Multipath Signal Model
We now consider a more general multiple-access signal model
where the users are asynchronous, and the channel exhibits multipath distortion
effects. In particular, the multipath channel impulse response of the kth user is modeled as in (1.10):
Equation 2.165
where L is the total number of
paths in the channel, and al,k and
tl,k are, respectively, the complex path gain and
the delay of the kth user's lth path, t1,k <
t2,k < ··· < tL,k. The
continuous-time signal received in this case is given by
Equation 2.166
where * denotes convolution and
sk(t) is the spreading waveform of the
kth user given by (2.2).
At the receiver, the received signal r(t) is filtered by a chip-matched filter and sampled
at a multiple (p) of the chip rate (i.e., the
sampling time interval is D = Tc/p = T/P, where 
is the
total number of samples per symbol interval). Let
be the maximum delay spread in terms of symbol intervals.
Substituting (2.2) into (2.166), the qth signal
sample during the ith symbol interval is given
by!
Equation 2.167
where 
. Denote
Then (2.167) can be
written in terms of vector convolution as
Equation 2.168
By stacking m successive sample
vectors, we further define the following quantities:
where the smoothing factor m is
chosen according to 
. Note that for such m, the matrix H is
a "tall" matrix [i.e., Pm
K(m + I)]. We can then write (2.168) in matrix form as
Equation 2.169
Sections
