# dsp.RLSFilter

Compute output, error and coefficients using recursive least squares (RLS) algorithm

## Description

The `dsp.RLSFilter`

System object™ filters each channel of the input using RLS filter implementations.

To filter each channel of the input:

Create the

`dsp.RLSFilter`

object and set its properties.Call the object with arguments, as if it were a function.

To learn more about how System objects work, see What Are System Objects?

## Creation

### Description

returns an adaptive
RLS filter System object, `rlsFilt`

= dsp.RLSFilter`rlsFilt`

. This System object computes the filtered output, filter error, and the filter weights for a
given input and desired signal using the RLS algorithm.

returns an RLS filter System object, `rlsFilt`

= dsp.RLSFilter(`len`

)`rlsFilt`

. This System object has the `Length`

property set to
`len`

.

returns an RLS filter System object with each specified property set to the specified value. Enclose each
property name in single quotes. Unspecified properties have default values.`rlsFilt`

= dsp.RLSFilter(`Name,Value`

)

## Properties

## Usage

### Description

`[`

shows the output of the RLS filter along with the error, `y`

,`e`

] =
rlsFilt(`x`

,`d`

)`e`

, between
the reference input and the desired signal. The filters adapts its coefficients until the
error `e`

is minimized. You can access these coefficients by accessing
the `Coefficients`

property of the object. This can be done only after
calling the object. For example, to access the optimized coefficients of the
`rlsFilt`

filter, call `rlsFilt.Coefficients`

after
you pass the input and desired signal to the object.

### Input Arguments

### Output Arguments

## Object Functions

To use an object function, specify the
System object as the first input argument. For
example, to release system resources of a System object named `obj`

, use
this syntax:

release(obj)

## Examples

## Algorithms

The `dsp.RLSFilter`

System object, when `Conventional RLS`

is selected, recursively
computes the least squares estimate (RLS) of the FIR filter weights.
The System object estimates the filter weights or coefficients,
needed to convert the input signal into the desired signal. The input
signal can be a scalar or a column vector. The desired signal must
have the same data type, complexity, and dimensions as the input signal.
The corresponding RLS filter is expressed in matrix form as **P**(*n*)
:

$$\begin{array}{l}k(n)=\frac{{\lambda}^{-1}P(n-1)u(n)}{1+{\lambda}^{-1}{u}^{H}(n)P(n-1)u(n)}\\ y(n)={w}^{T}(n-1)u(n)\\ e(n)=d(n)-y(n)\\ w(n)=w(n-1)+{k}^{*}(n)e(n)\\ P(n)={\lambda}^{-1}P(n-1)-{\lambda}^{-1}k(n){u}^{H}(n)P(n-1)\end{array}$$

where λ^{-1} denotes the reciprocal
of the exponential weighting factor. The variables are as follows:

Variable | Description |
---|---|

n | The current time index |

u(n) | The vector of buffered input samples at step n |

P(n) | The conjugate of the inverse correlation matrix at step n |

k(n) | The gain vector at step n |

k*(n) | Complex conjugate of k |

w(n) | The vector of filter tap estimates at step n |

y(n) | The filtered output at step n |

e(n) | The estimation error at step n |

d(n) | The desired response at step n |

λ | The forgetting factor |

**u**, **w**, and **k** are
all column vectors.

## References

[1] M Hayes, *Statistical Digital Signal Processing and
Modeling,* New York: Wiley, 1996.

[2] S. Haykin, *Adaptive Filter Theory,* 4th
Edition, Upper Saddle River, NJ: Prentice Hall, 2002.

[3] A.A. Rontogiannis and S. Theodoridis, "Inverse factorization
adaptive least-squares algorithms," *Signal Processing*, vol. 52, no. 1,
pp. 35-47, July 1996.

[4] S.C. Douglas, "Numerically-robust O(N^{2})
RLS algorithms using least-squares prewhitening," Proc. IEEE Int.
Conf. on Acoustics, Speech, and Signal Processing, Istanbul, Turkey,
vol. I, pp. 412-415, June 2000.

[5] A. H. Sayed, *Fundamentals of Adaptive
Filtering,* Hoboken, NJ: John Wiley & Sons, 2003.

## Extended Capabilities

## Version History

**Introduced in R2013a**