3.2.3. Recursive Least Squares Algorithm
Because the environment (e.g. mobileenvironment) is
time-variable, it is essential that the weight vector to be updated or adapted
periodically for an adaptive array network. As the necessary data to estimate
the optimal solution is noisy, an adaptive algorithm is exploited for updating
the weight vector periodically. In [22], it is reported that there are many
types of adaptive algorithms and the majorities are iterative. They utilized
the past information to minimize the computations required at each updatecycle.
In iterative algorithms, the current weight vector, W(n), is modified
by an incremental value to form a new weight vector,W(n+1) at each
iteration n. The RLS algorithm is summarized as follow [22]:
Initialization
(3.17)
W(0) = 0 (3.18)
Weight Update
(3.19)
(3.20)
(3.21)
(3.22)
Convergence coefficient
0<ë<1, where;
ä is a small positive number,
I is the MXM identity matrix,
ë is the forgetting factor
k(n) is the gain vector,
á(n) is the innovation,
W(n) is the weight vector,
P(n) is the inverse of the correlation matrix Ô(n),
u(n) is the input vector
d(n) is the desired response.
In the RLS method, the desired signal must be supplied using
either a training sequence
or decision direction. For the training sequence approach, a
brief data sequence is transmitted which is known by the receiver. The receiver
uses the adaptive algorithm to
approximate the weight vector in the training duration, then
retains the weights constant
while information is being transmitted. This technique
requires that the environment be
stationary from one training period to the next, and it
reduces channel throughput by requiring the use of channel symbols for
training. However, in the decision approach, the receiver uses recreated
modulated symbols based on symbol decisions, which are used as the desired
signal to adapt the weight vector [22].
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