Branko Kovačević, Zoran Banjac, Milan Milosavljević (auth.)-Adaptive Digital Filters-Springer-Verlag Berlin Heidelberg (2013).pdf

Branko Kovacˇevic´ Zoran Banjac Milan Milosavljevic´

Academic Mind University of Belgrade - School of Electrical Engineering University Singidunum Springer-Verlag Berlin Heidelberg 2013

Adaptive Digital Filters

Branko Kovacˇevic´ Zoran Banjac Milan Milosavljevic´ •

Adaptive Digital Filters

Academic Mind University of Belgrade - School of Electrical Engineering University Singidunum, Belgrade Springer-Verlag Berlin Heidelberg

Milan Milosavljevic´ University of Belgrade and Singidunum University Belgrade Serbia

Branko Kovacˇevic´ University of Belgrade Belgrade Serbia Zoran Banjac School of Electrical and Computing Engineering of Applied Studies Belgrade Serbia

ISBN 978-3-642-33560-0 DOI 10.1007/978-3-642-33561-7

ISBN 978-3-642-33561-7

(eBook)

Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2013935402 ISBN of Academic Mind : 978-86-7466-434-6 Ó Academic Mind Belgrade and Springer-Verlag Berlin Heidelberg 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer and Academic Mind. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Academic Mind is official publisher of University of Belgrade, School of Electrical Engineering (www.akademska-misao.rs) Springer is part of Springer Science+Business Media (www.springer.com)

Preface

The book Adaptive Digital Filters appeared as a result of years of cooperation between the Department for Signal Processing within the Institute of Applied Mathematic and Electronics, Belgrade, Serbia and the Division for Automation that has later evolved to the present Department for Signals and Systems within the School of Electrical Engineering, University of Belgrade. This cooperation started back in mid-1970s, with a goal to research the phenomenon of speech and has continued up until present days. Among important results of joint science research in the fields of modeling, analysis, processing, recognition, and transmission of speech signals are, besides mathematical algorithms, program packages, technical solutions, and electronic instruments, also numerous research papers, either published in leading international science journals or presented in proceedings of prestigious international science conferences. Since the very inception of their cooperation the Institute and the School introduced a custom to inform a wider circle of domestic researchers and experts, as well as the students of mathematics, electrical engineering, computing, and related areas about the most important results of their joint projects. This goal was usually achieved by publishing science monographs in Serbian language, and this was the way two science monographs appeared, ‘‘Speech signal processing and recognition’’ (a group of authors from the Institute and the School, published by the Center of High Military Schools, Belgrade, 1993) and ‘‘Robust digital processing of speech signals’’ (published by Academic Mind, Belgrade, 2000). It is important to mention here that the quoted publications were preceded by a number of master theses and doctoral dissertations. In this way, this book represents a continuation of the good practice introduced by the Institute and the School and presents the results of joint research within the past decade. The youngest author of this book, Dr. Zoran Banjac, finished his master’s thesis and doctoral dissertation in the course of his work within these projects. The aim and the stance of this book was maybe best defined by its reviewers, professors at the School of Electrical Engineering, the University of Belgrade, professors Ljiljana Milic´ and Dušan Drajic´, who wrote in the conclusion of their review: ‘‘The monograph Adaptive Digital Filters presents to our scientific and expert community an important discipline which has been underrepresented prior to the appearance of this book. The book first makes the reader acquainted with the basic terms of filtering and adaptive filtering, to further v

vi

Preface

introduce the reader into the field of advanced modern algorithms, some of which represent a contribution of the authors of the book. The work in the field of adaptive signal processing requires the use of a complex mathematical apparatus. The manner of exposition in this book presumes a detailed presentation of the mathematical models, a task done by the authors in a clear and consistent way. The chosen approach enables everyone with a college level of mathematics knowledge to successfully follow the mathematical derivations and descriptions of algorithms in the book. The algorithms are presented by flow charts, which facilitates their practical implementation. The book gives many experimental results and treats the aspects of practical application of adaptive filtering in real systems. The book will be useful both to students of undergraduate and graduate studies, and to all of those who did not have an opportunity to master this important science field during their formal education’’. The authors would like to express their gratitude to the referees for their useful suggestions and advices which contributed significantly to the quality of the book. The text of the book is divided into six chapters. The first, introductory chapter, considers generally three most often used theoretical approaches to the design of linear filters—the conventional approach, the optimal filtration, and the adaptive filtration. The further text analyzes only the third approach, i.e., the adaptive filtration. Chapter 2 presents the basic structures of adaptive filters. It also considers the criterion function for the optimization of the parameters of adaptive filters and analyzes the two basic numerical methods for the determination of the minimum of the criterion function: the Newton method and the method of steepest descent. After presenting the basic concept of adaptive filtering, it overviews the standard and the derived adaptive algorithms of the Least Mean Square Error—LMS type and Recursive Least Square—RLS algorithm, for the sake of further analysis and estimation of the possibilities to modify them in order to improve the characteristics of the mentioned adaptive algorithms. Also, potential advantages of the Infinite Impulse Response—IIR filters impose a need for their more intensive use, as well as for the analysis of the adaptation of the proposed solution for the systems with Finite Impulse Response—FIR systems, as well as for the IIR systems. This is the reason why in the second chapter care has been given to this problem too. An analysis of the ability of adaptive algorithms to follow nonstationary changes in the system, together with the synthesis of efficient algorithms based on variable forgetting factor, is presented in Chap. 3. A comparison has been made among a number of strategies for the choice of forgetting factor (extended prediction error, parallel adaptation, and Fortecue-Kershenbaum-Ydstie algorithm) against their ability to follow nonstationary changes and the complexity of the implementation of algorithms. The most convenient strategies for the choice of variable forgetting factor from the practical point of view were emphasized. Chapter 4 presents an original approach to the design of an FIR-type adaptive algorithm with a goal to increase the convergence speed in the parameter estimation process. The approach is based on an optimum approach to the

Preface

vii

construction of input signal, which belongs to the D-class of optimal experiment planning. The properties of the proposed algorithm were subsequently analyzed through the practical problem of local echo cancellation in scrambling systems. Besides that, a possibility has been shown to apply this approach in nonstationary environments through the application of a convenient strategy for the choice of variable forgetting factor. Robustification of adaptive algorithms against impulsive nonstationary noise in the desired response is considered in Chap. 5. An original robust algorithm based on the LMS approach is presented and an analysis is given of the possibility to apply D-optimal input to the robust recursive least squares algorithm in order to improve the convergence speed. After that a robust RLS algorithm is introduced with a recursive estimation of scaling factors for the case when besides the impulsive noise sudden changes of the system dynamics also occur. Besides that, another novel algorithm is presented which besides its robust properties against impulse noise also has the ability to track nonstationary changes of the values of estimated parameters. Contrary to the previous algorithm, the proposed approach is based on the design of robust detector of impulse noise, based on the design of robust median filter and the application of either robust or standard RLS-type procedure to the estimation of filter parameters, depending on the detection result. Chapter 6 is dedicated to the analysis of the possibility to apply the proposed adaptive digital filters for signal echo cancellation in telecommunication networks. Let us notice at the end that the algorithms and solutions considered in this book may find application in the wider area of adaptive signal processing, like adaptive cancellation of echo signal, adaptive noise cancellation, adaptive equalization, as well as in processing of signals with various physical nature (speech and image signals, biomedical signals, signals from radars, sonars, satellites, and other intelligent sensors. Belgrade, May 2012

The Authors

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Conventional Approach to the Design of Digital Filters . 1.2 Optimal Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Wiener Filter . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Adaptive Filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

1 1 9 9 16 27

2

Adaptive Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Structures of Digital Filters. . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Filters with Infinite Impulse Response (IIR Filters) . . . 2.2.2 Filters with Finite Impulse Response (FIR Filters) . . . 2.3 Criterion Function for the Estimation of FIR Filter Parameters . 2.3.1 Mean Square Error (Risk) Criterion: MSE Criterion . . 2.3.2 Minimization of the Criterion of Mean Square Error (Risk) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Adaptive Algorithms for the Estimation of Parameters of FIR Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Least Mean Square (LMS) Algorithm . . . . . . . . . . . . 2.4.2 Least Squares Algorithm (LS Algorithm). . . . . . . . . . 2.4.3 Recursive Least Squares (RLS) Algorithm . . . . . . . . . 2.4.4 Weighted Recursive Least Squares (WRLS) Algorithm with Exponential Forgetting Factor . . . . . . 2.5 Adaptive Algorithms for the Estimation of the Parameters of IIR Filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Recursive Prediction Error Algorithm (RPE Algorithm). . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Pseudo-Linear Regression (PLR) Algorithm . . . . . . . .

. . . . . . .

. . . . . . .

31 31 31 32 34 36 37

..

39

. . . .

. . . .

45 46 49 51

..

53

..

59

.. ..

67 72

ix

x

3

4

5

Contents

Finite Impulse Response Adaptive Filters with Variable Forgetting Factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Choice of Variable Forgetting Factor . . . . . . . . . . . . . . . . 3.1.1 Choice of Forgetting Factor Based on the Extended Prediction Error . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Fortescue–Kershenbaum–Ydstie Algorithm . . . . . . 3.1.3 Parallel Adaptation Algorithm (PA-RLS Algorithm) 3.1.4 Generalized Weighted Least Squares Algorithm with Variable Forgetting Factor . . . . . . . . . . . . . . 3.1.5 Modified Generalized Likelihood Ratio: MGLR Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Experimental Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Comparative Analysis of Recursive Algorithms for the Estimation of Variable Forgetting Factor (Analysis of RLS Algorithm with EGP, FKY and PA Strategy for the Calculation of Variable Forgetting Factor) . . . . . . . . . . . . . . . . . . . . . . . .

.... ....

75 75

.... .... ....

76 78 86

....

93

.... ....

96 101

....

101

Finite Impulse Response Adaptive Filters with Increased Convergence Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Definition of the Parameter Identification Problem . . . . . . . . 4.2 Finite Impulse Response Adaptive Filters with Optimal Input . 4.3 Convergence Analysis of Adaptive Algorithms . . . . . . . . . . . 4.4 Application of Recursive Least Squares Algorithm with Optimal Input for Local Echo Cancellation in Scrambling Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Definition of the Local Echo Cancellation Problem in Scrambling Systems . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Experimental Analysis . . . . . . . . . . . . . . . . . . . . . . . 4.5 Application of Variable Forgetting Factor to Finite Impulse Response Adaptive Filter with Optimal Input . . . . . . . . . . . . Robustification of Finite Impulse Response Adaptive Filters 5.1 Robust Least Mean Square Algorithm . . . . . . . . . . . . . . 5.1.1 Robustification of Least Mean Square Algorithm: Robust LMS Algorithm . . . . . . . . . . . . . . . . . . . 5.1.2 Stability Analysis of Robust Estimators . . . . . . . . 5.1.3 Simulation-Based Experimental Analysis . . . . . . . 5.2 Robust Recursive Least Squares Algorithm with Optimal Output . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Experimental Analysis . . . . . . . . . . . . . . . . . . . . 5.3 Adaptive Estimation of the Scaling Factor in Robust Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Experimental Analysis . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

109 110 112 115

..

131

.. ..

133 134

..

139

..... .....

147 149

..... ..... .....

152 155 158

..... .....

162 168

..... .....

170 177

Contents

5.4

6

xi

Robust Recursive Least Squares Algorithm with Variable Forgetting Factor and with Detection of Impulse Noise . . . . . . . 5.4.1 Experimental Analysis . . . . . . . . . . . . . . . . . . . . . . . . .

Application of Adaptive Digital Filters for Echo Cancellation in Telecommunication Networks . . . . . . . . . 6.1 Echo: Causes and Origins. . . . . . . . . . . . . . . . . . . . . 6.1.1 Echo in Speech Transmission . . . . . . . . . . . . . 6.1.2 Acoustic Echo . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Echo in Data Transfer . . . . . . . . . . . . . . . . . . 6.1.4 Basic Principles of Adaptive Echo Cancellation 6.2 Mathematical Model of an Echo Cancellation System . 6.3 Analysis of the Influence of Excitation Signal to the Performance of Echo Cancellation System for Speech Signal Transmission. . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

180 184

. . . . . . .

187 189 189 191 192 193 196

.......

197

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

205

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

209

Abbreviations

AECM APE BP BS EC EE EGP EPE ERLE FF FIR FKY HP IIR LAD LMS LP LS MAD MIMO ML MGLR MLMS MSE NEE NLMS ODE OE PA-RLS PLR RLMS RLS RMN

Asymptotic error covariance matrix Adaptive echo suppression Band pass filter Band stop filter Echo canceller Equation error Extended prediction error Extended prediction error Echo return loss enhancement Forgetting factor Finite impulse response Fortescue-Kershenbaum-Ydstie High pass filter Infinite impulse response Least absolute deviation Least mean square Low pass filter Least square Median absolute deviation Multiple input–multiple output Maximum likelihood Modified generalized likelihood ratio Median least mean square Mean square error Normalized estimation error Normalized least mean square Ordinary differential equation Output error Parallel adaptation recursive least square Pseudo linear regression Robust least mean square Recursive least square Robust mixed norm xiii

xiv

RPE RRLS RRLSO RW SISO SNR VFF WRLS

Abbreviations

Recursive prediction error Robust recursive least square RRLS algorithm with optimal input Random walk Single input–single output Signal to noise ratio Variable forgetting factor Weighted recursive least square

Chapter 1

Introduction

Electrical filters find numerous practical applications, often for very different purposes. One of the basic applications of filters is suppression of the influence of noise or interference, with a goal to extract useful components of the signal. A large subgroup is linear filters, their crucial property being a linear relation between the signals at their input and at their output. Basically, there are three theoretical approaches to the design of such filters: (1) conventional approach; (2) optimal filtering, the so-called Wiener or Kalman filtering and (3) self-adjusting filters or adaptive filtering. Although this book is mostly dedicated to the third approach to the design of linear filters, i.e. to the synthesis of self-adjusting or adaptive digital filtering, for the sake of completeness the introductory chapter will consider the other two approaches to the design of such systems.

1.1 Conventional Approach to the Design of Digital Filters The conventional approach is based on the design of frequency selective filters which require prior knowledge on the spectral contents of the filtered signal. Strictly speaking, the often used term ‘‘frequency-selective filters’’ suggests that these systems amplify frequency components from a given range, while suppressing the components from all other ranges. However, in a general case, all dynamical systems that modify signals in certain frequency ranges can be understood as filters [1–3]. A linear filter is basically a linear dynamic system with constant parameters, and the design itself reduces to the choice of the values of these parameters so that the system satisfies the predefined requirements regarding the bandwidth, amplification within the bandwidth, attenuation of frequency components outside the bandwidth, resonant frequencies, etc. Conventional filter design implies the following steps: (1) specifications of the required properties of a linear, time-invariant dynamic system; (2) approximation of these specifications, utilizing the properties of causal discrete systems (the response or output of such systems is equal to zero prior to switching on the input

B. Kovacˇevic´ et al., Adaptive Digital Filters, DOI: 10.1007/978-3-642-33561-7_1,  Academic Mind Belgrade and Springer-Verlag Berlin Heidelberg 2013

1

2

1 Introduction

or excitation signal); (3) system realization. Although these three steps are not independent, a special care is dedicated in literature to the second step, since the first step is primarily dependent on the field of application of the filter, while the third step is connected with the existing implementation technology. It is interesting to note that digital filters are often used to process signals obtained from continuous signals utilizing analog–digital (A/D) converters. When a digital filter is used to process analog signals, the specification of the digital filter and the effective continual filters (whose digital approximation is designed) is given in the frequency domain. This is primarily valid for frequency selective filters, like lowpass filter (LP), band-pass filter (BP) and high-pass filter (HP). If the sampling (discretization) period, T, is sufficiently short, no overlap will occur between the frequency components from different periods in the periodic frequency characteristics (spectrum) of the discretized continual filter (aliasing effect). Thus in Nyquist range jXj\ Tp, where X is the analog angular frequency, the digital filter will behave virtually identical to the desired continual filter with the frequency response   jXT  H e ; jXj\p=T Heff ðjXÞ ¼ ð1:1Þ 0; jXj  p=T In this case the specification that may be posed to the effective continual filter may also be posed as the requirements for the digital filter, by introducing substitution x ¼ XT (x is denoted as digital, and X analog angular frequency), i.e. Hðejx Þ becomes the specification of the digital filter within a (basic) period of infinite and periodic frequency response of the digital filter (represents an infinitely periodic function of the argument x, with a period 2p)  x   H ejx ¼ Heff j ; jxj\p ð1:2Þ T A typical characteristic of a digital filter is shown in Fig. 1.1, plotted for a normalized (digital) frequency 0  x  p. In order to satisfy the posed requirements, the real characteristics must fulfill the following:

Fig. 1.1 Specification of requested amplitudefrequency characteristics of an LP filter

H eff ( jΩ) 1 + δ1 1 − δ1

δ2

Bandpass Transition range range

ΩP

ΩS

Bandstop range

π /T

Ω

1.1 Conventional Approach to the Design of Digital Filters

3

   ð1  d1 Þ  H ejx   ð1 þ d1 Þ; jxj  xp ; xp ¼ Xp T   jx  H e   d2 ; xs  x  p; xs ¼ Xs T

ð1:3Þ ð1:4Þ

Many practically used filters are specified in the presented manner, without any limitation posed to their phase characteristics, stability (a system is stable if its output in a steady or equilibrium state which commences after its transient state vanishes is dictated by its excitation only) and causality (a causal system has a property that its output signal or response is equal to zero before the input or excitation signal is brought to it). At the same time it is known that an Infinite Impulse Response (IIR) filter must be causal and stable, while the basic assumption for a Finite Impulse Response (FIR) filter is that its phase is linear, since such a filter by itself represents a stable and causal system. In any case, according to the specified requirements one should choose a filter with frequency characteristics satisfying the posed limits, which represents a problem of functional approximation (Fig. 1.2). Traditional approach to the design of IIR filters is based on the transformation of the transfer function of a continuous filter satisfying given requirements into a corresponding discrete transfer function [1, 3]. This approach stems from the following: • Procedures for the design of continuous filters are well researched, numerous and they furnish good results; • Many methods for continuous filter design are very simple and their transformation to the digital domain significantly simplifies the numerical procedure of digital filter design; • Many simple methods used for the design of continuous filters do not furnish simple solutions in closed form if directly applied for digital filter design. The Butterworth filter and the Chebyshev filter are most often used for the design of continuous filters. The Butterworth LP filter is designed to satisfy the requirement to have a maximally flat amplitude-frequency characteristic in the passband. For an N-order filter this means that the first (2N  1) derivatives of the squared amplitude Fig. 1.2 Satisfactory amplitude-frequency characteristics of an LP filter

H (e j Ω ) 1 + δ1

1 − δ1

δ2 ΩP

ΩS

π /T

Ω

4

1 Introduction

characteristic at zero frequency should be equal to zero. Another important property of the Butterworth filter is the monotonicity of the amplitude characteristics both in the passband and the stopband range. A squared amplitude characteristic of a continual (analog) Butterworth filter is defined in the following manner [1, 3]: jHc ðjXÞj2 

1 1 þ ðjX=jXc Þ2N

ð1:5Þ

:

Some of these characteristics for different values of the parameter N (filter order) are shown in Fig. 1.3. With an increasing parameter N, the filter characteristic becomes sharper and the transition from the passband to the stopband becomes steeper. In other words, the characteristic is flatter in the passband, and in the stopband it is closer to zero if the filter order N is higher. According to the squared amplitude characteristic, one may conclude that Hc ðsÞHc ðsÞ ¼ 1þ

1  2N :

ð1:6Þ

s jXc

The zeroes of the polynomial in the denominator (the so-called poles of the filter) in the last expression are sk ¼ ð1Þ1=2N ðjXc Þ ¼ Xc eðjp=2N Þð2kþN1Þ ; k ¼ 0; 1; . . .; 2N  1:

ð1:7Þ

Thus these 2N zeroes are located at a circle with a radius Xc , in the s (complex frequency) plane. Therefore the poles of this complex function are symmetrically distributed around the imaginary axis and none of them can ever be on the imaginary axis. The angular distance between them is p=N. This means that one half of them, those located in the left half-plane of the s-plane (stable poles), should be associated with the function Hc ðsÞ, while the other half should be Fig. 1.3 Amplitudefrequency characteristics of Butterworth filters of N-th order, N = 2, 4 and 8

N=2 N=4 N=8

1

|Hc(jΩ )|2

0.8 0.6 0.4 0.2 0

Ωc

Ω

1.1 Conventional Approach to the Design of Digital Filters

5

associated with the function Hc ðsÞ. Thus the stable transfer function of a Butterworth filter can be written in the form H c ðsÞ ¼

K ; ðs  s1 Þ. . .ðs  sN Þ

ð1:8Þ

where the poles si are from the circle with the radius Xc from the left half-plane of the s-plane, while the amplification K is calculated from the condition that (unit amplification at zero frequency) Y Hc ð0Þ ¼ 1 ) K ¼ ð1ÞN si : ð1:9Þ i¼1;N

The amplitude-frequency characteristics (frequency response) of a Butterworth filter is monotonous both in the passband and in the stopband. A much more efficient solution, in the sense of satisfying the posed requirements by a lowerorder filter, is obtained by designing a filter whose amplitude-frequency characteristics is nonmonotonous. In this way one arrives to Chebyshev filters, where we discern two cases. One of them regards the amplitude-frequency characteristics that is nonmonotonous in the stopband (Chebyshev filter of the first kind), while the other type is characterized by an amplitude-frequency characteristics monotonous in the passband and nonmonotonous in the stopband (Chebyshev filter of the second kind). The squared amplitude characteristics of a Chebyshev filter of the first kind is jHc ðjXÞj2 ¼

1 1þ

e2 VN2 ðX=Xc Þ

;

ð1:10Þ

where VN ðxÞ denotes a Chebyshev polynomial of the N-th degree VN ð xÞ ¼ cosðN arccosð xÞÞ:

ð1:11Þ

It turns out that there is a recurrence relation between Chebyshev polynomials (Fig. 1.4) VNþ1 ð xÞ ¼ 2xVN ð xÞ  VN1 ð xÞ;

ð1:12Þ

where V0 ðxÞ ¼ 1 and V1 ðxÞ ¼ x. Taking into account the form of the Chebyshev it is clear that the value of the function jHc ðjXÞj2 varies in the range hpolynomials, i 1 1þe2

; 1 for frequencies in the range X 2 ½0; Xc .

The poles of a Chebyshev filter are located on an ellipse in the s-plane. The ellipse is defined by two circles with their radii equal to the major and the minor axis of the ellipse, respectively. The length of the minor axis is 2aXc , where  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  1=N a  a1=N ; a ¼ e1 þ 1 þ e2 : ð1:13Þ a¼ 2 The length of the major axis is 2bXc , where

6

1 Introduction

Fig. 1.4 Typical graph of the squared Chebyshev function

b¼

 1  1=N a þ a1=N : 2

ð1:14Þ

The poles of the filter are determined by a procedure in which in the first step one sketches 2 N half-lines in the s-plane with their starting point in the origin, they being mostly equidistant (each two half-lines form an angle equal to p=N radian) and with positions symmetric with regard to the real and the imaginary axis. In other words, if N is odd, these half-lines form an angle of ip=N; i ¼ 0; 1; . . .; 2N  1 with the positive part of the real axis, and an angle of ði þ 0:5Þ p=N; i ¼ 0; 1; . . .; 2N  1 if N is an even number. After that one determines the cross-sections of these half-lines with the circle whose center is in the origin and the radius is aXc . The real parts of these cross-sections, which are negative, simultaneously represent the real parts of the desired poles. The imaginary parts of the desired poles are determined in the cross-section of the corresponding halflines with a circle with its center in the origin and with a radius bXc . The procedure of the determination of the fourth-order filter poles (N = 4) is shown in Fig. 1.5. The obtained poles are simultaneously on the ellipse with its semi-axes aXc and bXc . In that case one adopts the following form for the transfer function of the Chebyshev filter H c ðsÞ ¼

K ; ðs  s1 Þðs  s2 Þ. . .ðs  sN Þ

ð1:15Þ

where K¼

XNc 2N1 e

:

ð1:16Þ

A Chebyshev filter of the second kind is characterized by the fact that its squared amplitude characteristic is given by the following expression

1.1 Conventional Approach to the Design of Digital Filters Fig. 1.5 Procedure for determination of the poles for a fourth order Chebyshev filter of the first kind

7

bΩc s1 s2

aΩc

s3 s4

jHc ðjXÞj2 ¼

1 1þ

1 ½e2 VN2 ðX=Xc Þ

:

ð1:17Þ

There are several ways to transform a continuous filter into a digital filter. One of them is the method of impulse invariance, another one is based on the bilinear transformation, while the third is the co-called matching method [3]. The impulse invariance method starts from the assumption that the impulse response of a digital filter should be equivalent to the values of the impulse response of a continuous filter in the moments of sampling h ½n ¼ Thc ðnTÞ:

ð1:18Þ

However, the matching method is much more often used; it is not based on generation of impulse response in time domain, but on the transformation of the filter transfer function. This method starts from the assumption that the transfer function of a continuous filter can be written as a sum of partial fractions Hc ðsÞ ¼

N X Ak : s  sk k¼1

ð1:19Þ

The corresponding impulse response is then determined as the inverse Laplace transform of the given transfer function [3], i.e.

8

1 Introduction

hc ð t Þ ¼

8 N