563489578934

Problems 489
6–54 A narrowband-signal-plus-Gaussian noise process is represented by the equation
r(t) = A cos(v c t + u c ) + x(t) cos(v c t + u c ) - y(t) sin(v c t + u c )
where A cos (v c t + u c ) is a sinusoidal carrier and the remaining terms are the bandpass noise with
independent Gaussian I and Q components. Let the noise have an RMS value of s and a mean
of 0. The signal-plus-noise process appears at the input to an envelope detector. Show that the
PDF for the output of the envelope detector is
R +A 2 )/(2s 2 )] RA
f(R) = c s 2 e-[(R2 I 0 P
s 2 Q , R Ú 0
0, R 6 0
where
I 0 (z) ! 1
2p L0
2p
e z cos u du
is the modified Bessel function of the first kind of zero order. f(R) is known as a Rician PDF in
honor of the late S. O. Rice, who was an outstanding engineer at Bell Telephone Laboratories.
6–55 Assume that
is a known signal. This signal plus white noise is present at the input to a matched filter.
(a) Design a matched filter for s(t). Sketch waveforms analogous to those shown in Fig. 6–16.
(b) Sketch the waveforms for the correlation processor shown in Fig. 6–18.
★ 6–56 A baseband digital communication system uses polar signaling with a bit rate of R = 2,000 bits/s.
The transmitted pulses are rectangular and the frequency response of channel filter is
where B = 6,000 Hz. The filtered pulses are the input to a receiver that uses integrate-and-dump
processing, as illustrated in Fig. 6–17. Examine the integrator output for ISI. In particular,
(a) Plot the integrator output when a binary “1” is sent.
(b) Plot the integrator output for an all-pass channel, and compare this result with that obtained
in part (a).
6–57 Refer to Fig. 6–19 for the detection of a BPSK signal. Suppose that the BPSK signal at the
input is
where
A
s(t) = c T t cos v ct, 0 … t … T
0, t otherwise
H c (f) =
r(t) = s(t) = a
7
p(t) = e e-t cos v c t, 0 6 t 6 T
0, t otherwise
and the binary data d n is the 8-bit string {+1, -1, -1, +1, +1, -1, +1, -1}. Use MATLAB to
n = 0
B
B + jf
d n p(t - nT)