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

94 ANALYSIS AND TRANSMISSION OF SIGNALS
To be more precise, the transfer function gain /H (f) I determines the gain of each input
frequency component, whereas LH (f) determines the delay of each component. Imagine a
system input x(t) consisting of multiple sinusoids (its spectral components). For the output
signal y(t) to be distortionless, it should be the input signal multiplied by a gain k and delayed
by tJ . To synthesize such a signal, y(t) needs exactly the same components as those of x(t),
with each component multiplied by k and delayed by tJ . This means that the system transfer
function H(f) should be such that each sinusoidal component encounters the same gain (or
loss) k and each component undergoes the same time delay of tJ seconds. The first condition
requires that
IH(f)I = k
We have seen earlier (Sec. 3.3) that to achieve the same time delay tJ for every frequency
component requires a linear phase delay 2rrftd (Fig. 3.18) through the origin
In practice, many systems have a phase characteristic that may be only approximately
linear. A convenient method of checking phase linearity is to plot the slope of LH (f) as a
function of frequency. This slope can be a function off in the general case and is given by
(3.58)
If the slope of eh is constant (that is, if eh is linear with respect to f), all the components
are delayed by the same time interval tJ . But if the slope is not constant, then the time delay
tJ varies with frequency. This means that different frequency components undergo different
amounts of time delay, and consequently the output waveform will not be a replica of the
input wavefonn (as in the example of the violin-cello duet). For a signal transmission to be
distortionless, t J (f) should be a constant t J over the frequency band of interest.*
Thus, there is a clear distinction between all-pass and distortionless systems. It is a common
mistake to think that flatness of amplitude response /H (f) I alone can guarantee signal quality.
A system that has a flat amplitude response may yet distort a signal beyond recognition if the
phase response is not linear Ctc1 not constant).
The Nature of Distortion in Audio and Video Signals
Generally speaking, a human ear can readily perceive amplitude distortion, although it is
relatively insensitive to phase distortion. For the phase distortion to become noticeable, the
* Figure 3.25 shows that for distortionless transmission, the phase response not only is linear but also must pass
through the origin. This latter requirement can be somewhat relaxed for bandpass signals. The phase at the origin
may be any constant [0h (fl = 0 o - 2:rcft d or 0 h (O) = 0 o J. The reason for this can be found in Eq. (3.37), which
shows that the addition of a constant phase 0 o to a spectrum of a bandpass signal amounts to a phase shift of the
carrier by 0 o . The modulating signal (the envelope) is not affected. The output envelope is the same as the input
envelope delayed by
1 d0 1, (f)
2:rc df
called the group delay or envelope delay, and the output carrier is the same as the input carrier delayed by
tg = - - --
0 h lf)
t p
= - 2:rcf
called the phase delay. where Jo is the center frequency of the passband.