Learning algorithms for adaptive digital filtering

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Learning algorithms for adaptive digital ltering

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Nambiar, Raghu (1993) Learning algorithms for adaptive digital ltering, Durham theses, Durham University. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/5544/

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Learning A l g o r i t h m s f o r A d a p t i v e

D i g i t a l Filtering:

Raghu Nambiar

School of Engineering and Computer Science

University of Durham

A thesis submitted in partial fulfilment of the require-ments of the Council of the University of Durham for the Degree of Doctor of Philosophy ( P h . D . ) .

January 1993

Abstract

In this thesis, we consider the problem of parameter optimisation in adaptive digital filtering. Adaptive digital filtering can be accomplished using both Finite Impulse Response (FIR) filters and Infinite Impulse Response Filters (IIR) filters. Adaptive FIR filtering algorithms are well established. However, the potential computational advantages of I I R filters has led to an increase in research on adaptive I I R filtering algorithms. These algorithms are studied in detail in this thesis and the limitations of current adaptive I I R filtering algorithms are identified. New approaches to adaptive I I R filtering using in-telligent learning algorithms are proposed. These include Stochastic Learning Automata, Evolutionary Algorithms and Annealing Algorithms. Each of these techniques are used for the filtering problem and simulation results are pre-sented showing the performance of the algorithms for adaptive I I R filtering. The relative merits and demerits of the different schemes are discussed. Two practical applications of adaptive I I R filtering are simulated and results of us-ing the new adaptive strategies are presented. Other than the new approaches used, two new hybrid schemes are proposed based on concepts from genetic algorithms and annealing. I t is shown with the help of simulation studies, that these hybrid schemes provide a superior performance to the exclusive use of any one scheme.

TO My VA1ZEMTS

To whom much more is owed than can be mentioned here.

Acknowledgments

It would be presumptuous to think that a thesis is the sole effort of a single individual. This is my sincere effort to thank all those who have, either directly or indirectly, helped during my stay at Durham and in my study.

o To my parents and sister for all the love and support.

o To my supervisor Prof. Mars - his wit and good cheer always made things more tractable, and for his infinite patience especially during the last stages.

o To Dr. Tang - for his thought provoking questions during the initial phase. o To the British Council - for all the support especially financial, and in particular

to Angie Stephenson - my program adviser at the British Council - she was a true friend.

o To Prof. Sengupta - all this would have not materialised if it had not been for his help.

o To Ritu, Bipul, A m i t , Manju, Bipul, Rashmi, Rajeev, Nithya and John - for providing a touch of home in the cold climes of Britain.

o To Shyam Sunder - for being such an accommodating host during my visits to the United States.

e To John, Alan and David in the lab - for all the good times.

o To numerous friends at the halls of residence - none is mentioned by name lest I offend those who I have forgotten by oversight.

o To Sylvia - for all the help during the three years.

Q To Jamie, Gemma, Neil and Trisha - for .... well, they wanted to be in the acknowledgments !!.

Declaration

I hereby declare that this thesis is a record of work undertaken by myself, that it has not been the subject of any previous application for a degree, and that all sources of information have been duly acknowledged.

(c) C o p y r i g h t 1993, Raghu N a m b i a r

The copyright of this thesis rests with the author. No quotation from i t should be published without his written consent, and information derived from it should be acknowledged.

Contents

1 I n t r o d u c t i o n 1 1.1 Why Adaptive Filtering ? 1

1.2 Outline of Thesis 4

2 A d a p t i v e D i g i t a l F i l t e r i n g 8

2.1 Introduction 8 2.2 Adaptive FIR Filtering 12

2.3 Adaptive I I R Filtering 13 2.3.1 Introduction 13 2.3.2 Different Formulations of Estimation Error 14

2.3.3 Adaptive Algorithms 16 2.4 Alternative Realizations 19

2.4.1 Parallel Form 20 2.4.2 Cascade Form 21 2.4.3 Lattice Form 22 2.5 Applications of Adaptive I I R Filtering 22

2.5.1 Adaptive Noise Cancelling 23 2.5.2 Adaptive Equalization 24

2.6 Discussion 25 3 Stochastic L e a r n i n g A u t o m a t a 32

3.1 Introduction 32 3.2 Stochastic Learning Automata 33

3.2.2 The Environment . . . 35 3.2.3 Norms of Behaviour 37 3.3 Learning Algorithms 38

3.3.1 Standard Learning Algorithms 38 3.3.2 Discretised Learning Algorithms 41

3.3.3 Estimator Algorithms 42 3.3.4 S-Model Learning Schemes 48

3.4 Interconnected Automata 51 3.4.1 Hierarchical Learning Automata 51

3.4.2 Automata Games 52 3.5 Discussion 53 4 A d a p t i v e D i g i t a l F i l t e r i n g using Stochastic L e a r n i n g A u t o m a t a 56 4.1 Introduction 56 4.2 Simulation Configuration 57 4.2.1 Introduction 57 4.2.2 Using Stochastic Learning Automata 57

4.2.3 Different Categories of Modeling 60

4.3 Simulation Results 62 4.3.1 Introduction 62 4.3.2 Results using P-Model Learning Algorithms 64

4.3.3 Results using S-Model Learning Algorithms 66

4.3.4 Other Categories 70 4.3.5 Automata Games and Hierarchical Schemes 71

4.4 Conclusions 72

5 Genetic and E v o l u t i o n a r y O p t i m i s a t i o n 90

5.1 Introduction 90 5.2 Genetic Algorithms 94

5.2.1 Introduction 94 5.2.2 Standard Genetic Operations 97

5.2.4 Adaptive Extensions of Genetic Algorithms 103

5.3 Evolutionary Strategies 104 5.3.1 Introduction 104 5.3.2 Standard Evolutionary Strategies 105

5.3.3 Improved Evolutionary Strategies 109

5.4 Evolutionary Programming I l l

5.4.1 Introduction I l l 5.4.2 Salient Features 112 5.4.3 Adaptive Extensions to Evolutionary Programming 113

5.5 Discussion 114 6 A d a p t i v e D i g i t a l F i l t e r i n g using G e n e t i c and E v o l u t i o n a r y O p t i m i -sation 116 6.1 Introduction 116 6.2 Simulation Configuration 117 6.2.1 Genetic Algorithms 117 6.2.2 Evolutionary Strategies and Programming 120

6.3 Simulation Results 121 6.3.1 Genetic Algorithms 121

6.3.2 Evolutionary Strategies 130 6.3.3 Evolutionary Programming 132 6.3.4 Applications using the Adaptive I I R Filter 134

6.4 Conclusions 137

7 S i m u l a t e d and Genetic A n n e a l i n g 171

7.1 Introduction 171 7.2 Simulated Annealing 173

7.3 Fast Simulated Annealing 176 7.4 Very Fast Simulated Reannealing 177

7.5 Genetic Annealing 178 7.5.1 Introduction 178 7.5.2 Hybrid Scheme - I 179

7.5.3 Hybrid Scheme - I I 181 7.6 Simulation Configuration and Results 182

7.7 Conclusions 184

8 Conclusions and F u r t h e r W o r k 199

8.1 Conclusions 199 8.2 Further Work 201

8.2.1 Use of Genetic Algorithms in Non-stationary Environments . . 201

8.2.2 Parallel Implementation 202 8.2.3 Genetic Algorithms and Neural Networks 203

8.2.4 Theoretical Analysis using Natural Genetics 203

8.2.5 Hybrid Schemes 204

A p p e n d i x A 205

A p p e n d i x B 207

B i b l i o g r a p h y 210

List of Figuires

1.1 Conventional and Adaptive Filtering Configurations 6 1.2 Direct and Inverse System Modeling Configurations 7

2.1 Digital Filter 26 2.2 Adaptive Digital Filter 26

2.3 System Identification Configuration 27

2.4 Equation Error Formulation 27 2.5 Equation Error Identifier 28 2.6 Output Error Formulation 28 2.7 Parallel Form Realization 29 2.8 Lattice Form Realization 29 2.9 Adaptive Noise Canceling Configuration 30

2.10 Adaptive Equalization Configuration 31

3.1 Stochastic Learning Automata 54 3.2 Hierarchical Stochastic Learning Automata 55

4.1 System Identification Configuration incorporating Stochastic Learning

Automata 74 4.2 Discretisation of the Parameter Space 74

4.3 The New Scheme of Error Estimation 75 4.4 Performance of Standard Learning Algorithms 76

4.5 Performance of Discretised Learning Algorithms 77 4.6 Performance of Estimator Learning Algorithms 78

4.8 Performance of Discretised Pursuit Algorithms 80 4.9 Performance of S-LRI Learning Algorithms (Old Normalisation) . . . 81

4.10 Performance of S-LRI Learning Algorithms (New Normalisation) . . 82 4.11 Performance of Estimator Learning Algorithms (S-Model) (Old

Nor-malisation) 83 4.12 Performance of Estimator Learning Algorithms (S-Model) (New

Nor-malisation) 84 4.13 Performance of Relative Reward Learning Algorithms (S-Model) (Old

Normalisation) 85 4.14 Performance of Relative Reward Learning Algorithms (S-Model) (Old

Normalisation) 86 4.15 Performance of Relative Reward Learning Algorithms (S-Model) (New

Normalisation) 87 4.16 Performance of Relative Reward Learning Algorithms (S-Model) (New

Normalisation) 88 4.17 Performance of P-Model Learning Algorithms (Category ( I V ) Model) 89

6.1 Comparison between Genetic and Random Search Algorithms . . . . 138 6.2 Comparison between Genetic and Random Search Algorithms . . . . 139 6.3 Comparison between Genetic and Stochastic Learning Automata A l

-gorithms 140 6.4 Different Order Filters 141

6.5 Effect of Mutation 142 6.6 Effect of Crossover 143 6.7 Effect of Population Size 144 6.8 Effect of Coding Schemes 145 6.9 Effect of the Number of Bits 146 6.10 Effect of New Crossover Schemes (pm = 0.075) 147

6.11 Effect of New Crossover Schemes ( pm = 0.025) 148

6.12 Effect of Improved Selection Operations 149 6.13 Effect of the Ranking Selection Scheme 150

6.14 Effect of the Ranking Elitist Selection Scheme 151

6.15 Effect of Measurement Noise 152 6.16 Results using Self Adaptive Genetic Algorithm 153

6.17 Effect of Standard Deviation in ESs 154 6.18 Effect of the Number of Parents/Offspring 155

6.19 Effect of Parents in Evolutionary Programming 156 6.20 Effect of the Number of Competitions in EP 157 6.21 Effect of the Number of Competitions in EP 158 6.22 Adaptive Noise Canceling - Sum of Sinusoids 159 6.23 Adaptive Noise Canceling - Square Wave 160 6.24 Adaptive Noise Canceling - PRBS Input 161 6.25 Adaptive Noise Canceling - PRBS Input 162 6.26 Evolution of the Adaptive Noise Canceling 163 6.27 Evolution of the Adaptive Noise Canceling 164 6.28 Evolution of the Adaptive Noise Canceling 165 6.29 Evolution of the Adaptive Noise Canceling 166 6.30 Results from the Adaptive Equalisation Experiment 167

6.31 Results from the Adaptive Equalisation Experiment 168 6.32 Results from the Adaptive Equalisation Experiment 169 6.33 Results from the Adaptive Equalisation Experiment 170

7.1 Results using Classical Simulated Annealing 186 7.2 Results using Fast Simulated Annealing 187 7.3 Comparative Results using Classical and Fast Simulated Annealing

(Decay Parameter = 0.9) 188 7.4 Results using Hybrid Scheme - I (Decay Parameter = 100) 189

7.5 Results using Hybrid Scheme - I (Decay Parameter = 100) 190 7.6 Results using Hybrid Scheme - I (Decay Parameter = 50) 191 7.7 Results using Hybrid Scheme - I (Decay Parameter = 50) 192 7.8 Results using Hybrid Scheme - I (Decay Parameter = 15) 193 7.9 Results using Hybrid Scheme - I (Decay Parameter = 15) 194

7.10 Results using Hybrid Scheme - I I (pm = 0.075, Decay = 0.9) 195

7.11 Results using Hybrid Scheme - I I (pm = 0.075, Decay = 0.7) 196

7.12 Results using Hybrid Scheme - I I (pm = 0.025, Decay = 0.9) 197

List of A b b r e v i a t i o n

A R M A Auto Regressive Moving Average D P A Discretised Pursuit Algorithm E P Evolutionary Programming ESs Evolutionary Strategies F I R Finite Impulse Response F S A Fast Simulated Annealing G A s Genetic Algorithms

H A R F Hyperstable Adaptive Recursive Filter I I R Infinite Impulse Response

L M S Least Mean Square L R S Linear Random Search m G A s Messy Genetic Algorithms M S E Mean Square Error

M S O E Mean Square Output Error P L R PseudoLinear Regression R L M S Recursive Least Mean Square R L S Recursive Least Square R P E Recursive Prediction Error SA Stochastic Automaton

S H A R F Simple Hyperstable Adaptive Recursive Filter S L A Stochastic Learning Automata

S P R Strictly Positive Real

V F S R Very Fast Simulated Reannealing

Chapter 1

I n t ro duct

ion

1.1 W h y A d a p t i v e F i l t e r i n g ?

rJ1 he term filtering a signal refers to processing the signal in such a manner, so as to

extract relevant information from i t . This could relate to enhancing certain desired components or on the other hand the removal of interfering noisy components. The earliest filters were usually of the analogue type. However the advent of digital elec-tronics and the subsequent rapid developments i n integrated circuit technology meant that digital filters were a cheaper and more reliable alternative to the conventional analogue niters. There are a number of advantages of digital filters over the analogue filters, these include easy modification of signal processing functions by means of soft-ware, higher order of precision and operational characteristics which remain stable over a wide range of conditions.

A digital filter operates with discrete samples of the input signal and is composed of adders, multipliers all implemented in digital logic. This results in a much better control over the accuracy of the operation than is possible in an analogue filter. In an analogue filter, tolerances in the components make it extremely difficult for a system designer to control the precision of the filter.

There are however many digital signal processing applications where the charac-teristics of a digital filter cannot be specified a priori. In such applications, the digital filter characteristics must be adaptable, so that the filter can adjust to different envi-ronments. This is achieved by using adjustable coefficients for the digital filter. Such

1.1 W h y A d a p t i v e F i l t e r i n g ?

a filter is referred to as an adaptive filter. Conventional digital filtering operates in an open-loop fashion; the filter characteristics are fixed and there is no feedback from the output. Adaptive filters on the other hand function in a closed-loop fashion - the digital filter characteristics are modified by means of a feedback mechanism which monitors the output of the filter. The feedback mechanism uses an adaptive algo-rithm to modify the filter coefficients. The adaptive algoalgo-rithm usually uses the input signal, the output signal and a reference signal to generate an error signal which is used in the feedback mechanism. This is illustrated in Figure [1.1] which shows both conventional and adaptive filter configuration.

Adaptive digital filtering can be achieved using either Finite Impulse Response (FIR) or Infinite Impulse Response (IIR) filters. In F I R filters, the output of the filter is a linear function of the delayed and current values of the input signal. These filters are well-behaved and are generally free of stability problems since as they possess only adjustable zeroes. However, to achieve a given degree of modeling accuracy, a high order F I R filter is required. This increases the computational load as the number of multiplications and additions are increased. The output of an I I R filter on the other hand is generated using a linear function of the delayed and current values of the input signal as well as delayed values of the output signal. Using an I I R filter results in a better model using a lesser number of coefficients than a F I R filter providing a similar performance. This is however countered by the fact that I I R filters possess adjustable poles as well as zeroes and thus are prone to stability problems caused by the migration of the poles during the adaptive process. More details of these issues are presented in Chapter 2.

The applications of adaptive filtering are many - the following table shows some important application areas:

1.1 W h y A d a p t i v e F i l t e r i n g ?

F u n c t i o n A p p l i c a t i o n s

Equalisation Telecommunications

Noise Cancelling Medical Electronics,

A i r c r a f t cockpit communications M u l t i p a t h Compensation Microwave Radio,

T V ghost suppression

Stabilization Space Applications

Modeling I n d u s t r i a l control applications

Of the m a n y configurations i n w h i c h an adaptive d i g i t a l f i l t e r may be used, two i m p o r t a n t configurations are the direct system modeling and the inverse system mod-eling configurations. T h e y have been used i n this thesis t o simulate different appli-cations using adaptive filters. I n the direct system modeling configuration (Figure [1.2]), the adaptive f i l t e r produces an o u t p u t signal t / ( n ) , w h i c h is an estimate of a desired response y(n). I n other words, the adaptive f i l t e r models the characteristics of the u n k n o w n f i l t e r . This configuration is used i n applications such as adaptive noise cancellation. Inverse system modeling configuration, (Figure [1.2]), consists of the adaptive filter generating an o u t p u t signal which is an estimate of the i n p u t sig-nal x(n). I n such a configuration, the i n p u t signal is distorted b y a process which is modeled by the u n k n o w n filter. T h e adaptive filter models the inverse of the un-k n o w n filter thereby restoring the degraded signal. T h i s configuration has f o u n d use i n applications such as adaptive equalisation. M o r e details of b o t h configurations are given i n Chapter 2.

Thus the m a i n m o t i v a t i o n i n studying adaptive d i g i t a l filtering is t h a t i n real w o r l d applications, the characteristics of a system being modeled may be unknown and t i m e varying. Using an adaptive filter makes i t possible t o model a large variety of systems under different operating conditions.

1.2 O u t l i n e o f T h e s i s

1.2

Outline of Thesis

T h e next chapter (Chapter 2) provides an in-depth review of adaptive d i g i t a l f i l t e r i n g and especially concentrates on adaptive I I R f i l t e r i n g algorithms. Brief details of the different alternative realizations used i n the simulation experiments are presented. T h e manner i n which the s t a b i l i t y issue of high order I I R filters was handled using these alternative realizations are discussed. T w o applications of adaptive I I R f i l t e r i n g - adaptive noise cancellation and adaptive equalisation, are explained. These have been used as testbeds i n the research to demonstrate the efficacy of the proposed new approaches t o adaptive f i l t e r i n g w h i c h have been examined i n this thesis.

Chapter 3 and 4 explain the theory and applications of using Stochastic Learn-ing A u t o m a t a algorithms ( S L A ) f o r adaptive I I R f i l t e r i n g . T h e basic theory and the learning algorithms are covered i n Chapter 3. B o t h the P-Model and S-Model schemes are examined i n detail. A new normalisation scheme f o r the S-Model algorithms is proposed and f r o m the simulation results is shown t o p e r f o r m better t h a n the stan-dard S-Model normalisation schemes. A brief m e n t i o n is made of the automata games approach and a scheme of hierarchical automata. T h e original reason f o r using the a u t o m a t a approach f o r the problem of adaptive f i l t e r i n g was t h a t the technique had shown capability of global o p t i m i s a t i o n when searching a noisy, stochastic m u l t i m o d a l surface. T h e results of using the automata algorithms are presented i n Chapter 4. The simulation configuration is explained as well as the manner i n w h i c h an automaton is used t o optimise the parameters f o r the adaptive filtering p r o b l e m . T h e advantages and shortcomings of each learning scheme is detailed. A n explanation is given why the S-Model learning algorithms performed poorly as compared t o P-Model schemes. T h e chapter concludes w i t h a discussion on the v i a b i l i t y of Stochastic Learning A u -t o m a -t a as a -t o o l for adap-tive digi-tal fil-tering. A l -t h o u g h -the S L A algori-thms provide a p o w e r f u l set of results, their u t i l i t y f o r adaptive filtering is l i m i t e d , m a i n l y due to the fact t h a t the iterations required for convergence when adapting a high order filter is very large and i m p r a c t i c a l .

Thus a new approach, especially one i n w h i c h dimensionality was not a hindering factor, was examined. T h i s new scheme can be broadly classified as evolutionary

op-1.2 O u t l i n e o f T h e s i s

t i m i s a t i o n , though three specific paradigms of evolutionary o p t i m i s a t i o n were exam-ined. Chapter 5 presents a detailed overview of the technique of simulated evolution used as an o p t i m i s a t i o n t o o l . T h e different paradigms covered include genetic algo-r i t h m s , evolutionaalgo-ry stalgo-rategies and evolutionaalgo-ry p algo-r o g algo-r a m m i n g . T h e basic algoalgo-rithms are explained along w i t h i m p r o v e d schemes which result i n a better performance. Chapter 6 presents the use and results of the evolutionary o p t i m i s a t i o n schemes for adaptive I I R f i l t e r i n g , concentrating on the use of genetic algorithms. T w o practi-cal applications of adaptive I I R f i l t e r i n g - adaptive noise cancellation and adaptive equalisation, are simulated w i t h the evolutionary strategy being used as the adaptive a l g o r i t h m .

Some l i m i t a t i o n s of the evolutionary schemes were observed d u r i n g the simulation studies. One of these, was the fact t h a t there was no established stopping criterion w h i c h could be used to t e r m i n a t e f u r t h e r iterations. T h i s led to an a t t e m p t , where the behaviour of evolutionary schemes was m o d i f i e d by incorporating concepts f r o m other established optimisations algorithms. Specifically the o p t i m i s a t i o n strategy of simulated annealing was used.

Chapter 7 presents the theory and results obtained i n using the simulated an-nealing approach f o r adaptive I I R f i l t e r i n g . B o t h the classical anan-nealing approach and the more recent fast annealing approach are applied to the adaptive I I R f i l t e r i n g p r o b l e m . Results obtained using the annealing approaches show t h a t although the m e t h o d was able t o locate the exact global o p t i m u m , the t i m e samples required for convergence was very large, thus reducing the practical use of the scheme. T w o new schemes are proposed w h i c h combine concepts of genetic algorithms and simulated annealing. T h e m o t i v a t i o n behind these schemes was to use the convergence speed of the evolutionary schemes and a stopping criterion derived f r o m the annealing algo-r i t h m . Thus, these schemes palgo-resent a stopping calgo-ritealgo-rion f o algo-r genetic algoalgo-rithms which otherwise were stopped by unsatisfactory heuristic methods.

Chapter 8 presents the overall conclusions for the research. T h e m a i n results of all the different approaches used f o r the f i l t e r i n g problems are compared. Finally a discussion is provided of promising areas for f u t u r e research.

C h a p t e r 1 F i g u r e s

Input Signal Output S

Digital Filter Digital Filter

c

Conventional Digital Filtering

z

Input Signal Adaptive

Digital Filter Output Signal Adaptive Digital Filter 1

7

Adaptive Digital Filtering

C h a p t e r 1 F i g u r e s y(n) Unknown Filter Input Signal s(n) Adaptive Filter A y(n)

Direct System Modeling

Input Signal s(n) Unknown Filter y(n) A y(n) Adaptive Filter

Inverse System Modeling

Chapter 2

A d a p t i v e D i g i t a l F i l t e r i e

2.1 I n t r o d u c t i o n

rJ1 his chapter gives a broad overview of adaptive d i g i t a l f i l t e r i n g concentrating more on adaptive I I R f i l t e r i n g . T h e interest and research i n adaptive f i l t e r i n g can be gauged f r o m the large number of books [TJL87, SD88, H M 8 4 , WS85, CG85, Ale86] which have been published on the subject. The basic direct f o r m configuration is discussed along w i t h the alternative realizations. Different error f o r m u l a t i o n s used f o r adaptive I I R filtering and the l i m i t a t i o n s of the existing adaptive algorithms are detailed and discussed.

D i g i t a l filters have f o u n d extensive applications i n m a n y diverse areas of engi-neering such as communications, control, signal processing etc. [WS85, P M 8 8 ] . The attractive feature of d i g i t a l filters is their availability as dedicated signal processing hardware i n the f o r m of integrated circuits. A d i g i t a l filter operates w i t h discrete samples of the signal and is m a i n l y composed of adders, m u l t i p l i e r s and delays a l l implemented i n d i g i t a l logic. T h e m a i n advantages of using d i g i t a l filters are t h e r m a l stability, precision and adaptability.

T h e f u n d a m e n t a l equation describing the i n p u t - o u t p u t relationship of a general d i g i t a l filter is given by

M N

2 . 1

where

I n t r o d u c t i o n

y ( n ) = O u t p u t sample at instant n

x ( n ) = I n p u t sample at instant n

x ( n - i ) = I n p u t sample delayed by i t i m e samples y ( n - j ) = O u t p u t sample delayed by j t i m e samples

a,j = Feedback filter coefficients

bi = Feedforward filter coefficients

T h e equivalent block diagram is shown i n Figure (2.1). A n equivalent f o r m of E q u a t i o n [2.1] is given below:

y(n) = B(n)x(n) + A(n)y(n) (2.2) w here B{n) =

X!

(

) = J2

>

~

As shown i n Equation [2.1], the o u t p u t y(n) can be regarded as an autoregressive m o v i n g average ( A R M A ) process driven by the i n p u t x(n). T h e coefficients aj, bi

determine the characteristics of the filter.

D i g i t a l filters can be classified i n t o two m a i n groups:

9 F i n i t e Impulse Response ( F I R ) Filters

2 . 1 I n t r o d u c t i o n

T h e equation describing an I I R filter is given by Equation [2.1], while the block diagram is as shown i n Figure (2.1). T h e i n f i n i t e nature of the impulse response of an I I R filter is because of the dependence of the o u t p u t y(n) on previous o u t p u t samples as shown i n Equation [2.1]. As a result of this recursion, the s t a b i l i t y of the filter is guaranteed only under certain conditions and forms an i m p o r t a n t issue i n the analysis and design of adaptive I I R algorithms.

The o u t p u t of an F I R filter is dependent only on the past and current i n p u t samples and is given by

M

y (n) =

5^

0

1=0

T h i s f o r m can be obtained f r o m Equation [2.1] by equating coefficients Oj's to zero. S i m i l a r l y the block diagram of a F I R filter can be obtained f r o m Figure [2.1] by m a k i n g the feedback coefficients a j ' s equal t o zero.

T h e m a i n advantage of an I I R filter over a F I R f i l t e r is t h a t , as an I I R filter re-quires considerably fewer coefficients t o model a system t h a n an equivalent F I R f i l t e r , there is a significant saving i n the c o m p u t a t i o n a l overheads. For the same number of coefficients, an I I R filter can provide better performance. A desired frequency response can be better approximated by a filter possessing b o t h poles and zeroes ( I I R filter) t h a n a filter having only zeroes ( F I R filter). T h i s is another significant advantage i n using I I R filters i n place of F I R filters.

A n i m p o r t a n t feature of d i g i t a l filters w h i c h has been mentioned before is t h a t of adaptability. T h i s property is significant when the operating environment of the filter is changing and the filter has t o m o d i f y its behaviour i n order t o track the change. T h e filter w h i c h is used i n such a situation is called an adaptive d i g i t a l filter. I n such a filter composed of either an I I R filter or a F I R filter, the coefficients a,- and are variable and can be altered u n t i l the o u t p u t satisfies a specified criteria. A block diagram of an adaptive d i g i t a l f i l t e r is shown i n Figure [2.2] [Shy89a]. I t consists of the f o l l o w i n g :

o A F I R or I I R filter w i t h adjustable coefficients 0 ( n ) .

ap-2 . 1 I n t r o d u c t i o n

proximates a desired response d(n).

Thus the adaptive filtering problem can be succinctly expressed as: Given x(n) and d(n), the coefficients of the adaptive filter have to be chosen such that a performance measure based on the estimation error is minimised. T h e estimation error e(n) (Figure [2.2]) is defined as

e(n) = d(n) - y(n) (2.4)

A c o m m o n l y used configuration i n adaptive control is the system identification configuration i n which an adaptive system is used t o model an u n k n o w n system. This configuration is also frequently used i n adaptive signal processing. Thus, the adaptive d i g i t a l filtering problem using the system i d e n t i f i c a t i o n configuration (Fig-ure [2.3]) is as follows: T h e i n p u t signal is applied b o t h t o the u n k n o w n system and the adaptive system. T h e unknown system o u t p u t forms the desired response f o r the adaptive system, w h i c h uses the estimation error as defined i n E q u a t i o n [2.4] above t o update its coefficients. I n most applications there is the presence of additive mea-surement noise w h i c h is shown i n (Figure [2.3]) by v(n). I n the system identification configuration, the desired response d(n), is generated by the same i n p u t x(n) which drives the adaptive system. Thus, some characteristics of the signal d(n) may be known i f the properties of the d r i v i n g signal x(n) is k n o w n . T h e desired response need not always be generated i n this manner and depends upon the application i n which the adaptive system is used. Thus the adaptive filtering problem can be cast as an o p t i m i s a t i o n problem, where a suitable f u n c t i o n of e(n) is t o be minimised.

A c o m m o n l y used criterion i n adaptive filtering is t o minimise the M e a n S q u a r e O u t p u t E r r o r $ which is defined as

= E [ e2( n ) ] (2.5)

where

E = Statistical Expectation Operator.

Recursive algorithms using this criteria are referred t o as S t o c h a s t i c G r a d i e n t algorithms [Shy89a]. Another criteria w h i c h has been used f r e q u e n t l y minimizes the

2.2 A d a p t i v e F I R F i l t e r i n g

sum of the squares of the estimation error e(n), i.e.

(2.6)

These algorithms are referred t o as the R e c u r s i v e L e a s t S q u a r e s algorithms. Adap-tive algorithms effecAdap-tively search a performance surface defined by the criterion used. The o p t i m u m set of coefficients are then the coefficients corresponding to the global m i n i m u m on the performance surface.

I n adaptive F I R f i l t e r i n g using the system i d e n t i f i c a t i o n configuration (Figure [2.3]), the adaptive f i l t e r is of the F I R type. T h e estimation error e ( n ) , w h i c h is the difference between the desired response and the o u t p u t of the adaptive f i l t e r is used i n the criterion to update the f i l t e r coefficients. T h e c r i t e r i o n usually used for adaptation is the m i n i m i z a t i o n of the M e a n S q u a r e E s t i m a t i o n E r r o r w h i c h is defined as

where 6,'s are the set of coefficients of the adaptive F I R f i l t e r . I t has been proved t h a t the f u n c t i o n $ is a quadratic unimodal function of the adaptive f i l t e r coefficients [WS85]. Thus there exists an unique set of coefficients of the adaptive f i l t e r at which the error reaches the m i n i m u m value w h i c h is the global m i n i m u m . T h i s facilitates the use of p o w e r f u l gradient algorithms which can converge to the o p t i m u m set of coefficients rapidly. I n particular a commonly used stochastic gradient a l g o r i t h m is the L e a s t M e a n S q u a r e ( L M S ) a l g o r i t h m first proposed i n [WH60]. Complete details of the L M S a l g o r i t h m are given i n [WS85].

C u r r e n t l y F I R f i l t e r s are more practical t o use and are widely used i n adaptive f i l t e r i n g . T h e m a i n reason f o r this is t h a t since F I R filters contain only adjustable zeroes, i t is free f r o m the s t a b i l i t y problems associated w i t h f i l t e r s having b o t h poles and zeroes ( I I R F i l t e r s ) . However, interest i n using I I R f i l t e r s as the adaptive f i l t e r has been increasing, p r o m p t e d m a i n l y by the reduced c o m p u t a t i o n a l demands when

2,2

A d a p t i v e F I R

Filterin.

2.3 A d a p t i v e I I R F i l t e r i n g

using an I I R f i l t e r .

2o3

A d a p t i v e

I I R

F i l t e r i n g

2.3.1 I n t r o d u c t i o n

The non-recursive nature of the F I R f i l t e r results i n a heavy c o m p u t a t i o n a l load when using adaptive F I R filters. Modeling a system w i t h an I I R f i l t e r can be achieved to a higher degree of precision using a m u c h lower order f i l t e r t h a n an equivalent F I R f i l t e r . For example, a f i f t h order I I R f i l t e r requiring nine m u l t i p l i c a t i o n s and eight additions matches an unknown system as well as a 64th order F I R f i l t e r requiring 64 m u l t i p l i c a t i o n s and 63 additions. T h i s has led t o exploring the possibility of using I I R filters as the adaptive element and as a consequence research into adaptive I I R f i l t e r i n g algorithms has been quite intensive i n the past decade. T h o u g h the algo-r i t h m s algo-relating t o adaptive I I R f i l t e algo-r i n g aalgo-re not as thoalgo-roughly analysed and developed as adaptive F I R f i l t e r i n g algorithms, they nevertheless f o r m a substantial set of re-sults. W o r k i n adaptive I I R filtering algorithms have been carried out by various researchers [SEA76, W h i 7 5 , Fei76, PAS80a, Joh79, T L J 7 8 , L T J 8 0 ] . The m a i n work which has been carried out i n adaptive I I R filtering has concentrated on the issues of global o p t i m a l i t y , stability and the rate of convergence of the adaptive algorithms. New algorithms have been devised w h i c h solve some of the problems stated above b u t are usually constrained by a set of conditions. T w o i m p o r t a n t review papers which present the current results i n adaptive I I R filtering are [Joh84, Shy89a]. Using an I I R filter as the adaptive element i n an adaptive scheme has the following implications [CG85]:

o Feedback i n the filter structure itself allows a low order filter to have a long d u r a t i o n impulse response.

o T h e I I R filter structure is not stable for a l l choices of coefficients, thus stability f o r m s an i m p o r t a n t aspect i n the analysis.

2.3 A d a p t i v e I I R F i l t e r i n g

o Use of gradient algorithms result i n increased c o m p u t a t i o n a l complexity than is the the case w i t h F I R filters.

o Presence of the poles i n the f i l t e r structure complicates the convergence analysis.

T h e adaptive I I R f i l t e r i n g problem has been approached i n two ways, the difference being the manner i n which the estimation error ( E q u a t i o n [2.4]) has been f o r m u l a t e d . T h i s is explained i n the next section.

2.3.2 D i f f e r e n t Formulations of E s t i m a t i o n E r r o r

E q u a t i o n E r r o r F o r m u l a t i o n

T h e equation error approach has been used i n adaptive control where i t is referred to as the series-parallel model. T h e E q u a t i o n E r r o r approach was proposed i n [Men73] and has been used f o r adaptive f i l t e r i n g [Goo83]. I n this f o r m u l a t i o n , the feedback coefficients of the I I R f i l t e r are updated i n an all-zero, non-recursive f o r m which are then copied t o a second f i l t e r which is implemented i n an all-pole f o r m as shown i n Figure (2.4) [Shy89a]. Essentially this f o r m u l a t i o n is of the adaptive F I R f i l t e r type where the F I R f i l t e r has two inputs. T h i s can be seen i n Figure (2.5) which shows the setup when the equation error f o r m u l a t i o n is used i n the system i d e n t i f i c a t i o n configuration [LTJ80]. W i t h reference t o Figure (2.4), the defining equation for the equation error approach is given by

M N

y

(n) = ^2 k

(

-

) + Y l

^

~ ' (

)

t=0 j = l

F r o m E q u a t i o n [2.8], i t can be seen t h a t the o u t p u t ye( n ) is obtained f r o m delayed

samples of the input x(n) and the desired response d(n) and not f r o m the past o u t p u t samples ye(n). Thus the o u t p u t ye(n) is a linear f u n c t i o n of the coefficients ( a j , 6 , ) . Hence gradient calculations are simplified when using gradient-based algorithms. The equation error is given by

2.3 A d a p t i v e I I R F i l t e r i n g

as is shown i n Figure (2.4). Expanding the above equation and using Equation [2.2], the equation error can be w r i t t e n as

Thus as ee( n ) is generated using the difference between t w o expressions/equations, i t is referred t o as the equation-error f o r m u l a t i o n . Since the equation error ee( n ) is a linear f u n c t i o n of the f i l t e r coefficients, the M e a n S q u a r e O u t p u t E r r o r (Equation [2.5]) is a quadratic f u n c t i o n of the f i l t e r coefficients w i t h a single global m i n i m u m . Thus the performance of the equation error adaptive I I R f i l t e r is similar t o the adap-tive F I R f i l t e r especially w i t h respect to the convergence and s t a b i l i t y of the coefficient updates. However the l i m i t a t i o n of the equation error approach is t h a t i n the pres-ence of measurement noise which is invariably present (Figure [2.3]), the a l g o r i t h m converges t o a solution t h a t is biased away f r o m the true values. I n a system iden-t i f i c a iden-t i o n coniden-texiden-t, iden-this corresponds iden-to incorreciden-t esiden-timaiden-tes of coefficieniden-ts 9 such that

E[9(n)] = 9V + bias i n the l i m i t n —> oo where 9 is the coefficient vector and 9* is the o p t i m a l set of coefficients of the adaptive f i l t e r i n g problem. I t has been shown t h a t this bias is eliminated i f the measurement noise is zero. A numerical example regarding the effect of noise on the bias is given i n [ShyS9a].

O u t p u t E r r o r F o r m u l a t i o n

This error f o r m u l a t i o n has also been used extensively i n adaptive control and is re-ferred t o as the parallel model. T h e O u t p u t E r r o r f o r m u l a t i o n is as shown i n Figure [2.6] and is characterized by the recursive equation

ee(n) = d(n) - ye(n) = d(n) — [(A(n)d(n) + B(n)x(n)](see f o o t n o t e1) = [d(n)(l - A(n)] - [B{n)x(n)] (2.10) N M Vo{n) = k x x(n - i ) + V _a dj x y0(n - j ) . (2.11)

2.3 A d a p t i v e I I R F i l t e r i n g

T h e current o u t p u t y0(n) depends on the past o u t p u t samples adding complexity to the adaptive algorithms. As shown i n Figure [2.6], the o u t p u t error is given by

eD( n ) is a nonlinear f u n c t i o n of the f i l t e r coefficients. Thus the M e a n S q u a r e O u t -p u t E r r o r need not be a quadratic f u n c t i o n of the f i l t e r coefficients and can have m u l t i p l e o p t i m a . T h i s results i n s u b o p t i m a l performance when using gradient tech-niques as the a l g o r i t h m could converge t o a local o p t i m u m depending on the i n i t i a l values of the coefficients. A specific numerical example is detailed i n [JL77].

2.3.3 A d a p t i v e A l g o r i t h m s

This section presents a brief overview of adaptive I I R algorithms. T h e adaptive al-gorithms relating to adaptive I I R filtering are more involved and less complete than F I R filter adaptive algorithms. T h e two formulations of the estimation error explained above lead t o adaptive algorithms w i t h different characteristics. T h e equation error approach has been accepted widely as an alternative to the c o m p u t a t i o n a l l y inten-sive o u t p u t error f o r m u l a t i o n b u t lead to biased estimates of the coefficient vector. However there exists an argument which suggests t h a t the o u t p u t error f o r m u l a t i o n is the correct approach as the adaptive filter is only operating on x ( n ) t o generate y ( n ) w h i c h is the estimate of the desired response d ( n ) . O n the other hand, the equation error approach uses the past values of the desired response d(n) as well as x(n) to estimate the current value of d(n). T h e o u t p u t error f o r m u l a t i o n has been adopted i n all the simulation results presented.

A s i m p l i f i e d f o r m of an adaptive a l g o r i t h m for I I R filters is as follows

e0(n) = d(n) - y0(n) (2.12) 6(n + 1) = 0(n) - ^ ( n ) [ V , J ( 0 ( n ) ) ] (2.13) where A*(n) = V , J ( 0 ( n ) ) = T h e parameter of the a l g o r i t h m Gradient of the error f u n c t i o n

2.3 A d a p t i v e I I R F i l t e r i n g

T h e t w o popular classes of adaptive algorithms f o r I I R f i l t e r i n g are the Least Squares approach and Gradient Search algorithms. Least Square techniques use the i n p u t data samples recursively t o m i n i m i z e a least squares c r i t e r i o n . Detailed analysis of the least squares m e t h o d is given i n [Hay86]. Gradient based algorithms require the gradient at a point on the error surface t o be measured, the next point searched being i n the direction of the negative of the gradient. T w o such algorithms are the Recursive P r e d i c t i o n E r r o r ( R P E ) and the Recursive Least Mean Square ( R L M S ) algorithms [LS83, Shy89a]. These algorithms use an instantaneous values of the estimation er-ror leading t o noisy estimates of the gradient b u t result i n asymptotically unbiased coefficients values. Another a l g o r i t h m f o r adapting I I R filters is the Pseudolinear regression ( P L R ) a l g o r i t h m which is a simpler version of the R P E a l g o r i t h m derived by using an approximate expression f o r the gradient [Shy89a]. Development of fast algorithms f o r the gradient techniques have reduced m u c h of the c o m p u t a t i o n a l load. T h e m a i n problem w i t h gradient techniques is s u b o p t i m a l performance when deal-ing w i t h m u l t i m o d a l error surfaces. T h e i n i t i a l interest i n adaptive I I R algorithms was sparked off by Feintuch i n 1976 who suggested a simple a l g o r i t h m f o r adapting I I R filter coefficients [Fei76]. T h i s was a direct application of the F I R f i l t e r L M S a l g o r i t h m on an I I R filter structure. However this a l g o r i t h m was shown t o converge t o false m i n i m a by Johnson and Larimore [JL77] who also showed the Mean Square O u t p u t E r r o r ( M S O E ) performance surface could be m u l t i m o d a l i f the adaptive filter was of insufficient order w i t h respect t o the u n k n o w n system. T h i s was later con-firmed by P a r i k h and A h m e d [PA78] who showed the i n a b i l i t y of the recursive L M S to i d e n t i f y a reduced order example proposed by t h e m . Further work on adaptive I I R filters was carried out by Stearns [Ste81], who stated a u n i m o d a l i t y conjecture for the system i d e n t i f i c a t i o n conditions. Soderstrom and Stoica [SS82] subsequently added t o the set of conditions p u t f o r w a r d by Stearns f o r an u n i m o d a l error surface. These conditions are as follows:

e T h e adaptive filter is of sufficient order t o able t o model the u n k n o w n system. ® T h e i n p u t signal is w h i t e .

2.3 A d a p t i v e I I R F i l t e r i n g

o T h e order of the adaptive f i l t e r numerator exceeds t h a t of the u n k n o w n system denominator.

T h e last condition was p u t f o r w a r d by Soderstrom and Stoica. Fan and Jenkins [FJ86] proposed a new adaptive a l g o r i t h m w h i c h has the characteristics of b o t h the o u t p u t error and equation error f o r m u l a t i o n . They used the system identification config-u r a t i o n and classified the error sconfig-urfaces for sconfig-uch a configconfig-uration w i t h a stationary stochastic setting into four cases depending on the order of the adaptive f i l t e r and the nature of the i n p u t excitation. These four case are:

o Class ( I ) : Sufficient Order Modeling - W h i t e Noise I n p u t

o Class ( I I ) : Sufficient Order Modeling - Coloured Noise I n p u t

o Class ( I I I ) : Reduced Order Modeling - W h i t e Noise I n p u t o Class ( I V ) : Reduced Order Modeling - Coloured Noise I n p u t

I t can be seen t h a t b o t h complexity and practical reality increase as we move down t h e above list. More recently extensive work has been done by Fan and Nayeri [FN89], wherein they proved Steam's conjecture f o r first and second order filters even w i t h o u t Soderstrom and Stoica's additional constraint. They also showed t h a t the M S O E error surface could be m u l t i m o d a l even when the adaptive f i l t e r was of sufficient order (Class ( I ) ) or when the order is over estimated.

A different approach i n designing adaptive I I R algorithms was based on the con-cept of Hyperstability and was detailed i n [Joh79]. T h e resulting a l g o r i t h m was re-ferred t o as the Hyperstable A d a p t i v e Recursive F i l t e r ( H A R F ) a l g o r i t h m . Hypersta-b i l i t y was a concept which was associated w i t h the analysis of closed loop nonlinear t i m e varying control systems [Pop73]. T h e a l g o r i t h m had provable convergence prop-erties b u t was c o m p u t a t i o n a l l y intensive especially for real t i m e applications. This led t o a simplified version of the a l g o r i t h m referred t o as the Simple Hyperstable A d a p t i v e Recursive F i l t e r ( S H A R F ) a l g o r i t h m [LTJ80]. T h e S H A R F a l g o r i t h m had convergence properties similar t o H A R F a l g o r i t h m b u t under weaker conditions. A f u r t h e r constraint of this approach was t h a t i t relied on a Strictly Positive Real (SPR)

2.4 A l t e r n a t i v e R e a l i z a t i o n s

condition f o r global convergence. T h i s condition effectively reduced the operating re-gion of the adaptive f i l t e r by restricting the pole positions.

R a n d o m Search algorithms were another technique used to search performance surfaces. They made use of a random process t o generate new points and made no assumptions about the nature of the error surfaces. T h i s approach was used for F I R f i l t e r i n g [ W M 7 6 ] , where the proposed linear random search (LRS) a l g o r i t h m was compared t o L M S . A whole chapter dedicated to different adaptive algorithms is given i n [WS85].

A l l the adaptive algorithms detailed i n this section use the direct f o r m structure. A drawback w i t h the direct f o r m realization is the sensitivity of the structure to the quantization of the coefficients which w o u l d result i n any i m p l e m e n t a t i o n . Another shortcoming w i t h the direct f o r m approach is t h a t the s t a b i l i t y check involves ad-d i t i o n a l c o m p u t a t i o n a l overheaad-ds. As a result, alternative realizations which have been derived f r o m the direct f o r m configuration have been used extensively i n a l l the s i m u l a t i o n experiments conducted i n this thesis and are detailed i n the next section.

2A

A l t e r n a t i v e R e a l i z a t i o n s

T h e direct f o r m realization of an I I R f i l t e r is as given by E q u a t i o n [2.1] and is repeated here f o r ease of reference.

M N y (n) = b* x x(n ~ +

X!

(

~

<=o i = i

A n o t h e r possible way of characterizing the above class of systems is t o use the transfer f u n c t i o n approach. The transfer f u n c t i o n f o r the above equation is given by

H(z) = J (2-14)

1

- L j = i

i

which is a r a t i o of two polynomials. F r o m the above equation, the poles and zeroes of the system f u n c t i o n H ( z ) can be obtained. T h e b u i l t - i n feedback structure of the

2.4 A l t e r n a t i v e R e a l i z a t i o n s

I I R filter leads t o problems of stability. T h i s is especially t r u e i n the case of adaptive filters as d u r i n g the adaptation one or more poles could move outside the u n i t circle i n the z-plane resulting i n an unstable filter. Thus adaptive algorithms need some f o r m of s t a b i l i t y check w h i c h may prove t o be c o m p u t a t i o n a l l y expensive i f i t involves factorizing the denominator at each i t e r a t i o n . A n o t h e r l i m i t a t i o n of the direct f o r m structures is the large sensitivities caused by the poles inadvertently slowing down the convergence rate. A way t o resolve this problem is t o decompose the direct f o r m structure i n t o alternative realizations like the parallel or cascaded forms which have lower coefficient sensitivities and a r i t h m e t i c quantization effects. T h e parallel or cascaded realizations are composed of smaller order filters arranged i n parallel or series w h i c h as a whole realize the transfer f u n c t i o n given by E q u a t i o n [2.14]. These realizations also allow easier i m p l e m e n t a t i o n of the s t a b i l i t y check.

A different alternative realization w h i c h does not d i r e c t l y follow f r o m the direct f o r m structure as given i n E q u a t i o n (2.1) is the lattice configuration. T h e advantage w i t h the lattice configuration is t h a t there exists a unique set of lattice coefficients f o r each direct f o r m I I R filter. T h e s t a b i l i t y check is also incorporated very easily i n the adaptive lattice algorithms.

2.4.1 Parallel F o r m

A parallel f o r m realization of an Pth order I I R filter can be obtained by p e r f o r m i n g a p a r t i a l f r a c t i o n expansion of H ( z ) as given i n Equation [2.14]. T h i s results i n

P/2 Hp(z) = j2Hi(z) (2.15) t= i where = ito + ^ + fr,*-2 ( 2 16 ) , w 1 + am-1 + ai2z-2 v ;

T h e parallel f o r m is usually composed of second order filters having the transfer f u n c t i o n as given i n E q u a t i o n [2.16]. T h e use of second order sub-systems prevents the use of complex a r i t h m e t i c as would be the case i f first order filters were used. T h e s t a b i l i t y check is incorporated by ensuring t h a t the denominator coefficients of the

2.4 A l t e r n a t i v e R e a l i z a t i o n s

second order sub-system lie inside the stability triangle [Shy89a]. T h i s realization is shown i n Figure (2.7) when used i n an adaptive f i l t e r i n g setup.

T h e instantaneous o u t p u t error is the given by P/2

e(n) = d ( n ) - j > ( « ) (2.17) «=i

Detailed analysis of the parallel f o r m adaptive I I R f i l t e r is given i n [NJ89, Shy89b]. I n [Shy89b] a frequency domain i m p l e m e n t a t i o n of the parallel f o r m I I R f i l t e r is pre-sented based on the discrete Fourier t r a n s f o r m . T h e discussion includes a study of the M S O E surface and the convergence properties. I n [NJ89], the different M S O E surfaces f o r alternative realizations like the parallel and cascade forms are examined and analysed. The m a i n conclusions drawn f r o m the analysis is t h a t whenever a direct f o r m I I R f i l t e r w i t h a u n i m o d a l M S O E surface is transformed i n t o an alter-native realization using either a parallel or cascaded f o r m , the M S E surface of the new structure may have a d d i t i o n a l stationary points w h i c h are either new equivalent m i n i m a or saddle points which are unstable solutions i n the parameter space.

2.4.2 Cascade F o r m

T h e cascade f o r m of E q u a t i o n [2.14] is given by

P/2

Hc(z) =

i[Hi(z) (2.18)

where H{(z) is as given i n E q u a t i o n [2.16]. T h e analysis of the cascade f o r m is similar t o t h a t of the parallel f o r m and is given i n [NJ89]. T h e c o m p u t a t i o n of the gradient i n the cascade f o r m is more involved as the o u t p u t of each section depends on the o u t p u t of the previous sections. I t has been shown t h a t the cascade f o r m has slower convergence rate t h a n other realizations. A detailed analysis of the adaptive recursive f i l t e r i n g using the cascade f o r m is presented i n [ T C C 8 7 ] .

2.5 A p p l i c a t i o n s o f A d a p t i v e I I R F i l t e r i n g

2.4.3 L a t t i c e F o r m

T h e lattice f o r m has been used i n adaptive signal processing f o r linear prediction and noise cancellation [Gri78, M V 7 8 ] . A d a p t i v e I I R f i l t e r i n g using the lattice f o r m has been discussed i n [Hor76, PAS80b]. A t h r o u g h exposition of the basic lattice structure is given i n [CG85].The m a i n advantages of using the lattice structure are stability check by inspection, cascading of identical sections and good numerical round-off characteristics. The lattice f o r m of a d i g i t a l f i l t e r is entirely different f r o m the forms w h i c h have been listed before. Each stage of a lattice structure is characterized by having a pair of i n p u t and o u t p u t terminals. T h e lattice structure equivalent t o a direct f o r m f i l t e r given by E q u a t i o n [2.14], is shown i n Figure [2.8]. T h e a l g o r i t h m to convert f r o m a direct f o r m f i l t e r t o a l a t t i c e f o r m is given i n A p p e n d i x A .

A n advantage over the parallel and cascaded f o r m is the M S O E surface f o r the l a t t i c e configuration used i n the adaptive f i l t e r i n g , does not possess any saddle points. Convergence properties of an adaptive lattice f i l t e r are similar t o t h a t obtained f o r a direct f o r m f i l t e r [Shy87]. Some recent results regarding stable and efficient adaptive lattice algorithms are presented i n [Reg92].

2c5

A p p l i c a t i o n s o f A d a p t i v e

I I R

F i l t e r i n g

To give a complete picture, the new approaches t o adaptive I I R f i l t e r i n g have been tested i n t w o i m p o r t a n t applications, b o t h w h i c h use an adaptive I I R f i l t e r . These are adaptive noise cancelling and adaptive equalization. A d a p t i v e noise cancelling as the t e r m indicates, is used to remove the d i s t o r t i o n f r o m a signal w h i c h has been corrupted by extraneous noise sources and restore the signal t o its original state. Previous work i n these areas has been done w i t h success using F I R filters, however the need f o r real t i m e processing requires the use of I I R filters. A d d i t i v e noise canceling has been used i n a variety of engineering areas such as biomedical measurements and antenna b e a m - f o r m i n g .

I n m o d e r n telecommunications, the transmission of data over large distances is of v i t a l importance. This is usually achieved using transmission lines or radio waves. Currently, d i g i t a l transmission is becoming more prevalent, w i t h the analogue

2.5 A p p l i c a t i o n s o f A d a p t i v e I I R F i l t e r i n g

voice/data source being digitised at the source and then t r a n s m i t t e d as a sequence of bits. A t the receiver, these bits are then converted back t o the analogue i n f o r m a t i o n . T h e m a i n problem w i t h this mode of transmission, is t h a t d u r i n g the transmission, the signals get corrupted and transformed. C o r r u p t i o n may occur due t o addition of background t h e r m a l noise or impulse noise. Transformation usually occurs as a result of the f i n i t e b a n d w i d t h of the transmission channel and could be frequency translation or t i m e dispersion. I n a m o d e m t r a n s m i t t e r , a number of bits are en-coded i n t o symbols and t r a n s m i t t e d . Due t o the finite b a n d w i d t h of the transmission channels, the effect of each symbol extends beyond the t i m e interval used t o represent t h a t symbol. T h e distortion caused by the resulting overlap is t e r m e d as intersymbol interference ( I S I ) . Equalization is a broad t e r m f o r techniques w h i c h overcomes this problem by compensating f o r t h e m at the receiver end.

2.5.1 Adaptive Noise Cancelling

T h e simulation configuration t o demonstrate the adaptive noise cancelling is taken f r o m the paper by Larimore et. a. [LTJ80]. The setup is shown i n Figure [2.9]. I t is desired t o estimate the signal s(n) which has been corrupted because of the additive

uncorrelated noise process vl(n). Thus the p r i m a r y signal source denoted by z(n) is

given by

z{n) = s{n) + vl(n) (2.19)

To compensate f o r the noise vl(n), usually a sensor is used which measures only the noise process as is shown at the t o p of Figure[2.9]. Thus a reference measurement,

v2(n) is available, which is correlated t o original noise process vl(n). B y means

of proper f i l t e r i n g , the configuration i n Figure [2.9] should be able t o reduce the interference caused by the noise process and provide a good estimate of the signal

s(n). As is shown i n Figure [2.9], the system i d e n t i f i c a t i o n configuration has been

employed. T h i s setup could be rearranged as an o u t p u t error identifier as has been shown i n [LTJ80]. T h e n , m i n i m i s i n g the mean square o u t p u t error, leads to the cancellation of the correlated signals w h i c h are present i n y(n) and z(n). Since, i t is the noise component of two signals y(n) and z(n) w h i c h are correlated, i t gets

2.5 A p p l i c a t i o n s o f A d a p t i v e I I R F i l t e r i n g

cancelled, resulting i n o u t p u t error approaching the undistorted signal s(n). This fact is of paramount importance, because i f the original signal s(n) is i n some manner correlated to the noise process v(n), then the o u t p u t error identifier would lead to the cancellation of the desired signal itself.

2.5.2 A d a p t i v e Equalization

I n modern d i g i t a l c o m m u n i c a t i o n , data is t r a n s m i t t e d using analogue channels. As a result of the f i n i t e b a n d w i d t h of the channel, the t r a n s m i t t e d signals are invariably distorted. Once such f o r m of d i s t o r t i o n is intersymbol interference caused as a result of t i m e dispersion or m u l t i p a t h effects. To overcome the effects of this d i s t o r t i o n , the received signals are passed thorough an equalizer which compensate f o r the distortion and recovers the original symbols which were t r a n s m i t t e d . One widely used f o r m for the equalizer has been the linear transversal equalizer w h i c h is i n effect an F I R filter. I t has been shown however t h a t this k i n d of structure is not suitable f o r n o n - m i n i m u m phase channel compensation.

T h e system shown i n Figure [2.10] is used f o r the experimental configuration. The i n p u t signal x(n) is modeled using an independent binary r a n d o m sequence, the bits being represented by + 1 and - 1 . T h e effect of the channel are modeled using a F I R filter w i t h real coefficients. T h e o u t p u t of this filter is given by

y(n) = a0x(n) + a\x{n — 1) + • • • + aMx{n — M)

M

= J2atx(n-l) (2.20)

where ( d o , . . . , a ^ ) are the coefficients of the F I R filter w h i c h models the transmission channel characteristics. T h e additive noise v(n) is of u n i t y power and zero mean. Thus the signal which is presented t o the equalizer is the noise corrupted signal y. The f u n c t i o n of the equalizer is t o use the values of y(n),..., y(n — K) t o produce the best estimate of of x(n), where K is the order of the equalizer. I n most cases, because of the n o n - m i n i m u m phase characteristics of equalizer only a delayed estimate of the original sequence rc(n)is obtained. More details of the i m p l e m e n t a t i o n are given i n

D i s c u s s i o n 2.6

Chapter 6, where the evolutionary a l g o r i t h m is used f o r adaptive equalization.

2 06

Discussion

This chapter presented an overview of adaptive d i g i t a l f i l t e r i n g and i n particular adaptive I I R f i l t e r i n g . A d a p t i v e F I R f i l t e r i n g is a mature field w i t h well analysed algorithms w i t h respect t o rate of convergence and o p t i m a l i t y . However the area of adaptive I I R f i l t e r i n g is s t i l l evolving. T h e m a i n l i m i t a t i o n s of the current adaptive I I R algorithms are either the c o m p u t a t i o n a l complexity or the f a i l u r e of the algorithm when dealing w i t h m u l t i m o d a l error surfaces. A problem w h i c h arises when modeling high-order I I R filters is one of stability. Ensuring s t a b i l i t y of the I I R filter kernel for all choices of filter coefficients is c o m p u t a t i o n a l l y expensive. Other adaptive tech-niques like random search algorithms have been used to solve this problem b u t have not given encouraging results. I n the next chapter we present a different approach which is based on Stochastic Learning A u t o m a t a . Stochastic Learning A u t o m a t a are techniques w h i c h make use of probabilistic transitions and have been shown by simulations to e x h i b i t global o p t i m a l i t y .

C h a p t e r 2 F i g u r e s Output y(n) Input s(n) a 1 ° 1 1

>

< 3

>

C h a p t e r 2 F i g u r e s Unknown System

o

Adaptive System y(n)

Figure 2.3: System I d e n t i f i c a t i o n Configuration

Input x(n) Desired Response d(n) B(n) A(n) i t Y Equation Error e (n) 1 - A(n) Copy Weights ye(") Output y(n)

C h a p t e r 2 F i g u r e s

Unknown ARMA Plant

^Input x(n) ^ Equation Error^

e(n)

Equation Error Identifier

Figure 2.5: Equation E r r o r Identifier

I n p u t x(n) B(n) 1 - A ( n ) O u t p u t yQ (n) Output Error eQ( n ) Desired Response d(n) Figure 2.6: O u t p u t Error F o r m u l a t i o n

C h a p t e r 2 F i g u r e s H (z) 1 Desired Response yy 00 d(n) H (z)

Input s(n) Output y(n)

Estimation Error e(n)

t= H . (z)

P/2

Figure 2.7: Parallel F o r m Realization

Input x(n) K n-l 0 n < V ' - 0 * ©

<0

C h a p t e r 2 F i g u r e s Signal Source O : ; . 0 Interfering Noise •'. Source Primary Measurement - o

"O

Signal Process s(n) + Primary Signal z(n) O G(p) + vl(n) Noise Process v(n) G(r) v2(n) G(r) Signal Estimate s(n) > y(n) Noise Canceller B) Lumped Model

C h a p t e r 2 F i g u r e s

Noise v(n) Input x(n)

FIR Filter

Limited

Chapter 3

Stochastic Learning A u t o m a t a

3.1 I n t r o d u c t i o n

rj p he process by w h i c h biological organisms learn has been a fascinating area of research f o r well over a century. The focus of research has been m a i n l y two pronged - to understand the principles involved d u r i n g the learning process of biological sys-tems and to develop methodologies whereby these principles could be incorporated i n t o machines. Learning can be regarded as a change brought about i n a system performance as a result of past experience [NT89]. A n i m p o r t a n t characteristic of a

learning system is its a b i l i t y t o improve its performance w i t h t i m e . I n a s t r i c t l y m a t h

-ematical context, the goal of a learning system can be said t o be the o p t i m i z a t i o n of a f u n c t i o n a l w h i c h may not be k n o w n completely. Thus an approach t o this problem is t o reduce the objective of the learning system t o an o p t i m i z a t i o n problem defined on a set of parameters and use established techniques t o arrive at the o p t i m a l set of parameters. T h i s chapter is concerned w i t h the learning methods based on Stochastic

Learning Automata.

T h e concept of Stochastic Automata was first introduced by the pioneering work of T s e t l i n i n the early 1960s i n the Soviet U n i o n who was interested i n the modeling of the behaviour of biological systems [Tse62]. Subsequent research has considered the use of the learning paradigms i n engineering systems. T h i s has led t o extensive work using automata as models of learning w i t h applications i n telephone routeing, p a t t e r n recognition, object p a r t i t i o n i n g and adaptive c o n t r o l [NT74, L a k 8 1 , N T 8 9 ,

3.2 S t o c h a s t i c L e a r n i n g A u t o m a t a

O M 8 8 , SN69, F M 6 6 ] . A Learning Automata could be regarded as an abstract object having a finite number of actions. I t operates by selecting an action f r o m a finite set of actions which is then evaluated by a random environment. The response f r o m the environment is used by the automaton t o select the next action. B y this process, the a u t o m a t o n learns asymptotically t o select the o p t i m a l action. The manner i n which the automaton uses the response f r o m the environment t o select its next action is determined by the specific learning a l g o r i t h m used. T h e next section gives details of the components of a Stochastic Learning Automata.

3o2

Stochastic L e a r n i n g A u t o m a t a

A Stochastic Learning A u t o m a t o n ( S L A ) comprises of t w o m a i n b u i l d i n g blocks: o A Stochastic A u t o m a t o n w i t h a finite number of actions and a Random

envi-ronment w i t h which the automaton interacts.

o T h e Learning A l g o r i t h m s by which the automata learns the o p t i m a l action.

3.2.1 Stochastic A u t o m a t a

A n A u t o m a t o n can be regarded as a finite state machine. M a t h e m a t i c a l l y i t can described by a q u i n t i p l e

SA = { a , / 3 , F , G , 0 }

(3.1)

where