# rls

(To be removed) Construct recursive least squares (RLS) adaptive algorithm object

rls will be removed in a future release. Use comm.LinearEqualizer or comm.DecisionFeedback instead.

## Syntax

alg = rls(forgetfactor)
alg = rls(forgetfactor,invcorr0)

## Description

The rls function creates an adaptive algorithm object that you can use with the lineareq function or dfe function to create an equalizer object. You can then use the equalizer object with the equalize function to equalize a signal. To learn more about the process for equalizing a signal, see Equalization.

alg = rls(forgetfactor) constructs an adaptive algorithm object based on the recursive least squares (RLS) algorithm. The forgetting factor is forgetfactor, a real number between 0 and 1. The inverse correlation matrix is initialized to a scalar value.

alg = rls(forgetfactor,invcorr0) sets the initialization parameter for the inverse correlation matrix. This scalar value is used to initialize or reset the diagonal elements of the inverse correlation matrix.

### Properties

The table below describes the properties of the RLS adaptive algorithm object. To learn how to view or change the values of an adaptive algorithm object, see Equalization.

PropertyDescription
AlgTypeFixed value, 'RLS'
ForgetFactorForgetting factor
InvCorrInitScalar value used to initialize or reset the diagonal elements of the inverse correlation matrix

Also, when you use this adaptive algorithm object to create an equalizer object (via the lineareq function or dfe function), the equalizer object has an InvCorrMatrix property that represents the inverse correlation matrix for the RLS algorithm. The initial value of InvCorrMatrix is InvCorrInit*eye(N), where N is the total number of equalizer weights.

## Examples

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This example configures the recommended comm.LinearEqualizer System object™ and the legacy lineareq feature with comparable settings.

Initialize Variables and Supporting Objects

d = randi([0 3],1000,1);
x = pskmod(d,4,pi/4);
r = awgn(x,25);
sps = 2; %samples per symbol for oversampled cases
nTaps = 6;
txFilter = comm.RaisedCosineTransmitFilter('FilterSpanInSymbols',nTaps, ...
'OutputSamplesPerSymbol',4);
'InputSamplesPerSymbol',4,'DecimationFactor',2);
x2 = txFilter(x);
r2 = rxFilter(awgn(x2,25,0.5));
filterDelay = txFilter.FilterSpanInSymbols/2 + ...
rxFilter.FilterSpanInSymbols/2; % Total filter delay in symbols

To compare the equalized output, plot the constellations using code such as:

% plot(yNew,'*')
% hold on
% plot(yOld,'o')
% hold off; legend('New Eq','Old Eq'); grid on

Use RLS Algorithm with Linear Equalizer

Configure lineareq and comm.LinearEqualizer objects with comparable settings.

eqOld = lineareq(5,rls(0.95),pskmod(0:3,4,pi/4))
eqOld =
EqType: 'Linear Equalizer'
AlgType: 'RLS'
nWeights: 5
nSampPerSym: 1
RefTap: 1
SigConst: [0.7071 + 0.7071i -0.7071 + 0.7071i -0.7071 - 0.7071i 0.7071 - 0.7071i]
InvCorrInit: 0.1000
InvCorrMatrix: [5×5 double]
Weights: [0 0 0 0 0]
WeightInputs: [0 0 0 0 0]
ResetBeforeFiltering: 1
NumSamplesProcessed: 0
eqNew = comm.LinearEqualizer('NumTaps',5,'Algorithm','RLS', ...
'ForgettingFactor',0.95,'Constellation',pskmod(0:3,4,pi/4),'ReferenceTap',1)
eqNew = comm.LinearEqualizer with properties:
Algorithm: 'RLS'
NumTaps: 5
ForgettingFactor: 0.9500
InitialInverseCorrelationMatrix: 0.1000
Constellation: [0.7071 + 0.7071i -0.7071 + 0.7071i -0.7071 - 0.7071i 0.7071 - 0.7071i]
ReferenceTap: 1
InputDelay: 0
InputSamplesPerSymbol: 1
TrainingFlagInputPort: false
InitialWeightsSource: 'Auto'
WeightUpdatePeriod: 1

Call the equalizers. When ResetBeforeFiltering is set to true, each call of the equalize object resets the equalizer. To get the equivalent behavior call reset after each call of the comm.LinearEqualizer object.

yOld1 = equalize(eqOld,r,x(1:100));
yOld2 = equalize(eqOld,r,x(1:100));

yNew1 = eqNew(r,x(1:100));
reset(eqNew)
yNew2 = eqNew(r,x(1:100));

Configure lineareq and comm.LinearEqualizer objects with comparable settings. For the comm.LinearEqualizer object, set the initial inverse correlation matrix to eye(5)*0.2.

eqOld = lineareq(5,rls(0.95),pskmod(0:3,4,pi/4))
eqOld =
EqType: 'Linear Equalizer'
AlgType: 'RLS'
nWeights: 5
nSampPerSym: 1
RefTap: 1
SigConst: [0.7071 + 0.7071i -0.7071 + 0.7071i -0.7071 - 0.7071i 0.7071 - 0.7071i]
ForgetFactor: 0.9500
InvCorrInit: 0.1000
InvCorrMatrix: [5×5 double]
Weights: [0 0 0 0 0]
WeightInputs: [0 0 0 0 0]
ResetBeforeFiltering: 1
NumSamplesProcessed: 0
eqNew = comm.LinearEqualizer('NumTaps',5,'Algorithm','RLS', ...
'ForgettingFactor',0.95,'Constellation',pskmod(0:3,4,pi/4),'ReferenceTap',1, ...
'InitialInverseCorrelationMatrix',eye(5)*0.2)
eqNew = comm.LinearEqualizer with properties:
Algorithm: 'RLS'
NumTaps: 5
ForgettingFactor: 0.9500
InitialInverseCorrelationMatrix: [5×5 double]
Constellation: [0.7071 + 0.7071i -0.7071 + 0.7071i -0.7071 - 0.7071i 0.7071 - 0.7071i]
ReferenceTap: 1
InputDelay: 0
InputSamplesPerSymbol: 1
TrainingFlagInputPort: false
InitialWeightsSource: 'Auto'
WeightUpdatePeriod: 1

Call the equalizers. When ResetBeforeFiltering is set to true, each call of the equalize object resets the equalizer. To get the equivalent behavior call reset after each call of the comm.LinearEqualizer object.

yOld1 = equalize(eqOld,r,x(1:100));
yOld2 = equalize(eqOld,r,x(1:100));

yNew1 = eqNew(r,x(1:100));
reset(eqNew)
yNew2 = eqNew(r,x(1:100));

## Algorithms

Referring to the schematics presented in Equalization, define w as the vector of all weights wi and define u as the vector of all inputs ui. Based on the current set of inputs, u, and the current inverse correlation matrix, P, this adaptive algorithm first computes the Kalman gain vector, K

$K=\frac{Pu}{\left(ForgettingFactor\right)+{u}^{H}Pu}.$

where H denotes the Hermitian transpose.

Then the new inverse correlation matrix is given by

(ForgetFactor)-1(P – KuHP)

and the new set of weights is given by

w + K*e

where the * operator denotes the complex conjugate.

## Compatibility Considerations

expand all

Warns starting in R2020a

## References

[1] Farhang-Boroujeny, B., Adaptive Filters: Theory and Applications, Chichester, England, John Wiley & Sons, 1998.

[2] Haykin, S., Adaptive Filter Theory, Third Ed., Upper Saddle River, NJ, Prentice-Hall, 1996.

[3] Kurzweil, J., An Introduction to Digital Communications, New York, John Wiley & Sons, 2000.

[4] Proakis, John G., Digital Communications, Fourth Ed., New York, McGraw-Hill, 2001.

### Topics

Introduced before R2006a