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

3.4 Signal Transmission Through a Linear System 93
Figure 3.25
Linear
time-invariant
system frequency
response for
distortion less
transmission.
o • • ·-
• • ·--------
• •
_____ ; :,n
--
But because
Y (J) = X (J)H (J)
we therefore have
H(J) = k e - J 2 :rcftd
This is the transfer function required for distortionless transmission. From this equation it
follows that
IH(J)I = k
81, (f) = -2nftd
(3.57a)
(3.57b)
This shows that for distortionless transmission, the amplitude response IH(J)I must be a
constant, and the phase response 81,(f) must be a linear function off going through the origin
f = 0, as shown in Fig, 3.25. The slope of0h(J) with respect to the angular frequency w = 2nf
is -td , where tJ is the delay of the output with respect to the input.*
All-Pass vs. Distortionless System
In circuit analysis and filter designs, we sometimes are mainly concerned with the gain of a
system response_ An all-pass system has a constant gain for all frequencies [i.e., IH (J)I = k],
without the linear phase requirement. Note from Eq. (3.57) that a distortionless system is
always an all-pass system, whereas the converse is not true. Because it is very common for
beginners to be confused by the difference between all-pass and distortionless systems, now
is the best time to clarify.
To see how an all-pass system may lead to distortion, let us consider an illustrative example.
Imagine that we would like to transmit a recorded music signal from a violin-cello duet. The
violin contributes to the high frequency part of this music signal, while the cello contributes to
the bases part. When this music signal is transmitted through a particular all-pass system, both
parts have the same gain. However, suppose that this all-pass system would cause a I -second
extra delay on the high-frequency content of the music (from the violin). As a result, the
audience on the receiving end will hear a "music" signal that is totally out of sync even though
all signal components have the same gain and all are present. The difference in transmission
delay for components of different frequencies is contributed by the nonlinear phase of H (J)
in the all-pass filter.
* In addition, we require that 0 h (O) either be O (as shown in Fig. 3.25) or have a constant value n:rc (n an integer),
that is, e h (f) = mr - 2:rcft d . The addition of the excess phase of n:rc may at most change the sign of the signal.