LMS Algorithm
We first consider the least-mean-squares (LMS) algorithm for
recursive estimation of m1 based
on (2.32). Define
Equation 2.35
as a projection matrix that projects any signal in 
onto the orthogonal space of s1.
Note that m1 can be decomposed
into two orthogonal components:
Equation 2.36
with
Equation 2.37
Using the decomposition above, the constrained optimization
problem (2.32) can then be converted to
the following unconstrained optimization problem:
Equation 2.38
The LMS algorithm for adapting the vector x1 based on the cost function (2.38) is then given by
Equation 2.39
where m
is the step size and where the stochastic gradient g (x1[i])
is given by
Equation 2.40
Substituting (2.40) into
(2.39), we obtain the following LMS
implementation of the blind linear MMSE detector. Suppose that at time i, the estimated blind detector is m1[i]=
s1 + x1[i].
The algorithm performs the following steps for data detection and detector
update.
Algorithm 2.2: [LMS blind
linear MMSE detector—synchronous CDMA]
Equation 2.41
Equation 2.42
Equation 2.43
The convergence analysis of Algorithm 2.2 is given in [183]. An alternative stochastic
gradient algorithm for blind adaptive multiuser detection is developed in [237], which employs the
technique of averaging to achieve an accelerated convergence rate (compared with
the LMS algorithm). An LMS algorithm for blind adaptive implementation of the
linear decorrelating detector is developed in [501]. Moreover, a comparison of the
steady-state performance (in terms of output mean-square error) shows that the
blind detector incurs a loss compared with a training-based LMS detector [183, 389, 390]. A two-stage adaptive detector
is proposed in [63],
where symbol-by-symbol predecisions at the output of a first adaptive stage are
used to train a second stage, to achieve improved performance.
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