Synchronous CDMA Signal Model
We start by considering the most basic multiple-access signal
model: a baseband K-user time-invariant
synchronous additive white Gaussian noise (AWGN) system, employing periodic
(short) spreading sequences and operating with a coherent BPSK modulation
format. (An approach to adaptive detection in (long) aperiodic code DS-SS
systems is developed in [61].) As noted in Chapter 1, the continuous-time waveform
received by a given user in such a system can be modeled as follows:
Equation 2.1
where M is the number of data
symbols per user in the data frame of interest; T
is the symbol interval; Ak, 
,
and sk(t) denote, respectively, the received complex
amplitude, the transmitted symbol stream, and the normalized signaling waveform
of the kth user; and n(t) is the baseband
complex Gaussian ambient noise with independent real and imaginary components
and with power spectral density s2. It is assumed that for each user k, is a collection of independent
equiprobable ±1 random variables, and the symbol streams of different users are
independent. For the direct-sequence spread-spectrum format, each user's
signaling waveform is of the form
Equation 2.2
where N is the processing gain,
is a signature sequence of ±1's assigned to the kth user, and y(·) is a chip waveform of duration Tc = T/N and unit energy 
At the receiver, the received signal r(t) is filtered by a
chip-matched filter and then sampled at the chip rate. The sample corresponding
to the jth chip of the ith symbol is thus given by
Equation 2.3
The resulting discrete-time signal corresponding to the ith symbol is then given by
Equation 2.4
Equation 2.5
with
where 
is a complex Gaussian random
variable with independent real and imaginary components; and 
[Here
Nc(·, ·) denotes a complex Gaussian
distribution and IN denotes an N x
N identity matrix.] 
and 
.
Suppose that we are interested in demodulating the data bits of
a particular user, say user 1, 
, based on the received waveforms
. A linear receiver for this purpose can be described by a weight
vector 
such that the desired user's data bits are demodulated
according to
Equation 2.6
Equation 2.7
Note that the linear equalizers and multiuser detectors
discussed in Chapter 1 can
all be written in this form, as will be seen below. In case the complex
amplitude A1 of the desired user is
unknown, we can resort to differential detection. Define the differential bit
as
Equation 2.8
Then using the linear detector output (2.6), the following differential detection rule can be
used:
Equation 2.9
Substituting (2.4) into
(2.6), the output of the linear receiver
w1 can be written as
Equation 2.10
In (2.10), the first
term on the right-hand side contains the useful signal of the desired user, the
second term contains the signals from other undesired users—the multiple-access interference (MAI), and the last term
contains the ambient Gaussian noise. The simplest linear receiver is the
conventional matched filter, where w1 = s1. As noted in Chapter 1, such a matched-filter receiver
is optimal only in a single-user channel (i.e., K
= 1). In a multiuser channel (i.e., K > 1),
this receiver may perform poorly since it makes no attempt to ameliorate the
MAI, a limiting source of interference in multiple-access channels. Two popular
forms of linear detectors that are capable of suppressing the MAI are the linear
decorrelating detector and the linear minimum mean-square-error (MMSE) detector,
which are discussed next.
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