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

470 RANDOM PROCESSES AND SPECTRAL ANALYSIS
Example 9.3
Determine the PSD and the mean square value of a random process
x(t) = A cos (w e t+ E>)
where E> is an RV uniformly distributed over (0, 2n).
For this case R x ('r) is already determined [Eq. (9.7c)],
A 2
R x (i) = 2
COS W e i
(9.21a)
(9.21b)
Hence,
A 2
S x lf) = 4
[8(j +f c ) + 8(j - f e )]
- A 2
P x = x 2 =R x (O) = - 2
(9.21c)
(9.21d)
Thus, the power, or the mean square value, of the process x(t) = A cos (w e t+ E>) is A 2 /2.
The power P x can also be obtained by integrating S x (j) with respect to f.
Example 9.4
Amplitude Modulation
Determine the autocorrelation function and the PSD of the DSB-SC-modulated process
m(t) cos (w e t + E>), where m(t) is a wide-sense stationary random process, and E> is an
RV uniformly distributed over (0, 2n) and independent of m(t).
Let
Then
cp(t) = m(t) cos (w e t + E>)
R ,p
(i) = m(t) cos (w e t+ E>) · m(t + i) cos [w e (t + i) + E>]
Because m(t) and E> are independent, we can write [see Eqs. (8.64b) and (9.7c)]
R ,p
(i) = m(t)m(t + i) cos (w e t+ E>) cos [w c (t + i) + E>]
1
= 2
R m (i) COS W e i
(9.22a)
Consequently,*
From Eq. (9.22a) it follows that
1
S ,p
(J) = 4
[Sm lf + fe ) + Sm lf - fe)]
-- l . 1 - ­
rp 2 (t) = R ,p
(O) = 2
Rm (O) = 2
m 2 (t)
(9.22b)
(9.22c)
* We obtain the same result even if <p(t) = m(t) sin (w e t+ (0) .