Adaptive filters
Adaptive Filters
1
Adaptive Filters
A Tutorial for the Course Computational Intelligence http://www.igi.tugraz.at/lehre/CI Christian Feldbauer, Franz Pernkopf, and Erhard Rank
Signal Processing and Speech Communication Laboratory
Inffeldgasse 16c
Abstract
This tutorial introduces the LMS (least mean squares) and the RLS (recursive least-squares) algorithm for the design of adaptive transversal filters. These algorithms are applied for identification of an unknown system. Usage
To make full use of this tutorial you have to
1. Download the file AdaptiveFilter.zip which contains this tutorial and the accompanying Matlab program(s). 2. Unzip AdaptiveFilter.zip which will generate a subdirectory AdaptiveFilter/matlab where you can find the Matlab program(s).
3. Add the path AdaptiveFilter/matlab to the matlab search path with a command like addpath(’C:\Work\Ada if you are using a Windows machine or addpath(’/home/jack/AdaptiveFilter/matlab’) if you are using a Unix/Linux machine.
1
Introduction
Discrete-time (or digital) filters are ubiquitous in todays signal processing applications. Filters are used to achieve desired spectral characteristics of a signal, to reject unwanted signals, like noise or interferers, to reduce the bit rate in signal transmission, etc.
The notion of making filters adaptive, i.e., to alter parameters (coefficients) of a filter according to some algorithm, tackles the problems that we might not in advance know, e.g., the characteristics of the signal, or of the unwanted signal, or of a systems influence on the signal that we like to compensate. Adaptive filters can adjust to unknown environment, and even track signal or system characteristics varying over time. 2
Adaptive Transversal Filters
In a transversal filter of length N , as depicted in fig. 1, at each time n the output sample y[n] is computed by a weighted sum of the current and delayed input samples x[n], x[n − 1], . . .
N −1
c∗ [n]x[n −
1
Adaptive Filters
A Tutorial for the Course Computational Intelligence http://www.igi.tugraz.at/lehre/CI Christian Feldbauer, Franz Pernkopf, and Erhard Rank
Signal Processing and Speech Communication Laboratory
Inffeldgasse 16c
Abstract
This tutorial introduces the LMS (least mean squares) and the RLS (recursive least-squares) algorithm for the design of adaptive transversal filters. These algorithms are applied for identification of an unknown system. Usage
To make full use of this tutorial you have to
1. Download the file AdaptiveFilter.zip which contains this tutorial and the accompanying Matlab program(s). 2. Unzip AdaptiveFilter.zip which will generate a subdirectory AdaptiveFilter/matlab where you can find the Matlab program(s).
3. Add the path AdaptiveFilter/matlab to the matlab search path with a command like addpath(’C:\Work\Ada if you are using a Windows machine or addpath(’/home/jack/AdaptiveFilter/matlab’) if you are using a Unix/Linux machine.
1
Introduction
Discrete-time (or digital) filters are ubiquitous in todays signal processing applications. Filters are used to achieve desired spectral characteristics of a signal, to reject unwanted signals, like noise or interferers, to reduce the bit rate in signal transmission, etc.
The notion of making filters adaptive, i.e., to alter parameters (coefficients) of a filter according to some algorithm, tackles the problems that we might not in advance know, e.g., the characteristics of the signal, or of the unwanted signal, or of a systems influence on the signal that we like to compensate. Adaptive filters can adjust to unknown environment, and even track signal or system characteristics varying over time. 2
Adaptive Transversal Filters
In a transversal filter of length N , as depicted in fig. 1, at each time n the output sample y[n] is computed by a weighted sum of the current and delayed input samples x[n], x[n − 1], . . .
N −1
c∗ [n]x[n −
References: [1] Simon Haykin: “Adaptive Filter Theory”, Third Edition, Prentice-Hall, Inc., Upper Saddle River, NJ, 1996. [2] Bernard Widrow and Samuel D. Stearns: “Adaptive Signal Processing”, Prentice-Hall, Inc., Upper Saddle River, NJ, 1985. [3] Edward A. Lee and David G. Messerschmitt: “Digital Communication”, Kluwer Academic Publishers, Boston, 1988. [4] Steven M. Kay: “Fundamentals of Statistical Signal Processing—Detection Theory”, Volume 2, Prentice-Hall, Inc., 1998.