Nonlinear Group-Blind Multiuser Detection for Synchronous CDMA
In Section 3.2 we developed
linear receivers for detecting a group of 
users' data bits in the presence of
unknown interfering users. In this section we develop nonlinear methods for
joint detection of the desired users' data. The basic idea is to construct a
likelihood function for these users' data and then to perform a local search
over such a likelihood surface, starting from the estimate closest to the
unconstrained maximizer of the likelihood function, and along mutually
orthogonal directions where the likelihood function drops at the slowest rate.
The techniques described in this section were developed in [456].
Consider the signal model (3.2). Since the transmitted
symbols b 
{+1, –1}K 
, for the convenience of the development in
this section, we write (3.2) in terms of real-valued
signals. Specifically, denote (recall that S is real valued)
Equation 3.99
where u[i] ~ N (0, (s2/2)I2N) is
a real-valued noise vector. Then (3.2) can be written as
Equation 3.100
For notational simplicity in what follows, we drop the symbol
index i. In this case the maximum-likelihood
estimate of the transmitted symbols (of all users) is given by
Equation 3.101
where q is the stationary point of l(b):
Equation 3.102
In (3.101) the Hessian
matrix of the log-likelihood function is given by
Equation 3.103
It is well known that the combinatorial optimization problem
(3.101) is computationally hard (i.e.,
it is NP-complete) [519]. We next consider a local-search
approach to approximating its solution. The basic idea is to search the optimal
solution in a subset W of the discrete parameter set
{–1, +1}K that is close to the
stationary point q. More precisely, we approximate the solution to (3.101) by
Equation 3.104
In the slowest-descent method
[453, 454], the candidate set
W consists of the discrete parameters chosen such that
they are in the neighborhood of Q (Q 
K) lines in , which are
defined by the stationary point q and the Q
eigenvectors of 
corresponding to the Q smallest eigenvalues. The basic idea of this method
is explained next.
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