RLS Algorithm
The LMS algorithm discussed above has a very low computational
complexity, on the order of 
operations per update. However, its
convergence is usually very slow. We next consider the recursive least-squares
(RLS) algorithm for adaptive implementation of the blind linear MMSE detector,
which has a much faster convergence rate than the LMS algorithm. Based on the
cost function (2.32), at time i the exponentially windowed RLS algorithm selects the
weight vector m1[i] to minimize the sum of exponentially weighted
mean-square output values:
Equation 2.44
where 0 < l < 1 (1 - l << 1) is called the
forgetting factor. The solution to this
optimization problem is given by
Equation 2.45
with
Equation 2.46
Denote 
. Note that since
Equation 2.47
by the matrix inversion lemma we have
Equation 2.48
Hence we obtain the RLS algorithm for adaptive implementation
of the blind linear MMSE detector as follows. Suppose that at time (i - 1), F[i - 1] is available.
Then at time i, the following steps are performed
to update the detector m1[i] and to detect the differential bit b1[i].
Algorithm 2.3: [RLS blind
linear MMSE detector—synchronous CDMA]
Equation 2.49
Equation 2.50
Equation 2.51
Equation 2.52
Equation 2.53
The convergence properties of Algorithm 2.3 are analyzed in detail in [389].
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