Linear MMSE Detector and RLS Blind Adaptation Rule
Consider the following received signal model:
Equation 7.151
where AK, bK and sK denote, respectively, the received
amplitude, data bit, and the spreading waveform of the Kth user; i
denotes the NBI signal; and 
is the Gaussian noise. Assume that
user 1 is the user of interest, and for convenience we will use the following
notations: 
, and 
. The weight vector of the linear
MMSE detector is given by
Equation 7.152
where Rr is the autocorrelation matrix of the received
discrete signal r:
Equation 7.153
The output SINR is given by
Equation 7.154
where
Equation 7.155
The mean output energy associated with w, defined as the mean-square output value of
w applied to r, is
Equation 7.156
where the last equality follows from (7.155) and the matrix inversion lemma. The mean-square
error (MSE) at the output of w is
Equation 7.157
The exponentially windowed RLS algorithm selects the weight
vector w[i]
to minimize the sum of exponentially weighted output energies:
where 0 < l < 1 is a forgetting factor (1 - l << 1). The purpose of
l is to ensure that the
data in the distant past will be forgotten in order to provide tracking
capability in nonstationary environments. The solution to this constrained
optimization problem is given by
Equation 7.158
where
Equation 7.159
A recursive procedure for updating w[i] is as
follows:
Equation 7.160
Equation 7.161
Equation 7.162
Equation 7.163
In what follows we provide a convergence analysis for the
algorithm above. In this analysis, we make use of three
approximations/assumptions: (a) For large i,
Rr[i] is
approximated by its expected value [111, 301]; (b) the input data r[i] and the
previous weight vector w[i–1] are assumed to be independent [175]; (c) some fourth-order statistic
can be approximated in terms of a second-order statistic [175].
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