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CMA Equalizer

Equalize using constant modulus algorithm




The CMA Equalizer block uses a linear equalizer and the constant modulus algorithm (CMA) to equalize a linearly modulated baseband signal through a dispersive channel. During the simulation, the block uses the CMA to update the weights, once per symbol. If the Number of samples per symbol parameter is 1, then the block implements a symbol-spaced equalizer; otherwise, the block implements a fractionally spaced equalizer.

When using this block, you should initialize the equalizer weights with a nonzero vector. Typically, CMA is used with differential modulation; otherwise, the initial weights are very important. A typical vector of initial weights has a 1 corresponding to the center tap and zeros elsewhere.

Input and Output Signals

The Input port accepts a scalar-valued or column vector input signal. The Desired port receives a training sequence with a length that is less than or equal to the number of symbols in the Input signal.

You can configure the block to have one or more of the extra ports listed in the table below.

PortMeaningHow to Enable
Err output y(R -|y|2), where y is the equalized signal and R is a constant related to the signal constellation Select Output error.
Wts output A vector listing the weights after the block has processed either the current input frame or sample. Select Output weights.


Referring to the schematics in Equalizer Structure, 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.

Equalizer Delay

The delay between the transmitter's modulator output and the CMA equalizer output is typically unknown (unlike the delay for other adaptive equalizers in this product). If you need to determine the delay, you can use the Find Delay block.


Number of taps

The number of taps in the filter of the equalizer.

Number of samples per symbol

The number of input samples for each symbol.

When you set this parameter to 1, the filter weights are updated once for each symbol, for a symbol spaced (i.e. T-spaced) equalizer. When you set this parameter to a value greater than one, the weights are updated once every Nth sample, for a fractionally spaced (i.e. T/N-spaced) equalizer.

Signal constellation

A vector of complex numbers that specifies the constellation for the modulation.

Step size

The step size of the CMA.

Leakage factor

The leakage factor of the CMA, a number 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.

Initial weights

A vector that lists the initial weights for the taps.

Output error

If you check this box, the block outputs the error signal described in the table above.

Output weights

If you check this box, the block outputs the current weights.


[1] Haykin, Simon, Adaptive Filter Theory, Third Ed., Upper Saddle River, N.J., Prentice-Hall, 1996.

[2] Johnson, Richard C. Jr., Philip Schniter, Thomas. J. Endres, et al., "Blind Equalization Using the Constant Modulus Criterion: A Review," Proceedings of the IEEE, vol. 86, pp. 1927-1950, October 1998.

Introduced before R2006a

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