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

12.1 Linear Distortions of Wireless Multipath Channels 667
Figure 12.1
Simple illustration
of a
two-ray multipath
channel.
Base station
To understand the effect of multipath in this example, we denote the line-of-sight signal
arrival and the reflective arrival, respectively, as
s(t) = m(t) cos Wet
and
Here we assumed that the modulation is DSB with PAM message signal (Chapter 7)
m(t) = L akp(t - kT)
where Tis the PAM symbol duration. Note also that we use a1 and r1, respectively, to represent
the multipath loss and the delay relative to the line-of-sight signal. Hence, the receiver RF input
signal is
r(t) = m(t) cos Wet + a1 m(t - ri) cos W c (t - r1) + n e (t) cos Wet + n s (t) sin w e t (12. 1)
In Eq. (12. 1), n e (t) and n s (t) denote the in-phase and quadrature components of the bandpass
noise, respectively (Sec. 9.9). By applying coherent detection, the receiver baseband output
signal becomes
y(t) = LPF{2r(t) cos wet}
= m(t) + a1 (cos Wer1)m(t - r1) + n e (t)
= L akp(t - kT) + (a1 · cos W e r1) L akp(t - kT - r1) + nc (t)
k
k
= L ak [p(t - kT) + (a1 cos cvcr1 ) p(t - kT - r1)] + nc (t)
k
(12.2a)
( 12.2b)
By defining a baseband waveform
we can simplify Eq. (12.2b)
q(t) = p(t) + (a1 cos W e r1)p(t - r1)
y(t) = L ak q(t - kT) + n e (t)
k
(12.2c)
Effectively, this multipath channel has converted the original pulse shape p(t) into q(t). If p(t)
was designed (as in Chapter 7) to satisfy Nyquist's first criterion of zero ISi,
p(nT) = { 1 n = 0
0 n = ±1, ±2, ...
then the new pulse shape q(t) will certainly have ISi as
q(nT) = p(nT) + (a1 · cos We r1)p(nT - r1) f. 0 n = ±1, ±2, ...