Recursive Least-Squares (RLS) Adaptive Filters. Definition. With the arrival of new data samples estimates are updated recursively. Introduce a weighting factor to the sum-of-error-squares definition. two time-indices n: outer, i: inner. Weighting factor. Forgetting factor.
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Recursive Least-Squares (RLS)Adaptive Filters
ELE 774 - Adaptive Signal Processing
two time-indices
n: outer, i: inner
Weighting factor
Forgetting factor
: real, positive, <1, →1
=1 → ordinary LS
1/(1- ): memory of the algorithm
(ordinary LS has infinite memory)
w(n) is kept fixed during the
observation interval 1≤i ≤n for
which the cost function
(n) is defined.
ELE 774 - Adaptive Signal Processing
ELE 774 - Adaptive Signal Processing
Regularisation term
Smooths and stabilises the solution
: regularisation parameter
ELE 774 - Adaptive Signal Processing
then the time-average autocorrelation matrix of the input u(n) becomes
autocorrelation matrix
is always non-singular
due to this term.
(-1 always exists!)
ELE 774 - Adaptive Signal Processing
ELE 774 - Adaptive Signal Processing
inverse correlation
matrix
gain vector
Riccati
equation
ELE 774 - Adaptive Signal Processing
ELE 774 - Adaptive Signal Processing
is called the a priori estimation error,
is called the a posteriori estimation error. (Why?)
gain vector
a priori error
regularisation
parameter
ELE 774 - Adaptive Signal Processing
ELE 774 - Adaptive Signal Processing
ELE 774 - Adaptive Signal Processing
where
ELE 774 - Adaptive Signal Processing
where wo is the regression parameter vector and eo(n) is the measurement noise. The noise eo(n) is white with zero mean and variance so2 which makes it independent of the regressor u(n).
ELE 774 - Adaptive Signal Processing
(n) is an accumulation of the a priori error → hence, the input
→Smoothing (low-pass filtering) effect.
ELE 774 - Adaptive Signal Processing
=1
ELE 774 - Adaptive Signal Processing
and invoking Assumption I and simplifying we obtain
ELE 774 - Adaptive Signal Processing
^
ELE 774 - Adaptive Signal Processing
ELE 774 - Adaptive Signal Processing
ELE 774 - Adaptive Signal Processing
ELE 774 - Adaptive Signal Processing
ELE 774 - Adaptive Signal Processing