# Documentation

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# normlms

Construct normalized least mean square (LMS) adaptive algorithm object

## Syntax

`alg = normlms(stepsize)alg = normlms(stepsize,bias)`

## Description

The `normlms` 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 Adaptive Algorithms.

`alg = normlms(stepsize)` constructs an adaptive algorithm object based on the normalized least mean square (LMS) algorithm with a step size of `stepsize` and a bias parameter of zero.

`alg = normlms(stepsize,bias)` sets the bias parameter of the normalized LMS algorithm. `bias` must be between 0 and 1. The algorithm uses the bias parameter to overcome difficulties when the algorithm's input signal is small.

### Properties

The table below describes the properties of the normalized LMS adaptive algorithm object. To learn how to view or change the values of an adaptive algorithm object, see Access Properties of an Adaptive Algorithm.

PropertyDescription
`AlgType`Fixed value, ```'Normalized LMS'```
`StepSize`LMS step size parameter, a nonnegative real number
`LeakageFactor`LMS leakage factor, a real number between 0 and 1. A value of 1 corresponds to a conventional weight update algorithm, while a value of 0 corresponds to a memoryless update algorithm.
`Bias`Normalized LMS bias parameter, a nonnegative real number

## Examples

For an example that uses this function, see Delays from Equalization.

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### Algorithms

Referring to the schematics presented 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

`$\left(\text{LeakageFactor}\right)w+\frac{\left(\text{StepSize}\right){u}^{*}e}{{u}^{H}u+\text{Bias}}$`

where the * operator denotes the complex conjugate and H denotes the Hermitian transpose.

## References

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