSK holds.
2, the error probability approximation of Eq. (10.109b)
10.6-8 Use the approximation ofEq. (10.109b) for 16 PSK to compare the symbol error probabilities of
16-QAM and 16-PSK. Show approximately how many decibels of E i, / N (SNR) loss 16-PSK
incurs versus 16-QAM (by ignoring the constant difference in front of the Q- function).
10.6-9 Compare the symbol error probabilities of 16-PAM, 16-PSK, and 16-QAM. Sketch them as
functions of E h / N.
10.6-10 Show that for MPSK, the optimum receiver of the form in Fig. 10.19a is equivalent to a phase
comparator. Assume all messages equiprobable and an AWGN channel.
10.6-11 A ternary signaling has three signals for transmission:
m 0 : 0, m1 : 2p(t), m2 : -2p(t).
(a) If P(m 0 ) = P(m1 ) = P(m2) = 1/3. detem1ine the optimum decision regions and P eM of
the optimum receiver as a function of E. Assume an AWGN channel.
(b) Find P eM as a function of E/N.
(c) Repeat parts (a) and (b) if P(m 0 ) = 1/2 and P(m1 ) = P(m2) = 0.25,
10.6-12 A 16-ary signal configuration is shown in Fig. Pl0.6-12. Write the expression (do not evaluate
various integrals) for the P eM of the optimum receiver, assuming all symbols to be
equiprobable. Assume an AWGN channel.
10.6-13 A five-signal configuration in a two-dimensional space is shown in Fig. Pl0.6-13.
(a) Choose the 'Pl (t) = ,.f2ff-; cos w e t and <p2 (t) = ✓2/T 0 sin W e t and sketch the waveforms
of the five signals.