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

Problems 801
13.4-5 A cascade of two channels is shown in Fig. Pl3.4-5. The symbols at the source, at the output of
the first channel, and at the output of the second channel are denoted by x, y. and z. Show that
and
H (xlz) ::0: H (xly)
J(x; y) ::O: J(x; z)
This shows that the information that can be transmitted over a cascaded channel can be no greater
than that transmitted over one link. In effect, information channels tend to leak information.
Hint: For a cascaded channel, observe that
Hence, by Bayes' rule,
Figure
P. 13.4-5
X
y
z
13.5-1 For a continuous random variable x constrained to a peak magnitude M (-M < x < M),
show that the entropy is maximum when x is uniformly distributed in the range (-M,M)
and has zero probability density outside this range. Show that the maximum entropy is given
by log 2M .
13.5-2 For a continuous random variable x constrained to only positive values O < x < oo and a mean
value A, show that the entropy is maximum when
Show that the corresponding entropy is
H (x) = log eA
13.5-3 A television transmission requires 30 frames of 300,000 picture elements each to be transmitted
per second. Use the data in Prob. 13.1-2 to estimate the theoretical bandwidth of the AWGN
channel if the SNR at the receiver is required to be at least 50 dB.
13.7-1 ln a communication system over a frequency-selective channel with transfer function
the input signal PSD is
H ) (f
= I+ jn(f /200)
The channel noise is AWGN with spectrum Sn (f) = I o- 2 . Find the mutual information between
the channel input and the channel output.