(To be removed) Construct least mean square (LMS) adaptive algorithm object

lms will be removed in a future release. Use comm.LinearEqualizer or comm.DecisionFeedbackEqualizer instead.


alg = lms(stepsize)
alg = lms(stepsize,leakagefactor)


The lms 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 = lms(stepsize) constructs an adaptive algorithm object based on the least mean square (LMS) algorithm with a step size of stepsize.

alg = lms(stepsize,leakagefactor) sets the leakage factor of the LMS algorithm. leakagefactor must be between 0 and 1. A value of 1 corresponds to a conventional weight update algorithm, and a value of 0 corresponds to a memoryless update algorithm.


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

AlgTypeFixed value, 'LMS'
StepSizeLMS step size parameter, a nonnegative real number
LeakageFactorLMS leakage factor, a real number between 0 and 1


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You can equalize a signal by using the equalize function to apply an adaptive equalizer object to the signal. The equalize function also updates the equalizer.

In typical applications, an equalizer begins by using a known sequence of transmitted symbols when adapting the equalizer weights. The known sequence, called a training sequence, enables the equalizer to gather information about the channel characteristics. After the equalizer finishes processing the training sequence, it adapts the equalizer weights in decision-directed mode using a detected version of the output signal. To use a training sequence when invoking the equalize function, include the symbols of the training sequence as an input vector.

Note as an exception, that CMA equalizers do not use a training sequence. If an equalizer object is based on CMA, you should not include a training sequence as an input vector.

This code illustrates how to use equalize with a training sequence. The training sequence in this case is just the beginning of the transmitted message.

Set up parameters and signals.

M = 4; % Alphabet size for modulation
msg = randi([0 M-1],1500,1); % Random message
qpskMod = comm.QPSKModulator('PhaseOffset',0);
modmsg = qpskMod(msg); % Modulate using QPSK.
trainlen = 500; % Length of training sequence
chan = [.986; .845; .237; .123+.31i]; % Channel coefficients
filtmsg = filter(chan,1,modmsg); % Introduce channel distortion.

Equalize the received signal.

eq1 = lineareq(8, lms(0.01)); % Create an equalizer object.
eq1.SigConst = qpskMod((0:M-1)')'; % Set signal constellation.
[symbolest,yd] = equalize(eq1,filtmsg,modmsg(1:trainlen)); % Equalize.

Compute error rates with and without equalization

Determine the number of errors that occurred in trying to recover the modulated message with and without the equalizer. The symbol error rates show that the equalizer improves the performance significantly.

qpskDemod = comm.QPSKDemodulator('PhaseOffset',0);
demodmsg_noeq = qpskDemod(filtmsg); % Demodulate unequalized signal.
demodmsg = qpskDemod(yd); % Demodulate detected signal from equalizer.
errorCalc = comm.ErrorRate; % ErrorRate calculator
ser_noEq = errorCalc(msg(trainlen+1:end), demodmsg_noeq(trainlen+1:end));
ser_Eq = errorCalc(msg(trainlen+1:end),demodmsg(trainlen+1:end));
disp('Symbol error rates with and without equalizer:')
Symbol error rates with and without equalizer:
disp([ser_Eq(1) ser_noEq(1)]) 
         0    0.3230

Plot the signals

Create a scatter plot showing the signal before and after equalization, as well as the reference signal constellation for QPSK modulation. The points of the equalized signal are clustered more closely around the points of the reference signal constellation, indicating the signal improvement from equalization.

h = scatterplot(filtmsg,1,trainlen,'bx'); hold on;
legend('Filtered signal','Equalized signal',...
   'Ideal signal constellation');
hold off;

For more examples that use training sequences, see Adaptive Equalization.

If you invoke equalize multiple times with the same equalizer object to equalize a series of signal vectors, you might use a training sequence the first time you call the function and omit the training sequence in subsequent calls. Each iteration of the equalize function after the first one operates completely in decision-directed mode. However, because the ResetBeforeFiltering property of the equalizer object is set to 0, the equalize function uses the existing state information in the equalizer object when starting equalization operation for each iteration. As a result, the training affects all equalization operations, not just the first.

Notice in this code that the first call to equalize uses a training sequence as an input argument, and the second call to equalize omits a training sequence.

Set up the signal transmission

Create a signal, QPSK modulate it, then filter it through a distortion channel.

M = 4; % Alphabet size for modulation
msg = randi([0 M-1],1500,1); % Random message
qpskMod = comm.QPSKModulator('PhaseOffset',0);
modmsg = qpskMod(msg); % Modulate using QPSK

chan = [.986; .845; .237; .123+.31i]; % Channel coefficients
filtmsg = filter(chan,1,modmsg); % Introduce channel distortion

Set up equalizer

Specify equalizer parameters and create an lms equalizer object

trainlen = 500; % Length of training sequence
eqlms = lineareq(8, lms(0.01)); % Create an equalizer object
eqlms.SigConst = qpskMod((0:M-1)')'; % Set signal constellation parameter in the equalizer

Maintain continuity between calls to equalize.

eqlms.ResetBeforeFiltering = 0;

Equalize the received signal in pieces

Process the training sequence.

s1 = equalize(eqlms,filtmsg(1:trainlen),modmsg(1:trainlen));

Process some of the data in decision-directed mode.

s2 = equalize(eqlms,filtmsg(trainlen+1:800));

Process the rest of the data in decision-directed mode.

s3 = equalize(eqlms,filtmsg(801:end));

Concatenate the signal segments to get the full output of equalizer.

s = [s1; s2; s3]; 


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 weights, w, this adaptive algorithm creates the new set of weights given by

(LeakageFactor) w + (StepSize) u*e

where the * operator denotes the complex conjugate.

Compatibility Considerations

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Not recommended starting in R2019a


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

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

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

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

Introduced before R2006a