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

Problems 139
the autocorrelation function and the PSD are given by
1 oo
R y
(r) = ca 2 + 2 L C n
2 cos n2nfo r:
n=l
00
2 I 2
Sy (f) = Co 8(f) +
4 L C n [8(f - nfo ) + 8(f + nfo)]
n=l
Hint: Show that if g(t) = g1 (t)+ g2 (t), then R g (r:) = R g1 (r:)+ R g2 (r:) + R g1 g2 (r:)+ R g2g1 (r:).
where R g1 g2 (r:) = limr➔ oo O/T) J!{ 2 g1 (t)g2 (t + r:) dt. If g1 (t) and g2 (t) represent any
two of the infinite terms in y(t), then show that R g1g2
(r:) = R g2g1
(r:) = 0. To show this, use
the fact that the area under any sinusoid over a very large time interval is at most equal to the
area of the half-cycle of the sinusoid.
3.8-2 The random binary signal x(t) shown in Fig. P3.8-2 transmits one digit every T b seconds. A
binary 1 is transmitted by a pulse p(t) of width T b /2 and amplitude A; a binary O is transmitted by
no pulse. The digits 1 and O are equally likely and occur randomly. Determine the autocorrelation
function Rx(r) and the PSD Sxif).
Figure P.3.8-2
x(t)
1 0 0 1 0 1 0 1 0
D D I D D D D
t-
3.8-3 Find the mean square value (or power) of the output voltage y(t) of the RC network shown
in Fig. P3.5-4 with RC = 2n if the input voltage PSD Sxif) is given by (a) K; (b) IT(nf );
(c) [8(f + 1) + 8(f - I)]. In each case calculate the power (mean square value) of the input
signal x(t).
3.8-4 Find the mean square value (or power) of the output voltage y(t) of the system shown in Fig. P3.8-
4 if the input voltage PSD Sxif) = IT (rrf). Calculate the power (mean square value) of the
input signal x(t).
Figure P.3.8-4