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

714 DIGITAL COMMUNICATIONS UNDER LINEARLY DISTORTIVE CHANNELS
When the mobile speed vd is not zero, then {J; (t) are time-varying. As a result, the channel
is no longer linear time-invariant. Instead, the channel is linear time-varying. Suppose the
channel input is a pure sinusoid, x(t) = exp(Jcu p
t). The output of this time-varying channel
according to Eq. (12.77) is
K
K
L {J; (t) exp Ucu p
(t - r;)] = exp(Jcu p
t) · L {J; (t) exp(-}cu p r;)
i=O
(12.80)
This relationship shows that the channel response to a sinusoidal input equals a sinusoid of
the same frequency but with time-varying amplitude. Moreover, the time-varying amplitude
of the channel output also depends on the input frequency (cu p ). For these multipath channels,
the channel response is time-varying and is frequency dependent. In wireless communications,
time-varying channels are known as fading channels. When the time-varying behaviors
are dependent on frequency, the channels are known as frequency-selective fading channels.
Frequency-selective fading channels, which are characterized by time-varying ISi, are major
obstacles to wireless digital communications.
Flat Fading Channels
One special case to consider is when the multipath delays { r;} do not have a large spread. In
other words, let us assume
0 = ro < T] < · · · < TK
If the multipath delay spread is small, then TK « T and
T; 0 i = l, 2, . .. , K
In this special case, because p(t - r;) p(t), the received signal y(t) is simply
It y(t) = Sk a; exp [-}(cue + cu;)r;] exp (-jcu;t) p(t - kT - r;) )
L sk \ta; exp [-}(cue + cu;)r;] exp (-jcu;t) p(t - kT) )
k i=O
= \ta; exp [-}(cue + cu;)r;] exp (-jcu;t) ) L skp(t - kT)
i=O
k
= p(t) · L akp(t - kT)
k
where we have defined the time-varying channel gain as
K
p(t) = La; exp [-}(cue + cu;)r;] exp (-jcu;t)
i=O
(12.81)
(12.82)
Therefore, when the multipath delay spread is small, the only distortion in the received
signal y(t) is a time-varying gain p(t). This time-variation of the received signal strength