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

9.3 Power Spectral Density 469
It is helpful to repeat here, once again, that the PSD may not exist for processes that are
not wide-sense stationary. Hence, in our future discussion, random processes will be assumed
to be at least wide-sense stationary unless specifically stated otherwise.
Example 9 .2
Determine the autocorrelation function R x (r) and the power P x of a low-pass random process
with a white noise PSD Sx (f) = N /2 (Fig. 9.8a).
Figure 9.8
Bandpass white
noise PSD and its
autocorrelation
function.
1-
-4
2B 2B 2B
2B
3 4
2B 2B 2B 2B
7--,.
(b)
We have
Sx (f) = TI (; B
)
(9.20a)
Hence, from Table 3.1 (pair 18),
Rx (r) = NB sine (2nBr)
(9.20b)
This is shown in Fig. 9.8b. Also,
P x
=
X 2 = R x (O) = NB
(9.20c)
Alternately,
P x = 2 fo 00 Sx if) df
= 2 { B N df
l o 2
=
NB
(9.20d)