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

9.9 Bandpass Random Processes 493
Thus, the system response to the input 8(t - a) is 2ho(t - a) cos [w c (t - a)]. Clearly, this
means that the underlying system is linear time invariant, with impulse response
and transfer function
h(t) = 2ho(t) cos W e t
H (f) = Ho (f +J c ) + Ho (f - Jc)
The transfer function H(f) (Fig. 9.20c) represents an ideal bandpass filter.
If we apply the bandpass process x(t) (Fig. 9.19) to the input of this system, the output
y(t) at d will remain the same as x(t). Hence, the output PSD will be the same as the input
PSD
IH(f) l 2 S xlf) = S xlf)
If the processes at points b1 and b2 (low-pass filter outputs) are denoted by X c (t) and X s (t),
respectively, then the output x(t) can be written as
x(t) = X c (t) cos (w e t + 0) + X s (t) sin (w e t+ 0) (9.66)
where X c (t) and X s (t) are low-pass random processes band-limited to B Hz (because they are
the outputs of low-pass filters of bandwidth B). Because Eq. (9.66) is valid for any value of 0,
by substituting 0 = 0, we get the desired representation in Eq. (9.65).
To characterize X c (t) and X s (t), consider once again Fig. 9.20a with the input x(t). Let
0 be an RV uniformly distributed over the range (0, 2rr), that is, for a sample function, 0 is
equally likely to take on any value in the range (0, 2rr ). In this case x(t) is represented as in
Eq. (9.66). We observe that X c (t) is obtained by multiplying x(t) by 2 cos (w e t + 0), and then
passing the result through a low-pass filter. The PSD of 2x(t) cos (w e t + 0) is [see Eq. (9.22b)]
This PSD is Sx (f) shifted up and down by J c , as shown in Fig. 9.21a. When this is passed
through a low-pass filter, the resulting PSD of X c (t) is as shown in Fig. 9.21 b. It is clear that
1/1 S B
1/1 > B
(9.67a)
Figure 9.21
Derivation of
PSDs of
quadrature
components of a
bandpass
random process.
-B 0
(a)
B
2fc
f -----
SxJf) or Sx,(f)
---..
----
-B 0
(b)
B
f -----