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Recursive least-squares FIR adaptive filter

adaptfilt.rls will be removed in a future release. Use dsp.RLSFilter instead.


ha = adaptfilt.rls(l,lambda,invcov,coeffs,states)


ha = adaptfilt.rls(l,lambda,invcov,coeffs,states) constructs an FIR direct form RLS adaptive filter ha.

For information on how to run data through your adaptive filter object, see the Adaptive Filter Syntaxes section of the reference page for filter.

Input Arguments

Entries in the following table describe the input arguments for adaptfilt.rls.

Input Argument



Adaptive filter length (the number of coefficients or taps) and it must be a positive integer. l defaults to 10.


RLS forgetting factor. This is a scalar and should lie in the range (0, 1]. lambda defaults to 1.


Inverse of the input signal covariance matrix. For best performance, you should initialize this matrix to be a positive definite matrix.


Vector of initial filter coefficients. it must be a length l vector. coeffs defaults to length l vector with elements equal to zero.


Vector of initial filter states for the adaptive filter. It must be a length l-1 vector. states defaults to a length l-1 vector of zeros.


Since your adaptfilt.rls filter is an object, it has properties that define its behavior in operation. Note that many of the properties are also input arguments for creating adaptfilt.rls objects. To show you the properties that apply, this table lists and describes each property for the filter object.






Defines the adaptive filter algorithm the object uses during adaptation.


Vector containing l elements

Vector containing the initial filter coefficients. It must be a length l vector where l is the number of filter coefficients. coeffs defaults to length l vector of zeros when you do not provide the argument for input.


Any positive integer

Reports the length of the filter, the number of coefficients or taps. Remember that filter length is filter order + 1.



Forgetting factor of the adaptive filter. This is a scalar and should lie in the range (0, 1]. It defaults to 1. Setting forgetting factor = 1 denotes infinite memory while adapting to find the new filter. Note that this is the lambda input argument.


Matrix of size l-by-l

Upper-triangular Cholesky (square root) factor of the input covariance matrix. Initialize this matrix with a positive definite upper triangular matrix.


Vector of size (l,1)

Empty when you construct the object, this gets populated after you run the filter.


false or true

Determine whether the filter states get restored to their starting values for each filtering operation. The starting values are the values in place when you create the filter if you have not changed the filter since you constructed it. PersistentMemory returns to zero any state that the filter changes during processing. Defaults to false.


Double array

Vector of the adaptive filter states. states defaults to a vector of zeros which has length equal to (l + projectord - 2).


System Identification of a 32-coefficient FIR filter over 500 adaptation iterations.

x  = randn(1,500);     % Input to the filter
b  = fir1(31,0.5);     % FIR system to be identified
n  = 0.1*randn(1,500); % Observation noise signal
d  = filter(b,1,x)+n;  % Desired signal
P0 = 10*eye(32); % Initial sqrt correlation matrix inverse
lam = 0.99;            % RLS forgetting factor
ha = adaptfilt.rls(32,lam,P0);
[y,e] = filter(ha,x,d);
subplot(2,1,1); plot(1:500,[d;y;e]);
title('System Identification of an FIR Filter');
xlabel('Time Index'); ylabel('Signal Value');
subplot(2,1,2); stem([b.',ha.Coefficients.']);
xlabel('Coefficient #'); ylabel('Coefficient valUe');  grid on;

In this example of adaptive filtering using the RLS algorithm to update the filter coefficients for each iteration, the figure shown reveals the fidelity of the derived filter after adaptation.

Introduced in R2011a

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