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Sec. 2–6 Review of Linear Systems 87
which implies that, to have no distortion at the output of a linear time-invariant system, two
requirements must be satisfied:
1. The amplitude response is flat. That is,
|H(f)| = constant = A
(2–150a)
2. The phase response is a linear function of frequency. That is,
u(f) = lH(f) = -2pfT d
(2–150b)
When the first condition is satisfied, there is no amplitude distortion. When the second
condition is satisfied, there is no phase distortion. For distortionless transmission, both conditions
must be satisfied.
The second requirement is often specified in an equivalent way by using the time delay.
We define the time delay of the system as
T d (f) = -
By Eq. (2–149), it is required that
1
2pf u(f) = - 1
2pf lH(f)
T d (f) = constant
(2–151)
(2–152)
for distortionless transmission. If T d (f) is not constant, there is phase distortion, because the
phase response, u(f), is not a linear function of frequency.
Example 2–18 DISTORTION CAUSED BY A FILTER
Let us examine the distortion effect caused by the RC low-pass filter studied in Example 2–17.
From Eq. (2–145), the amplitude response is
1
|H(f)| =
(2–153)
31 + (f/f 0 ) 2
and the phase response is
u(f) = lH(f) = -tan -1 (f/f 0 )
(2–154)
The corresponding time delay function is
1
T d (f) =
(2–155)
2pf tan -1 (f/f 0 )
These results are plotted in Fig. 2–16, as indicated by the solid lines. See Example2_18.m
for detailed plots. This filter will produce some distortion since Eqs. (2–150a) and (2–150b) are
not satisfied. The dashed lines give the equivalent results for the distortionless filter. Several
observations can be made. First, if the signals involved have spectral components at frequencies
below 0.5f 0 , the filter will provide almost distortionless transmission, because the error
in the magnitude response (with respect to the distortionless case) is less than 0.5 dB and the
error in the phase is less than 2.1 (8%). For f 6 f 0 , the magnitude error is less than 3 dB and