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

492 RANDOM PROCESSES AND SPECTRAL ANALYSIS
This can be proven by considering the system in Fig. 9.2Oa, where H o(f) is an ideal
low-pass filter (Fig. 9.2Ob) with unit impulse response ho(t). First we show that the system
in Fig. 9.20a is an ideal bandpass filter with the transfer function H (f) shown in Fig. 9.2Oc.
This can be conveniently done by computing the response h(t) to the unit impulse input 8(t).
Because the system contains time-varying multipliers, however, we must also test whether it
is a time-varying or a time-invariant system. It is therefore appropriate to consider the system
response to an input 8(t - a). This is an impulse at t = a. Using the fact that [see Eq. (2.l Ob)]
f (t) 8 (t - a) = f (a )8 (t - a), we can express the signals at various points as follows:
Signal at a1 : cos (w e a + 0) 8(t - a)
a2 : sin (w e a + 0) 8(t - a)
bi : cos (w e a + 0)ho(t - a)
b2 : sin (w c a + 0)ho (t - a)
CJ : cos (w c a + 0) cos (w e t+ 0)ho(t - a)
c2 : sin (w c a + 0) sin (w e t + 0)ho(t - a)
d : ho(t - a) [cos (w c a + 0) cos (w e t+ 0) + sin (w e a + 0) sin (w e t + 0)]
= 2ho(t - a) cos [w c (t - a)]
Figure 9.20
(a) Equivalent
circuit of an
ideal bandpass
filter. (b) Ideal
low-pass filter
frequency
response.
(c) Ideal
bandpass filter
frequency
response.
x(t)
2 cos (w e
t+ 0) COS (w e t + 0)
a 1
a 2
Ideal
low-pass x c (t)
filter
HoU)
b i
Ideal
low-pass
filter
HoU)
x.(t)
b 2
t
CJ
C2
l
y(t)
d
2 sin (w e t + 0) sin (wJ + 0)
(a)
-B f -
(b)
H(f)
r2B 1
-.fc fc f-
(c)