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

9.3 Power Spectral Density 471
Hence, the power of the DSB-SC-modulated signal is half the power of the modulating
signal. We derived the same result earlier [Eq. (3.93)] for deterministic signals.
We note that, without the random phase 0, a DSB-SC amplitude-modulated signal
m(t) cos (w e t) is in fact not wide-sense stationary. To find its PSD, we can resort to the
time autocorrelation concept of Chapter 3.
Example 9.5
Random Binary Process
In this example we shall consider a random binary process for which a typical sample
function is shown in Fig. 9. 9a. The signal can assume only two states (values), 1 or - 1,
with equal probability. The transition from one state to another can take place only at node
points, which occur every T b seconds. The probability of a transition from one state to the
other is 0.5. The first node is equally likely to be situated at any instant within the interval
0 to T b from the origin. Analytically, we can represent x(t) as
x(t) = I>np(t - nT b - a)
n
Fi g
ure 9.9
Derivation of
autocorrelation
function and PSD
of a random
binary process.
I
t--+
T
(a)
(b)
-T b
T b 't:--+
-47T -27T 0 21T w-
(c)
T b
T b
I
I I I I
-2R b
-R b
0 R b
2R b J-
where a is an RV uniformly distributed over the range (0, T b ) and p(t) is the basic pulse
(in this case TI[(t - T b /2)/T b D- Note that a is the distance of the first node from the
origin, and it varies randomly from sample function to sample function. In addition, a n is
random, taking values 1 or - 1 with equal probability. The amplitudes at t represent RV x 1 ,
and those at t + r represent RV x2. Note that x 1 and x2 are discrete and each can assume only
(d)
T b
T b