# On adaptive least mean square FIR filters: new implementations and applications

##### Bilcu, R. C. (2004)

Bilcu, R. C.

Tampere University of Technology

2004

Tietotekniikan osasto - Department of Information Technology

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**Julkaisun pysyvä osoite on**

https://urn.fi/URN:NBN:fi:tty-200810021077

##### Tiivistelmä

Among many adaptive algorithms that exist in the open literature, the class of approaches which are derived from the minimization of the mean squared error between the output of the adaptive filter and some desired signal, seems to be the most popular. Probably the simplest algorithm belonging to this class is the Least Mean Squared (LMS) algorithm which has the advantage of low complexity and simplicity of implementation. One of the main concerns in all practical situations is to develop algorithms which provide fast convergence of the adaptive lter coefficients and in the same time good ltering performance. There are four main classes of applications where the adaptive filters were applied with success, namely: system identification, inverse modeling, prediction and interference canceling. In this thesis we develop new algorithms for the first two classes of applications although they can be implemented also for prediction and interference canceling.

In this thesis several new algorithms for adaptive filtering are introduced. The main goal is to improve the performances of the existing algorithms, in terms of convergence speed and filtering performance and also to introduce some new approaches. The new algorithms are classified into several classes each of them addressing a certain application.

It is well known that the LMS algorithm has a slow convergence for correlated inputs. Moreover its filtering performance and convergence speed are inversely related through a single parameter, the step-size. An adaptive filter implementing the LMS might have stability problems operating in a non-Gaussian environment due to the use of instantaneous gradient to update the coefficients. In applications belonging to the class of system identification, not only the values of the coefficients of the model are of interest, but also the length of the model. Therefore algorithms for length adaptation might be of equal interest. Another situation, that can appear in identification applications is when the coefficients of the model are time-varying. The adaptive algorithm should provide a mechanism to track the changes of the model.

This thesis contains three main parts which are concentrated on time domain implementations, transform domain implementations and applications respectively. At the beginning of the first part two new variable step-size LMS algorithms are introduced which show good convergence speed. The dependence between the speed of adaptation and fi ltering performance is reduced and the setup of the parameters is very easy as compared with other existing approaches. The problem of length estimation is addressed later on and an algorithm to iteratively adjust the length of the adaptive lter toward the length of the model is proposed. This algorithm is derived for system identification application. Next, the problem of tracking time-varying systems is discussed and the analytical expressions for the steady-state mean squared error and mean squared coefficient error are revised. Based on these expressions a new algorithm in which the step-size is iteratively modified toward the optimum is introduced. An important feature of the proposed algorithm is the fact that the user does not need to know any information about the statistics of the optimum model.

At the end of the first part the class of order statistics LMS algorithms is discussed and a new algorithm belonging to this class is introduced. The new algorithm uses an adaptive filter to smooth the gradient such that it does not require the knowledge of the noise distribution in order to be implemented.

The second part of the thesis is dedicated to the transform domain implementation of the LMS algorithm and its variants. First, three new algorithms belonging to the class of variable step-size LMS algorithm in transform domain are introduced. To the best of our knowledge the idea of step-size adaptation in transform domain, based on the output error was not addressed so far in the open literature. The existing approaches, assume a time-varying step-size due to the power estimates of the transform coefficients, whereas in our implementations, the step-size is adapted by the output error. We continue with the problem of time-varying modeling using the transform domain LMS and we introduce a new algorithm. The aim of this algorithm is to increase the convergence speed of its time domain counterpart and also to reduce its complexity.

At the end of the second part the scrambled LMS is briey presented and it is compared with the LMS and transform domain LMS. The chosen framework is the digital data transmission over a telephone line. The analytical expressions of the mean squared error and mean squared coefficient error are derived for this special case and a discusion about their convergence speed and steady-state error is given. The aim of this discusion is to provide some useful information about the utility of each algorithm.

In the first two parts of the thesis, computer experiments, showing the performances of all the proposed algorithms, are provided for system identi cation application. Since many of the algorithms can be implemented also in other applications, in the third part of this thesis, channel equalization, CDMA multiuser detection and echo cancellation applications are also addressed.

In this thesis several new algorithms for adaptive filtering are introduced. The main goal is to improve the performances of the existing algorithms, in terms of convergence speed and filtering performance and also to introduce some new approaches. The new algorithms are classified into several classes each of them addressing a certain application.

It is well known that the LMS algorithm has a slow convergence for correlated inputs. Moreover its filtering performance and convergence speed are inversely related through a single parameter, the step-size. An adaptive filter implementing the LMS might have stability problems operating in a non-Gaussian environment due to the use of instantaneous gradient to update the coefficients. In applications belonging to the class of system identification, not only the values of the coefficients of the model are of interest, but also the length of the model. Therefore algorithms for length adaptation might be of equal interest. Another situation, that can appear in identification applications is when the coefficients of the model are time-varying. The adaptive algorithm should provide a mechanism to track the changes of the model.

This thesis contains three main parts which are concentrated on time domain implementations, transform domain implementations and applications respectively. At the beginning of the first part two new variable step-size LMS algorithms are introduced which show good convergence speed. The dependence between the speed of adaptation and fi ltering performance is reduced and the setup of the parameters is very easy as compared with other existing approaches. The problem of length estimation is addressed later on and an algorithm to iteratively adjust the length of the adaptive lter toward the length of the model is proposed. This algorithm is derived for system identification application. Next, the problem of tracking time-varying systems is discussed and the analytical expressions for the steady-state mean squared error and mean squared coefficient error are revised. Based on these expressions a new algorithm in which the step-size is iteratively modified toward the optimum is introduced. An important feature of the proposed algorithm is the fact that the user does not need to know any information about the statistics of the optimum model.

At the end of the first part the class of order statistics LMS algorithms is discussed and a new algorithm belonging to this class is introduced. The new algorithm uses an adaptive filter to smooth the gradient such that it does not require the knowledge of the noise distribution in order to be implemented.

The second part of the thesis is dedicated to the transform domain implementation of the LMS algorithm and its variants. First, three new algorithms belonging to the class of variable step-size LMS algorithm in transform domain are introduced. To the best of our knowledge the idea of step-size adaptation in transform domain, based on the output error was not addressed so far in the open literature. The existing approaches, assume a time-varying step-size due to the power estimates of the transform coefficients, whereas in our implementations, the step-size is adapted by the output error. We continue with the problem of time-varying modeling using the transform domain LMS and we introduce a new algorithm. The aim of this algorithm is to increase the convergence speed of its time domain counterpart and also to reduce its complexity.

At the end of the second part the scrambled LMS is briey presented and it is compared with the LMS and transform domain LMS. The chosen framework is the digital data transmission over a telephone line. The analytical expressions of the mean squared error and mean squared coefficient error are derived for this special case and a discusion about their convergence speed and steady-state error is given. The aim of this discusion is to provide some useful information about the utility of each algorithm.

In the first two parts of the thesis, computer experiments, showing the performances of all the proposed algorithms, are provided for system identi cation application. Since many of the algorithms can be implemented also in other applications, in the third part of this thesis, channel equalization, CDMA multiuser detection and echo cancellation applications are also addressed.

##### Kokoelmat

- Väitöskirjat [4865]