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

668 DIGITAL COMMUNICATIONS UNDER LINEARLY DISTORTIVE CHANNELS
To generalize, if there are K + 1 different paths, then the effective channel response is
K
q(t) = p(t) + L [a; cos W e i;] p(t - r;)
i=l
in which the line-of-sight path delay is assumed to be ro = 0 with unit path gain ao = 1.
The ISi effect caused by the K summations in q(t) depends on (a) the relative strength of the
multipath gains {a;}; and (b) the multipath delays {r;}.
General QAM Models
For conserving bandwidth in both wire-line and wireless communications, QAM is an efficient
transmission. We again let the QAM symbol rate be 1/T and its symbol duration be T. Under
QAM, the data symbols {sk } are complex-valued, and the quadrature bandpass RF signal
transmission is
s(t) = [ Re{sdp(t - kT)] cos W e t + [
lm{sdp(t - kT)] sin W e t
(12.3)
Thus, under multipath channels with K + l paths and impulse response
the received bandpass signal for QAM is
o(t) + I: a;o(t - r;)
i=l
K
r(t) = s(t) + L a;s(t - r;) + n, (t) cos W e t + ns (t) sin W e t
i=I
(12.4)
Applying coherent detection, the QAM demodulator has two baseband outputs
LPF {2r(t) cos W e t} and LPF{2r(t) sin W e t}. These two (in-phase and quadrature) outputs are
real-valued and can be written as a single complex-valued output:
y(t) = LPF{2r(t) cos w e t} + j · LPF{2r(t) sin w e t}
(12.5a)
= L Re{sd [t(a; cos W e T;)p(t - kT - r;)]
k
i=O
+
lm{sk } [t(a; sin W e r;)p(t - kT - r;)]
- j · L Re{sd [ t (a; sin w e r;)p(t - kT - r;)]
k
i=O
+ j · L Im{sk } [t (a; cos W e r;)p(t - kT - r;)] + ne(t) + j ns(t)
k i=O
= LSk [ t a; exp(-Jwer;)p(t - kT - r;)] + ne(t) + J ns (t) (12.5b)
k i=O