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Sec. 4–5 Bandpass Filtering and Linear Distortion 251
|H(f)|
Signal
bandwidth
A
f c
f
(a) Magnitude Response
u(f)
u 0
f c
f
(b) Phase Response
Figure 4–4
Transfer characteristics of a distortionless bandpass channel.
Now it will be shown that Eqs. (4–27a) and (4–27b) are sufficient requirements for
distortionless transmission of bandpass signals. From Eqs. (4–27a) and (4–28), the channel
(or filter) transfer function is
H(f) = Ae j(-2pfT g+ u 0 ) = (Ae ju 0
)e -j2pfT g
(4–29)
over the bandpass of the signal. If the input to the bandpass channel is represented by
v 1 (t) = x(t) cos v c t - y(t) sin v c t
then, using Eq. (4–29) and realizing that
of the channel is
e -j2pfT g
causes a delay of T g , we find that the output
v 2 (t) = Ax(t - T g ) cos[v c (t - T g ) + u 0 ] - Ay(t - T g ) sin[v c (t - T g ) + u 0 ]