LINEARIZATION OF ELECTROACOUSTIC TRANSDUCERS · ‘‘vlak’’ zijn van de frekwentie responsie...

LINEARIZATIONOF

ELECTROACOUSTIC TRANSDUCERS

Hans Schurer

Copyright � 1997 by H. Schurer.All rights reserved. No part of this publication may be reproduced, stored in a retrieval system,or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording,or otherwise, without prior permission of the copyright owner.

First published 1997

Schurer, Hans

Linearization of Electroacoustic TransducersHans Schurer. - [S.l. : s.n.]. -I11.Thesis University of Twente Enschede. - With bibliogr., ref.ISBN 90-365-1032-5Subject headings: nonlinear digital signal processing; electroacoustic transducers / nonlinear compensators.

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LINEARIZATIONOF

ELECTROACOUSTIC TRANSDUCERS

PROEFSCHRIFT

ter verkrijging vande graad van doctor aan de Universiteit Twente,

op gezag van de rector magnificus,prof. dr. F.A. van Vught,

volgens besluit van het College voor Promotiesin het openbaar te verdedigen

op donderdag 6 november 1997 te 13:15 uur.

door

Hans Schurer

geboren op 26 september 1967te Nijeberkoop

Dit proefschrift is goedgekeurd door:

Prof. dr.-ing. O.E. Herrmann (promotor) enDr. ir. C.H. Slump (assistent–promotor)

v

Summary

Ideal electroacoustic transducers convert electrical energy into acoustical energy. Practicaltransducers, however, exhibit nonlinear system behavior resulting in acoustical distortions.These distortions deteriorate the original audio signal in the reproduction of music or speech.Besides this well known application area, electroacoustic transducers are widely used in otherapplications. In these areas the overall system behavior is also deteriorated. In active noisecontrol, for example, the generation of nonlinear distortion reduces the effect of the noisereduction as acoustical energy is introduced at undesired frequencies. In this field often acompromise is found in using more or larger transducers. In acoustical echo cancelingnonlinear distortion limits the achievable echo suppression. As in active noise control, it ishereby assumed that the transducer is a linear system, resulting in a controller which does notperform optimal.

In this thesis we identify the major physical sources of nonlinearity in the most widely usedelectroacoustic transducer: the electrodynamic loudspeaker. Three widely used acousticalloads are hereby considered: a closed cabinet, a vented cabinet and a horn. Nonlinearphenomena can of course be reduced by improvements in the transducer design. Thesechanges, however, lead to a more complex (often bigger in size) and therefore moreexpensive transducer. We propose a different solution.

Based on lumped element models, we derive three different concepts to eliminate nonlineartransfer behavior by means of a feedforward compensator digitally implemented on a generalpurpose Digital Signal Processor (DSP). Although a feedforward realization has importantadvantages like no need for sensors, it suffers from one disadvantage: it is more sensitive tomodel uncertainties. Therefore attention is given to the robustness of each compensatorconcept to modeling errors.

In the implementation of a digital distortion compensating algorithm, discretization is animportant step. Considerable attention is therefore given to the dynamic building blockswhich are needed in digital realization.

In addition to the analysis and synthesis of the linearizing compensators, real-timeimplementation on a floating point DSP is performed. From results obtained with thesecompensators we show that it is indeed feasible to compensate nonlinear transfer behavior ofan electroacoustic transducer by means of a feedforward compensator. The relative distortionsare reduced by 20%, in case of a loudspeaker in a closed cabinet, down to levels below 2%.Compensation of distortion in horn loudspeakers has resulted in reductions of 15%.

vii

Samenvatting

De ideale elektro-akoestische omzetter zet elektrische signaalenergie om in akoestischesignaalenergie zonder het signaal aan te tasten. In de praktische realisatie van dit soorttransducenten, beter bekend als ‘‘luidspreker’’, is dit niet haalbaar, en zal het akoestischesignaal altijd vervormt worden weergegeven. De meest bekende vervorming is het niet‘‘vlak’’ zijn van de frekwentie responsie (lineaire vervorming). Hiernaast treden er echter ookniet-lineaire vervormingen op. Dit resulteert in de toevoeging van hogere harmonischen enintermodulatieproducten aan het akoestische uitgangssignaal.

Deze vervormingen verslechteren de weergave van muziek en spraak. Naast dit bekendetoepassingsgebied van audio reproductie, wordt de elektro-akoestische omzetter ook in veelandere gebieden toegepast. In actieve lawaaibestrijding bijvoorbeeld, reduceert het optredenvan niet-lineaire vervormingen de effectiviteit van de lawaaireductie omdat extra akoestischeenergie wordt toegevoegd op ongewenste frekwenties. Daarom wordt er tegenwoordig eencompromis gesloten door meer of grotere omzetters te gebruiken. Indien de ruimte voor hetaanbrengen van luidsprekers echter beperkt is, voldoet deze oplossing niet meer. Inhands-free telecommunicatie apparatuur begrenst niet-lineaire vervorming de maximaalhaalbare akoestische echo reductie, en daarmee de verstaanbaarheid van spraak. Net als bij deactieve lawaaibestrijding neemt men hierbij aan dat de omzetter een lineair systeem is,hetgeen resulteert in een niet optimaal regelsysteem.

In dit proefschrift identificeren we de belangrijkste niet-lineaire effecten in de meesttoegepaste elektro-akoestische omzetter: de elektrodynamische luidspreker. De drie meestgebruikte akoestische configuraties worden hierbij beschouwt: de gesloten behuizing, debas-reflex behuizing en de hoorn. De oorzaken van niet-lineariteiten kunnen natuurlijkworden gereduceerd door het aanbrengen van verbeteringen in het ontwerp van de omzetter.Zulke aanpassingen resulteren echter in een ingewikkelder (meestal groter) en daaromduurdere omzetter. Daarom wordt in dit proefschrift een andere oplossing voorgesteld.

Op basis van elektrische netwerk elementen worden fysisch equivalente modellen ontwikkelt.Op basis van deze modellen worden drie concepten uitgewerkt om niet-lineair systeemgedragte reduceren door middel van een vooruitregeling geïmplementeerd op een Digitale SignaalProcessor (DSP). Alhoewel een vooruitgeregelde oplossing belangrijke voordelen heeft zoalshet ontbreken van sensoren, heeft het als nadeel een grotere gevoeligheid voor modelonzekerheid. Daarom wordt er aandacht besteed aan de robuustheid van de compensatorenhiervoor.

viii Samenvatting

Omdat de compensatoren worden afgeleid vanuit continue tijd modellen, is een belangrijkestap het discretiseren van het algoritme om implementatie op een DSP mogelijk te maken.Daarom wordt er tevens aandacht besteed aan de dynamische elementen voor een digitalerealisatie.

Na de analyse en de synthese van lineariserende compensatoren zijn de drie conceptenreal-time geïmplementeerd op een floating-point DSP. De resultaten behaald met decompensatoren zoals beschreven in dit proefschrift tonen aan dat het haalbaar is omniet-lineair systeemgedrag van elektro-akoestische omzetters te reduceren metvooruitgeregelde compensatoren. De relatieve vervorming wordt hierbij met maximaal 20%gereduceerd tot waarden beneden de 2% in geval van een luidspreker in een gesloten kast.Met compensatie van vervorming in hoornluidsprekers zijn reducties behaald van maximaal15%.

ix

Dankwoord

Voor u ligt het resultaat van ruim vier jaar onderzoek dat, alhoewel geschreven door éénauteur, niet tot stand zou zijn gekomen zonder de hulp van vele mensen die ik hierbij wilbedanken.

Om te beginnen bedank ik alle (ex-) collega’s van het laboratorium voor Netwerktheorie voorde prettige werksfeer, al hun hulp en tolerantie jegens de (soms) luidruchtige experimenten.In het bijzonder bedank ik professor Otto Herrmann voor het optreden als promotor en KeesSlump als assistent promotor.

De overige leden van de promotiecommissie: Prof. dr. ir. J. van Amerongen, Prof. dr. ir. P.Bergveld, Prof. dr.-ing. H.J. Butterweck, Dr. ir. W.F. Druyvesteyn, Prof. dr. ir. P.P.L. Regtien,Prof. dr. ir. H. Tijdeman, Prof. ir. A.J.M. van Tuijl worden bedankt voor het plaats nemen inde promotiecommissie en hun constructieve commentaar op de concept versie van hetproefschrift.

Erik Druyvesteyn van Philips Research ben ik zeer erkentelijk voor zijn support en het terbeschikking stellen van meetruimte en apparatuur. Hij was samen met Peter Munter tevensopdrachtgever/begeleider van het TWAIO-project dat werd uitgevoerd door Mark Boer enAlex Nijmeijer (AEMICS B.V.). Mark en Alex ben ik zeer erkentelijk voor de belangrijkebijdrage die zij geleverd hebben aan het onderzoek en voor de realisatie van een demonstratiesysteem.

Voormalig kamergenoot Mark Bentum bedank ik voor de prettige sfeer op de kamer en tevensvoor het becommentariëren van de concept versie van dit proefschrift. De andere leden vande ‘‘leescommissie’’, Arthur Berkhoff, Alex Nijmeijer en Kees Slump bedank ik eveneensvoor het grondig doorlezen van het concept. De diverse discussies gevoerd met WolfgangKlippel en Bert Wuite heb ik als zeer verhelderend ervaren en hebben zeker geleid tot eenbeter eindresultaat.

Vele studenten hebben de afgelopen jaren een bijdrage geleverd aan de uitbreiding van dekennis op het gebied van de elektro-akoestiek en gerelateerde digitale signaalbewerking. Eenaantal studenten hebben hierbij een bijdrage geleverd aan de totstandkoming van ditproefschrift (in alfabetische volgorde): Peter Annema, Erwin Balkema, Ton de Graaff enToine Werner.

Tijdens het onderzoek werd door Hans-Elias de Bree zijn ‘‘Microflown’’ uitgevonden. Vooralbij het onderzoek met betrekking tot de hoorn bleek dit een handige sensor en ik bedank hem

x Dankwoord

en zijn ‘‘Microflown-team’’ voor het leveren van de microflowns.

De Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO) en Shell Nederlandalsmede de Universiteit Twente ben ik erkentelijk voor het mogelijk maken van diverseconferentie bezoeken in het binnen- en buitenland.

Tenslotte bedank ik mijn ouders, familie en vrienden voor hun steun in de afgelopen jaren.

xi

Contents

Summary v. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Samenvatting vii. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Dankwoord ix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Contents xi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

List of Abbreviations xv. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 Introduction 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Electroacoustic transduction 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Problem description 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Objectives of the thesis 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4 Overview 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Modeling and control of nonlinear systems 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Nonlinear input-output descriptions 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2.1 NARMA modeling 12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Functional expansions 12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2.3 Block oriented modeling 15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Nonlinear state space 17. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Nonlinear control 17. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4.1 State measurement and observation 22. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Summary 23. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 The electrodynamic transducer 25. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The electrodynamic loudspeaker 25. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1.1 Closed-cabinet 27. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Vented-cabinet 30. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1.3 Horn 31. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Extensions to the basic model 36. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Nonlinearities in electrodynamic transducers 38. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3.1 Nonlinearities in the electro-magnetic part 38. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Nonlinearities in the mechanical part 40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xii Contents

3.3.3 Nonlinearities in the acoustical part 41. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3.4 Major nonlinearities 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Nonlinear lumped element models 46. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4.1 Closed-cabinet 46. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Vented-cabinet 46. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Horn 47. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5 Types of distortion 51. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Volterra linearizing compensators 57. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Synthesis of Volterra linearizing compensators 57. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Second-order Volterra linearizing compensators 61. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3 Parameter extraction 64. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Linear parameters 64. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3.2 Nonlinear parameter extraction 65. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Digital implementation of the 2nd-order Volterra compensator 69. . . . . . . . . . . . . . . . . . .

4.4.1 Discrete differentiator realization 70. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.2 Error analysis 73. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Experiment 75. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5 Sensitivity to (non-)linear parameters 77. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Linear parameter sensitivity 77. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Nonlinear parameter sensitivity 79. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5.3 Parameter uncertainty 80. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Conclusions 82. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Differential equation based compensators 83. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 General method 83. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2 Closed- and vented-cabinet 85. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Nonlinear compensator 85. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Experiment 89. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2.3 Novel mirror filter design 91. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Simulation and experiment 94. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2.5 Parameter uncertainty 97. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Horn loudspeaker 99. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3.1 Parameter extraction 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3.2 Nonlinear compensator 101. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Simulation 103. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3.4 Time domain compensators 105. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Simulation 108. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.6 Experiment 110. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4 Conclusions 114. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiiiContents

6 State space compensators 117. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Nonlinear state space models 117. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Direct method 119. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2.1 General theory 119. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Direct nonlinear compensator 122. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.3 Feedback linearization 124. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.3.1 General theory 124. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Closed cabinet 127. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.3.3 Vented cabinet 130. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Discretization and simulation 132. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.4.1 Digital integrator realizations 132. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.4.2 Simulation 137. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Parameter uncertainty 140. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.5 Experiment 141. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Comparison of the compensators 144. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Conclusions 145. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 Conclusions and recommendations 147. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Conclusions 147. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Recommendations for further research 150. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Appendices 151. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Nonlinear parameters 153. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B Loudspeaker parameters 155. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C Basics of differential geometry 157. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D Publications 161. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Bibliography 163. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Curriculum Vitae 173. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Levensloop 173. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Index 175. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

List of Abbreviations

1P One ParameterAD Analog to Digital

AEC Acoustical Echo CancelingBEM Boundary Element MethodBIBO Bounded Input Bounded Output

DA Digital to AnalogDC Direct Current

DSP Digital Signal ProcessorDFT Discrete Fourier Transform

ERLE Echo Return Loss EnhancementFEM Finite Element MethodFIR Finite Impulse ResponseHM HarMonicID Inverse DynamicsIF InterFerence

IIR Infinite Impulse ResponseIM InterModulation

KZK Khokhlov Zabolotskaya KuznetsovLD Linear Dynamic(s)

LMS Linear MultiStepNAR Nonlinear AutoRegressive

NARMA Nonlinear AutoRegressive Moving AverageNL NonLinear

NLMS Normalized Least Mean SquaresNLSSF NonLinear State Space Form

NMA Nonlinear Moving AverageNN Neural NetworkPR Positive Real

PVC PolyVinyl ChlorideR–K Runga-KuttaRLS Recursive Least mean Squares

SH SubHarmonicSISO Single Input Single OutputTHD Total Harmonic Distortion

TIF Total Interference Distortion

1

Chapter 1

Introduction

The problem of generated distortions, caused by nonlinear characteristics ofelectroacoustic transducers, in several application areas is presented. Main partof this thesis is devoted to modeling and compensation of this behavior by digitalpreprocessing of the input signal.A global approach to tackle the problem of generation of nonlinear distortion inelectroacoustic transducers is outlined. Finally, objectives and the outline of thisthesis are given.

1.1 Electroacoustic transduction

In electroacoustic transducers electrical energy is converted into acoustic energy.Electroacoustic transducers are used in several application areas. Well known is thereproduction of music and speech using a large variety of available audio equipment. Otherapplications are in hearing aids, active noise control, telecommunication equipment, i.e.where acoustical energy is desired. In all these applications several transduction types areused. A classification of the transduction principles, used in electroacoustic reproduction, are:

� Electrodynamic principle with either one or multiple voice coil(s) which is (are)positioned in a radial magnetic field of a permanent magnet, (widely used namesare for low frequencies (20Hz-1kHz): woofer, for mid-range frequencies(1-10kHz): squawker, for high frequencies (10-20kHz): tweeter), mounted invarious possible cabinets (closed, with single or multiple Helmholtz resonators,etc.) or loaded by an acoustic structure like a pipe, horn, etc.

� Electrostatic transduction, used in the electrostatic loudspeakers which are notwidely used because of their, in general, huge size.

� Electromagnetic transduction, widely used in hearing aid- and telephone-receivers.

� Piezoelectric transduction, widely used in small beepers for watches, telephonesetc.

2 Problem description

� Isodynamic transduction. Consists of a conducting membrane (often a printedpattern) with perpendicular to it, on one or either side, a system of magnetic rods.Used in headphones and horn loaded speakers.

� Electropneumatic transduction. Electrical energy is converted into acoustic bymodulation of an air flow.

� Ionic transduction. In this case acoustic energy is generated without mechanicalmoving parts. Using a high voltage that is amplitude modulated by the audiosignal, ionic discharge of air produces acoustic waves which are coupled into ahorn.

Because of simple construction and thus cheap production, the electrodynamic drivingprinciple is most widely used. This thesis will therefore focus on transducers which are basedon the first transduction type, denoted as ’loudspeaker’ in short in the rest of this thesis.Although we restrict our discussion to the loudspeaker, the electrodynamic transductionprinciple is used in widespread applications. Actuators in magnetic disc-drive read/writeheads, positioning the lens in CD-players, shakers for research in physics, and electric balanceinstruments are examples of such applications.

1.2 Problem description

In this section we address four application areas where the production of distortions, causedby the nonlinear system behavior of electroacoustic transducers, is undesirable. In the firstthree applications the electrodynamic transduction principle is most widely used. The finalapplication addressed, the hearing aid receiver, uses the electromagnetic principle. Althoughthis thesis is devoted to the electrodynamic transducer, application of the modeling andcompensation techniques to other transduction principles is rather straightforward.

Audio reproductionWith the increased quality of the audio reproduction chain for consumer use (from recordingup to reproduction), mainly due to the increased use of digital techniques, the loudspeaker asfinal element in this chain has become ’the weakest link’. Especially as there is a need forsmall loudspeakers with a sufficient low frequency sound pressure output. To still achieve agood low frequency response we need large cone excursions which yield excessivedistortions, due to nonlinear system behavior. In public address systems, where high soundpressure levels together with a good directivity of sound is needed, the use of horn loadedloudspeaker is widespread. Compared to the direct radiator loudspeaker it has a much higherefficiency. Disadvantage with this type of transducer are its greater size, higher price, andproduction of high acoustical distortions.

In both cases the nonlinear system behavior of the transducers result in distortions such asharmonic, intermodulation, interference, and in some cases sub-harmonic and Doppler

3Introduction

distortions. Despite the intensive research in the field of audibility of distortions in music andspeech, it is still not possible to give an exact threshold of the human ear to this kind ofdeteriorations. Additional complications with this kind of research are effects like frequencydependent sensitivity of the ear, auditory masking, and the fact that the ear itself is alsononlinear. Connected to the research presented in this thesis, several listening tests areperformed using different strategies in an attempt to find a threshold for nonlinear distortionat low frequencies [82]. From this research it was concluded that with a common two-wayloudspeaker system at normal listening conditions (72dB(A)), nonlinear distortions are hardlyaudible. At slightly higher levels, however, distortions do become audible and thereforereduction of them is desirable. The question how much distortion is allowed, and thus howmuch we need to reduce them, remains unanswered. It is beyond the scope of this thesis togive an answer to this question, and we will therefore reduce distortions as much as possible.

Microphone

Loudspeakers(n)

Room

e(n)

t(n)=s(n)+e(n)

- -

NLMS

NN

AEC

Nonlinearloudspeaker

Room +microphone

Figure 1.1: Acoustical echo canceling (AEC) problem in hands-free telephones (speaker-phones) orteleconferencing (upper scheme), and most commonly used solution to eliminate these acoustic echoesby means of adaptive direct modeling (NLMS). In this AEC we have included a possible method tocancel the nonlinear behavior by an adaptive neural network (NN) based nonlinear controller (lowerscheme) [9].

4 Problem description

Our objective is thus to reduce these distortions by means of digital (pre-) processing of theaudio input signal. Although we restrict ourselves, in the discussion of techniques whichperform this task, to actuators they are also applicable to sensors based on the electrodynamictransduction principle like for example the electrodynamic microphone.

Acoustical echo cancelingThe problem of nonlinear distortions is not restricted to reproduction of music in home orpublic address situations. It also appears to be a problem in Acoustical Echo Canceling (AEC)as well [9]. In Figure 1.1 the general acoustical echo canceling problem is depicted. Besidesthe near end speech signal s(n) also an echo e(n) caused by the reverberation of the room isdetected by the microphone. Speech intelligibility for the far-end listener is heavilydeteriorated by this effect. Normal (digital) echo cancelers operate by direct linear modelingof the path from loudspeaker input up till microphone output. Especially at higher loudnesslevels the performance of this method is limited. Well known problem with acoustical echocanceling is undermodeling of the room acoustics. For proper modeling in the order ofthousands of FIR filter taps are needed [32]. Next to this problem it appears thatnonlinearities in the loudspeaker are, after enclosure vibrations, the most important limitationto the maximum achievable Echo Return Loss Enhancement (ERLE). First attempts to reducethis effect have been performed by the use of a tapped delay line Neural Network (NN) whichmodels nonlinear loudspeaker behavior, besides the classical adaptive Normalized LeastMean Squares (NLMS) echo canceler as depicted in Figure 1.1 [9]. Results with thiscalculation intensive method, are not yet very satisfactory and better results may be expectedby using one of the compensators described in this thesis.

Active sound controlIn active control of sound, either to achieve different acoustical properties of a room/hall orfull cancelation of undesired sound (noise), generation of nonlinear distortions is also aproblem. In both cases the loudspeaker(s) injects acoustical energy into the system atfrequencies which are not predicted when we assume the loudspeaker to be a linear system.

Especially in the case of active control of (loud) noise caused by, for example, engines,exhaust systems, transformers, like depicted in Figure 1.2, we need excessive sound pressuregeneration by the secondary source. Active sound control is needed and performs best in thelower frequency regions (up to 500Hz). This is exactly the same region where distortions dueto (large) voice coil excursions are maximal. The controller W usually does not take intoaccount this effect. A compromise is found by using more or larger loudspeakers to generateenough sound pressure. In many applications, however, space to accommodate equipment islimited. Next to that, both solutions are more expensive and are avoided when we use alinearizing compensator like presented in this thesis.

5Introduction

W

Silentregion

Errormicrophone

ControllerPrimary noisesource & signal microphone

Secondarysource

Figure 1.2: Active noise control using a feedforward controller W.

Hearing aidsFinal example of application area where nonlinear distortion may be of importance, is inhearing aid receivers. Transducers used in this equipment are optimized to produce as muchacoustic sound pressure as possible from a small enclosure. Generation of nonlineardistortions at such high driving levels is hereby inevitable. Most hearing aid receivers arebased on the balanced armature electromagnetic transduction. First investigation of nonlineardistortion generation with this kind of receivers showed values up to 40% harmonic distortionin the lower frequency region [33]. Major physical causes for this distortion are effects whichalso occur in electrodynamic transducers (displacement dependent self-inductance,transduction constant, suspension stiffness and acoustic compliance) [117].

Not much research is performed on this topic until now, and it is therefore not yet knownwether distortions generated by the receiver are a real problem. The hearing impaired earitself is mostly also highly nonlinear and may therefore mask the distortions generated by thehearing aid receiver.

1.3 Objectives of the thesis

The major objective of the work presented in this thesis is to reduce or, if possible, eliminatedistortions generated by the nonlinear characteristics of the electrodynamic transducer. Lineardistortion, i.e. the ‘‘flatness’’ of the transfer function, is not considered in this thesis. We willnot attempt to accomplish this by changing or optimizing the transducer by means ofimprovements in the electromagnetic, mechanical, or acoustical design. This kind ofoptimizations are often performed by trial and error, and lead to a more complicated andtherefore more expensive transducer to manufacture. Our approach is to reduce nonlinear

6 Objectives of the thesis

distortions by an electric controller, digitally implemented on a Digital Signal Processor(DSP) which is depicted schematically in Figure 1.3. In this manner design problems arepartially shifted into the electrical domain and a compromise is possible between transducercomplexity and generation of nonlinear distortions.

DigitalControl

A

D AD

Audio-input

Sensor(s)

DSP-System

Sensor input(s) M

Loudspeaker

Figure 1.3: Digital control set-up to reduce the nonlinear behavior of a loudspeaker. Analog poweramplifier M drives the loudspeaker and is assumed to be linear.

The performance of this total system of sensor-controller-actuator is determined by a largenumber of factors. For a practical implementation we seek for the tradeoff betweenperformance, computational power and costs. In this thesis we will focus on controller designand implementation. The controller is the most important element in this signalpreconditioning part which also comprises the AD-, DA-convertor, power amplifier andsensor(s). Major issues which have to be considered in this controller design are:

� In consumer applications costs is a very important issue. It is mainly determinedby the choice of the DSP, AD-, DA-convertors and sensors; fixed point arithmeticis still significantly cheaper compared to floating point arithmetic. Throughout thisthesis algorithms are implemented on a floating point DSP (TMS320C30), givinga high numerical accuracy and dynamic range.

� Sampling frequency, limited by the complexity of the controller and theprocessing power of the DSP. Often it is determined, however, by the (digital)input data stream. If necessary we can use multi-rate techniques to lighten the loadon the DSP. Speed of response is highly determined by the sampling rate and isdue to the frequency band of interest much higher than generally used in thecontrol world.

� Causality and stability must be guaranteed. Stability can be achieved either in alocal or in a global sense.

� Low and high frequency characteristics. The DC-behavior of the power amplifieris of great influence on the performance of the, especially second-order, distortionreduction. The highest frequency at which we achieve distortion reduction islimited by the choice of the sample rate.

� Robustness, i.e. the sensitivity of the controller performance to effects which arenot taken into account in the design. This can be disturbance, noise, unmodeleddynamics and parameter uncertainty. Parameter sensitivity is an important issue inloudspeakers as accurate determination is difficult in practice. In case we use

7Introduction

sensors to obtain output, input or state measurement, the quality of these sensors(noise level, dynamic range, bandwidth, durability etc.) have a high impact on theperformance of the controller.

� Hardware design (Analog, Digital or Hybrid), analog is fast, digital is moreflexible and environment independent. It is obvious that we have chosen toimplement the controller digitally for reasons of flexibility, reliability, and lessproblems with noise when using floating point implementation. An additionaladvantage of digital realization is that we can implement more complex andadvanced control strategies without the hardware extensions.

We will focus on the controller, especially on its complexity and errors introduced by digitalimplementation and model uncertainty. Roughly speaking we have the choice to implementthe controller as a feedforward controller, a feedback controller or a combination of thesetwo. In this thesis feedforward controllers aiming to eliminate nonlinear dynamics from thetransducer are used mostly. Well known advantages of feedforward control compared tofeedback solutions are

� No need for a sensor. Use of sensors makes the system more expensive and thequality of the sensors highly influences the performance of the control.

� No chance of instability due to a badly designed feedback loop or wear-out of thesensor. Therefore we do not need additional provisions to protect the loudspeakerif an element in the feedback circuit clips or fails.

Disadvantage, however, is

� More sensitive to model uncertainties.

Considerable attention is given to this disadvantage in comparison between the differentcontrollers. Model uncertainties are the differences between the model and the real physicalsystem. They are separated into two groups: parametric uncertainties (parameter errors) andnon-parametric uncertainties, like unmodeled dynamics, noise and disturbances. It isimportant to have a sense of feeling for the influence of such modeling errors. It is anobjective of this thesis to give insight into the influence of the parameter uncertainty on theperformance of the distortion reduction. From this we deduce how accurate we need todetermine linear and nonlinear parameters. Unmodeled dynamics are more difficult tocharacterize for nonlinear systems, in contrast with linear systems where frequency domaincharacterizations can be systematically applied.

1.4 Overview

In this thesis algorithms, or in less general terms, linearizing controllers that eliminate theproduction of nonlinear distortions in electrodynamic transducers are studied. Linearizationmust not be confused with the well known technique of linearization around an operating

8 Overview

point by computing the Jacobian of the system matrix. Complete elimination of distortionswill never be achieved in practice. Reduction of the relative distortions down to a few percentis therefore our objective. It is stressed that we will only focus on the transducer. Influencesof the room or other elements in the audio reproduction chain are not considered.

All techniques are based on real-world continuous-time models, and an important steptowards implementation is discretizing the controller. Considerable attention will be given tothe sensitivity of the performance to digital realization of the controllers.

In chapter 2 we will give an overview of all the most important modeling and controlmethods for nonlinear systems. Modeling is important for the reason that controllers aredirectly derived from a system description. In the next chapter we will review the well knownlinear lumped element model for the electrodynamic loudspeaker. Three widespread usedacoustical loads are used throughout this thesis as cases to test the theory: closed cabinet,vented cabinet and horn loaded. Especially to modeling of the latter element considerableattention is given because no (standard) model exists.

In the successive three chapters three different modeling methods are used for derivation of anonlinear controller. The electrodynamic loudspeaker is a weakly nonlinear system in normaluse. The Volterra series expansion, a well known functional expansion, is very suitable tomodel this class of nonlinear systems. In chapter 4 the Volterra series expansion is thereforeused to model the loudspeaker and derive two different second-order compensation circuits.Considerable difference in performance is observed, in theory as well as in practice, betweenthese two realizations. Next, in chapter 5, more direct methods based on the nonlinearintegro-differential equation are discussed. Finally techniques based on nonlinear state-spacemodels are presented in chapter 6.

In chapter 7 we finish our discussion by concluding remarks and recommendations for furtherresearch.

9

Chapter 2

Modeling and control of nonlinearsystems

A nonlinear model description is the basis for a linearizing controller. Therefore asurvey of the field of nonlinear modeling and control is made in this chapter.Some of the nonlinear system descriptions are used in successive chapters, whileothers are less suitable as a starting point for the derivation of a linearizingcontroller.

2.1 Introduction

Starting point for the design of a linearizing controller is the choice of a controller type.Open-loop control does not use any measured signals from the system. It is fast but sensitiveto disturbance signals and model uncertainty. Feedback control uses signals from the system,is therefore slower but less sensitive to model uncertainty. Finally there is feedforwardcontrol, which uses measured disturbance signals, and is therefore fast and only sensitive tounmeasured disturbances. Open-loop control is equal to feedforward control if we only usethe input signal as measurement signal. Combination of these three controller types has led toa wide range of controller configurations.

A controller is always based on a model of the system. Modeling of a physical systemconsists of several steps. In logical order, one starts with synthesis and analysis of a modelwhich can be done by several techniques. This yields an a-priori model which may beaccurate enough. If not, identification has to be performed by doing experiments, eitheroff-line or on-line. Identification on its turn consists of two steps. The first step is structuredetermination (linear/nonlinear order, time delay), and the next step is determination of thenonlinear and linear parameters.

Final step in controller design is wether we need adaptation of the controller or not. Adaptive

10 Nonlinear input–output descriptions

control is necessary if control can not wait for identification or if the process is time variant.In this thesis we assume that our process, i.e. the electrodynamic loudspeaker under normaldriving conditions, is time invariant and therefore adaptive control is not considered.Major part of the developed controllers, which are considered in this thesis, are feedforwardcontrollers. They calculate their control action from the input signal alone and signalssynthesized from the input. As our goal is to reduce the nonlinear dynamics of the system andleave the linear behavior unharmed, the controllers are better denoted as compensators in thecontext of this thesis. Therefore, in the rest of this thesis, the algorithms are named this way.

Feedforward control with nonlinear systems is more important than it is in linear control.Very often it is even impossible to control a nonlinear system stably without feedforwardcontrol. It has the advantage of being fast, and additional sensors to measure state variables ofthe process are superfluous. These sensors make the system more expensive and their qualitydetermines in great depth the performance of the controller. Next to this disadvantage thecontroller may become instable due to aging and wear of the sensor. In using excessivefeedback, sensors may clip, or even introduce distortion themselves. In this case there is alsothe danger that the system runs into saturation, for example due to the limited driving rangeof the amplifier or the loudspeaker.

In this chapter we present an overview of all major nonlinear modeling techniques. This is thebasis for the nonlinear controllers which are reviewed later. System identification of theelectrodynamic transducer in particular is discussed in the next chapter. Our approach is touse as much as possible expert knowledge about the pertinent transducer. Advantages are ahighly transducer related controller and insight in the importance of certain parameters.Disadvantage is that in general we need to perform more effort at forehand, compared to theso called ‘‘black-box’’ methods, to come to a usable model [67].

2.2 Nonlinear input-output descriptions

The purpose of this paragraph is to give an overview of the modeling and identificationtechniques. For each technique a more or less complete description is given with itsadvantages and disadvantages compared to other techniques. All the techniques we discussare based on the assumptions that the system:

� output is a smooth continuous functional or mapping of the input (i.e.discontinuities are excluded),

� time-invariant,

� causal, and depends to an arbitrarily small extent on the remote past of the input.

The first assumption restricts our systems to the so-called soft-nonlinear systems orquasi-linear systems, and excludes hard-nonlinear systems or essential nonlinear systems[59]. The nonlinearities of the soft-nonlinear systems are described by a converging Taylor

11Modeling and control of nonlinear systems

series. A response to a sinusoidal signal of systems belonging to this class, will containharmonics of the input frequency of which the amplitudes, when the input amplitude isdecreased, decrease faster to zero compared to the fundamental amplitude and inverselyproportional to the order of the harmonic frequency. Hard-nonlinear systems are described bynonlinear functions which are discontinuous in one or more points. Linearization ofhard-nonlinear systems is possible, however, for example by addition of signals with a certainamplitude probability-density function which is determined by the nonlinear function [92].

Because our interest is audio signals, we shall consider only Single-Input Single-Output(SISO) systems throughout the remainder of this chapter. We classify nonlinear modelingtechniques into two major categories, either in discrete time (system is described bydifference equations) or in continuous time (systems described by differential equations). Inthe case of a nonlinear difference equation of order K the model description of the system hasthe general form (L � K)

y(n) � F�y(n� 1), ...,y(n� K), u(n), ...,u(n� L)� (2.1)

where u(n) and y(n) are the system its input and output, F{} is the system function which isnonlinear in our case.Differential equation modeling is the continuous time counterpart of the difference equationmodeling. For most physical systems it is possible to determine the differential equation fromknown physical properties (ideal physical model). Nonlinear differential equations of order kare obtained when parameters depend on certain variables in the system, and are generallygiven by (l � k)

y(t) � F�y. (t),y..(t), ��,y(k)(t), u(t),u. (t), ��, u(l)(t)� (2.2)

where u( j ) and y( j ) denote the j –th time derivative of the input u, respectively output y.

In this thesis we will also discuss two other kinds of nonlinear equations; partial differentialequations and integro-differential equations. Partial differential equations are generallydivided into hyperbolic, parabolic and elliptic. Integro-differential equations are useful in amethod where we separate linear and nonlinear part of a differential equation from each other.The linear part is then described in the s-domain.

Resulting models are either black-box models or physically parameterized models. Black-boxmodels use no a-priory knowledge of the system and their parameters have no physicalmeaning. Physically parameterized models are the result of laborious modeling where asmuch as possible physical insight about the behavior of the process is put into a model.Between these two extremes there is a zone which is called semi-physical modeling orgrey-box modeling [67][69][70]. Important physical insight is used, but it does not leadtowards a full physically parameterized model.

12 NARMA modeling

2.2.1 NARMA modeling

Equation (2.1) includes nonlinear autoregressive (NAR), nonlinear moving average (NMA)and autoregressive/moving average (NARMA) model descriptions, which all belong to theblack-box models. Where a NAR-model contains only past information of y(n), aNMA-model only past information of u(n) while a NARMA-model uses both.

The problem with this kind of modeling consists of finding the correct combinations of x andy terms and cross-products of any number of these terms, including powers; i.e. determinationof model order and structure. It is important to select a subset of terms from all possible termsto achieve an accurate model of the nonlinear system. A special case of the NARMA model isthe functional expansion which contains only polynomial combinations of u(n).For determination of the correct subset in ARMA modeling several algorithms weresuggested in the literature. A good overview of orthogonal and fast orthogonal searchmethods for linear systems is given by Korenberg [59], which can also be used for nonlinearidentification. A criterion to stop the search is in most cases the decrease in the mean squareerror. Orthogonalization of the subsets is often done using a classical or modifiedGram-Schmidt-method to speed up the identification process.

In the case of transducers for audio reproduction this technique could be a valuable modelingtool which indeed has already been used [119][38]. The results of this research did not givebetter results compared with parameterized models based on physical knowledge. Also, alinearizing controller based on a NARMA model does not have an optimum size. Because wedo not use any a-priori knowledge of the transducer the number of coefficients is often muchgreater than with other modeling techniques, which is considered as the major disadvantageof this modeling technique.

2.2.2 Functional expansions

A functional is an operation on a function for which the result is a number. The probablymost well known functional series are the Volterra series. They were first studied at the end ofthe nineteenth century by the mathematician Vito Volterra [103]. The first application of theseseries to nonlinear systems, however, was by Nobert Wiener. The purpose of this section is togive a brief introduction to these series and other functional expansions. The Volterra seriesare discussed in more depth in chapter 4.


Discrete Volterra seriesThe truncated discrete time Volterra series is represented by a special case of the right side ofequation (2.1):

y(n) � ho � �N�1

m�0

h1(m)u(n� m)

�N�1

m1�0

�N�1

m2�0

h2 �m1, m2� u�n� m1�u�n� m2�

�

�N�1

m1�0

�� N�1

mi�0

hi �m1, . . . ,mi� u�n� m1� �� u�n� mi�

(2.3)

where i is the order of the Volterra series, (N–1) is the memory length and hi is the ith (alwayssymmetric) Volterra kernel. We will not consider the questions of convergence anduniqueness of the Volterra series expansions of nonlinear systems. The those interested werefer to [75] and the references herein.

Since the Volterra series are included in equation (2.1), the identification of the kernels canuse the difference equation identification techniques, either in time domain or in frequencydomain. Identification in time domain is often done by applying a white noise, not necessarilygaussian, sequence towards the system [25][44] or by a multi-tone sinusoidal excitation [18].Persistence of the excitation for system identification is in general a very important issue.Alternative input signal for identification of discrete Volterra series is by means of pseudorandom multilevel sequences. It can be proven that exactly p+1 distinct levels are sufficientin systems with nonlinearities of polynomial degree of order p [86].

Discrete frequency domain representation, and identification of the resulting frequencydomain Volterra kernels, is also possible [18][46][80][81]. Frequency domain kernels areobtained from the multi-dimensional discrete Laplace transform. There are, however, variousother techniques for determination of the Volterra kernels. An example of such a technique isthe use of an adaptive Volterra filter which is an adjustable realization of equation (2.3)[75][114]. Fast implementation is the lattice structure which has several advantages above thedirect implementation. The kernels are adapted on-line according to an LMS or a RecursiveLeast mean Square (RLS) algorithm in order to fit the desired response which is e.g. obtainedfrom the real system.

A special case of the Volterra series expansion is the recursive version, which is similar to theNARMA approach, and is called the IIR-Volterra or bilinear structure [75][95]. Just as withlinear IIR filters the recursive nonlinear approach can model nonlinear systems with muchmore parsimony than their FIR counterparts. But also here, stability may be a problem in thedesign of such filters. First practical applications also encountered this problem [29], but still

14 Functional expansions

it can be a promising technique, especially in real-time applications with limited processingcapabilities. Application to loudspeaker linearization has also been attempted, although goodresults are not reported in literature [30][31].

Continuous Volterra seriesThe continuous time Volterra series expansion is a generalization of the functional expansionfor linear systems, and is for order n, generally given by:

y(t) � h0� ��

��

h1(�)u(t � �)d�

� ��

��

��

��

h2��1, �2u�t � �1u�t � �2d�1d�2� ��

� ��

��

��

��

hn��1, . . . ,�nu�t � �1��u(t � �n)d�1��d�n

(2.4)

where u(t) and y(t) are the system input and output respectively and hn(�1,...,�n) are thegeneralized impulse responses, also called kernels. The second term is the well knownconvolution integral for a linear system and for a causal system the kernels are zero for �i


in the amplitude c. From this, just as with a Taylor series expansion, it is found that theVolterra series representation of a physical system may converge only for a limited range ofthe input amplitude. This also clarifies the fact that the Volterra series is seen as a Taylorseries with memory [103].

Main disadvantage in system identification using Volterra series is that its kernels are notorthogonal. This means that once we have determined a certain kernel the whole procedure ofdetermination has to be repeated for a higher order kernel. For kernels higher than the secondor third order, depending on the linear order of the system, this yields an unacceptablecomputational load and it is therefore concluded that the Volterra series are only suitable inmodeling weak nonlinear systems. Simplifications which reduce the identification complexityare proposed. Measuring kernels by a limited multi-tone excitation was suggested by Boyd etal. [12].

Even though the restrictions are clear, we will use the Volterra series expansion to model thenonlinear loudspeaker because it is a weakly nonlinear system. Therefore a third-orderexpansion is often sufficient to model the electrodynamic loudspeaker. Important feature ofthe series is that it is quite easy to derive a nonlinear compensator to eliminate thenonlinearities described by this functional expansion. Furthermore, its frequency domainrepresentation makes it possible to evaluate the influence of certain parameters and optimizethem directly on measured distortion components.

Other functional expansionsOther functional expansions are the Fliess and Wiener G-functionals, where the latter, and allexpansions in complete orthonormal series of functions of it, have the advantage of beingorthogonal [103]. Example of such functions are the Laguerre functions, which are causal anddigitally realizable. Determination of these series requires even more complicatedcomputation than with the Volterra series. We will limit our research to the use of the Volterraseries, although it is worthwhile to investigate if other (orthogonal) functional expansions arealso applicable in designing nonlinear linearizing controllers and whichadvantages/disadvantages they offer with respect to the Volterra series.

2.2.3 Block oriented modeling

A wide class of nonlinear systems can be modeled by interconnections of nonlinear static(NL) and linear dynamic (LD) sub-processes. Well known examples are the Wiener model(cascade of LD and NL) and Hammerstein model (cascade of NL and LD). The latter hasfound wide application due to its property of only nonlinear processing of the input. In thisfield of block oriented modeling there is also a tendency of using more physical knowledge ofthe process. Extension of the simple cascaded block approach, and identification of itsparameters, is connecting them in parallel. Korenberg for example [60], introduced animproved version of the Wiener model which is called the parallel cascade identification. This

16 Block oriented modeling

method is discussed shortly as an example, for a more complete overview of the numerousmethods of block oriented modeling, see [17].

A very effective method for estimating the kernels of nonlinear systems with a high order isthe parallel cascade identification. It is based on the fact that every nonlinear system,comprising the properties mentioned in the beginning of section 2.2, can be uniformlyapproximated by a finite sum of cascades each comprising a static nonlinear and a dynamicallinear system. In the parallel cascade method the dynamic linear system in each path isdefined using a slice of the crosscorrelation function. The static nonlinear system is thereafterdetermined from a best-polynomial-fit on the residue, i.e. the difference between the outputof the parallel cascade system up till then, and the real output. The essence of parallel cascadeidentification is to approximate the nonlinear system output using a first cascade, determinatethe residue between the real system and the cascade outputs, then approximate the residueusing a second cascade, and so on see Figure 2.1. The resulting system is a number of Wienermodels in parallel.

The parallel cascade identification provides kernel estimation with smaller error compared tothe Lee-Schetzen [103] method of estimating Wiener kernels. Additional advantage is that theparallel cascade method does not require the use of an input source having Gaussianproperties, nor an input having some kind of special probability density or autocorrelationproperties.

x1 (n)

x2 (n)

xi (n)

z1 (n)

z2 (n)

zi (n)

y(n) ��i

j�1

zj (n)

u(n) LD1 NL1

LD2 NL2

LDi NLi

Figure 2.1: Process of parallel cascade identification as an example of block oriented modeling [60].

It seems that this method is attractive for linearization techniques. The structure shows greatmodularity and it is not very difficult to obtain an open-loop linearizing controller fromFigure 2.1. For weakly nonlinear systems we only need to cascade the nonlinear system withthis model, with as only difference a minus sign in the final summation. Next we need to setthe constant values in the NL-block to zero: these are the linear parameters of the systemwhich we do not want to compensate. In chapter 5 we will discuss a compensation techniquewhich results in a scheme similar to Figure 2.1. The only difference is that we will usephysical knowledge to determine the LD and NL blocks. As we will see the outputs of thelinear blocks are the state variables of the system, and the nonlinear blocks describe the


parameter dependencies on the variables. Result is a more transducer related compensatorwhich needs less blocks in comparison with the standard block oriented modeling approach.

2.3 Nonlinear state space

A wide class of nonlinear systems, especially those with moderate nonlinear behavior, can bedescribed by the state space model. In the most general form this model is, in continuous timerepresented by

x.(t) � F{ x(t),u(t)}

y(t) � H{ x(t),u(t)} (2.6)

or in discrete time

x(k� 1)� F{ x(k),u(k)}y(k) � H{ x(k),u(k)} (2.7)

with u � �m the input vector, y� �p the output vector, and x � �d the state vector of thesystem. The vector fields F and H are assumed to be smooth and are of course different forthe continuous and discrete time case if the same system is considered. As stated before wewill only consider SISO systems which have one input and output, thus m=p=1. Referringback to the systems comprising ‘‘soft’’ nonlinearities, it is possible to use parameterizedpower series of state vector x and input u. In this way we get a system description which islinear in the parameters, and if nonlinearities are not too strong it yields an approximationwith a reasonable number of parameters. Partial linear processes can be seen as special casesof equations (2.6) or (2.7). Examples are state-linear processes (x is excluded from functionsF{} and H{}), input-linear processes (u is excluded from functions F{} and H{}), and bothstate-linear/input-linear processes (only input-state multiplications in functions F{} and H{}).The latter are better known as the Bilinear process.

Advantages of the state-space description is that a lot of mathematical theory developed forlinear state-space models is also available for the nonlinear case nowadays. Examples arenonlinear observability, controllability, state space transformations, stability, and minimumphase property [36][83]. A nonlinear state-space description of the nonlinear electrodynamicloudspeaker is the basis for a linearizing controller in chapter 6.

2.4 Nonlinear control

First question one has to answer in control of nonlinear systems is: ”Do we really neednonlinear control?” In many cases linear control of a nonlinear system may suffice. In othercases it is not sufficient, as may become clear from the following example (taken from [73]).

18 Nonlinear control

Example 2.1 In this example we consider the following simple nonlinear system

x.1 � x2

x.2 � �x

32� u

y� x1.(2.8)

When we apply the linear state feedback control u � � k1x1� k2x2 based on the linearapproximation about x=0, the following closed loop system results

x.1 � x2

x.2 � � k1x1� k2x2� �x

32

y� x1(2.9)

which has an unstable limit cycle. If we choose the linear gains k1 � k2, k2 � k, it turns out

that, with for example �� 1�3, all state space trajectories starting in the region

�x � �2 | kx21� (1�k)x22 � 9� tend to infinity. These problems are omitted if we use the

nonlinear control u � � �x32� k1x1� k2x2 which cancels out the nonlinearity and if we then

apply the desired linear control law.

In general we have two categories in nonlinear control: stabilization (or regulation) andtracking (or servo). In case of stabilizing problems we design a controller (called stabilizer)so that the state of the closed-loop system is stabilized around an equilibrium point. Intracking problems we desire the output of the system to track a certain predeterminedtime-varying trajectory. Our control problem is clearly a tracking problem, i.e. we need acontroller (called tracker) which tracks the input signal (audio). In both controllers this caneither be done by static state feedback or dynamic state feedback, where in the latter casethere are dynamics in the control law. Although our objective is tracking, there is a slight difference in the way we approach theproblem compared with the control world. Our objective is, in principle, only to eliminate thenonlinearities of the system, leaving hereby the linear properties of the system untouched.Next to this difference we will analyze the performance of the controllers in the frequencydomain ( and measure resulting distortions), while a control engineer in most cases analyzesthe time domain response (step- or impulse-response, tracking or stabilization).

Digital linear control (equalizing) of loudspeakers is done by filters having a very high order.This is necessary because the order of the system in the complete pass-band is high due tobreak-up modes of the diaphragm, resonances of the cabinet etc. It would be superfluous touse such a high order model as a basis for a nonlinear controller, also because nonlineardistortions in direct radiator loudspeakers mainly exist in the lower frequency band, a regionwhere the loudspeaker can be modeled quite well with a simple low order model. This is, bythe way, in general an important issue: more accurate models are not always better because


they result in a more complex analysis and control design, resulting in a more computationaldemanding implementation. Higher order models comprise more parameters which mayresult in higher modeling errors if parameters are inaccurate.

In the next part we will give an overview of the methods to design a nonlinear controller.Contrary to linear control, there is no general method for designing nonlinear controllers.Often it depends very much on the particular system (model) which method from the richcollection of techniques is chosen.

Trial and errorUsing nonlinear analysis techniques like the phase plane method, the describing functionmethod and Lyapunov analysis, a nonlinear controller can be synthesized. Using a lot ofexpert knowledge and intuition this method may lead towards a correctly operating nonlinearcontroller. For more complex systems, however, the trial-and-error method often fails.

Gain-schedulingOriginally developed for trajectory control of aircraft, gain-scheduling is an attempt to use thewell known linear control techniques for control of nonlinear systems. First a number ofoperating points is chosen which must cover the full range of operation. Then, at each ofthese points a linear time-invariant approximation of the system is made and a linearcontroller is developed for each linearized system. Between each of these points parametersof the controllers are then interpolated (scheduled), resulting in a global controller. Advantageof gain-scheduling is its simplicity which has led to successful application to a number of‘‘difficult to control’’ systems. Main disadvantages are that there is no theoretical guaranteeof stability in nonlinear operation and the enormous amount of computational power neededto calculate all linear controllers.

Feedback linearizationThis technique was in fact already described in Example 2.1 in the beginning of this section.The method is based on differential geometrical techniques (see appendix C) and basicallytransforms a system into a linear system, after which the well known methods from linearcontroller design are applied. It can only be applied to a certain class of nonlinear systemswhich exhibit the so-called input-state linearizable or minimum phase property. Practicalrealization typically consist of two parts: first all dynamics (if possible or necessary) areremoved by state feedback after which the desired linear dynamics is applied. Robustness toparameter uncertainty and disturbances of this method is not guaranteed; it is considered inchapter 6. Feedback linearization is also used as a basis for robust and adaptive controllerswhich are shortly reviewed next.

20 Nonlinear control

Robust controlIn robust nonlinear control a controller is designed based on a model plus some kind ofcharacterization of the model uncertainties (for example the knowledge of minimum andmaximum values of parameters). Examples are sliding control and backstepping. The first isbased on the fact that it is much easier to control 1st-order systems being nonlinear oruncertain, than it is to control nth-order systems. In this technique an nth-order system isreplaced by a set of 1st-order problems, which can in principle achieve perfect performance inthe presence of arbitrary parameter uncertainties. This is achieved, however, at the price ofextreme high control activity. Integrator backstepping or simply backstepping, is recursivelyapplying the technique of ‘‘adding an integrator’’ to stabilize nonlinear systems [26].

Adaptive controlThe need for adaptation was already shortly mentioned in the introduction of this chapter. Theterm adaptive has a broad general meaning but currently adaptive controllers are based on aknown dynamic structure and adapt for uncertain constant or slowly time-varying parameters.It is important to recognize that adaptive controllers, wether they are developed for linear orfor nonlinear systems, are inherently nonlinear. For nonlinear systems several techniques havebeen developed which can be seen as an extension to static nonlinear controllers. It is, forexample, possible to design an adaptive feedback linearizing controller [73], or extend arobust controller with an adaptive part. Adaptive control is therefore seen as an alternativeand complementary approach to existing nonlinear control techniques.

Fuzzy logic and neural networks based controlDevelopment on the theory and application of fuzzy logic and neural networks has beentremendous in the last decade. Especially due to the availability of powerful computinghardware they have found widespread applications. In nonlinear system modeling and controlneural networks and fuzzy logic have also found their application, resulting in neural, fuzzyand neuro-fuzzy controllers. Like with adaptive control, both techniques must be seen as analternative and complementary approach to existing controllers.For example, in adaptive linearizing control for an electrodynamic loudspeaker a neuralnetwork was applied within the frame work of a controller to be discussed in chapter 5 [72].Results of this work, however, show no improvement towards the same controller without aneural network, i.e. with static off-line determined parameters. However, this field of controlengineering is still under development and application in linearizing control for otherapplications has shown the usefulness of neural nets and/or fuzzy logic.

Other techniquesAll methods described so far are techniques having their roots in control engineering, and arein most cases based on a state-space description of the system. In section 2.2, however, wehave seen several other system description methods, most of them related to the (digital)signal processing engineering. One of them is the Volterra series expansion, suitable to model


weakly nonlinear systems either in discrete or continuous time. Based on such an nth-orderexpansion it is possible to synthesize a pth-order pre- or post-inverse system (p � n). It can beshown that the pth-order post-inverse of a nonlinear system is the same as its pth-orderpre-inverse [103]. Such an inverse system, when concatenated with the nonlinear system,cancels p kernels and will in such a way reduce distortions. This technique is the subject ofchapter 4. Finally there are techniques which can not be directly classified into one of thegroups we have discussed so far. They are highly dedicated to the specific system to becontrolled and are based directly on the nonlinear (integro-) differential equation, or aspecific physical description of nonlinear parts of a device. These techniques are subject ofchapter 5.

At the end of this section we address the question of how accurate a compensator which hasto cancel distortions, needs to be. Intuitively it is clear that to cancel a primary signal (i.e. thedistortion components), we need to apply a signal from a secondary source having exactly thesame magnitude but an opposite sign, i.e. exactly 180o shifted in phase. This thought iscorrect with linear systems where the superposition principle holds. In this case we cancalculate how much magnitude and phase error are allowed. This is depicted in Figure 2.2where the reduction of a primary sound source due to application of a secondary sound sourceas a function of the magnitude and phase errors is given [22]. To obtain a reduction of 20dB,for example, a magnitude error of 0.91dB and a phase error of 5.75o are maximally allowed.For nonlinear systems, however, we can not use Figure 2.2 directly because we synthesizedistortion products which are processed by the nonlinear system itself.

0100

200300

−10−5

05

10

−20

−15

−10

−5

0

5

10

(φp − φs) [degrees]20log10

|pp| |ps|

___ [dB]

∆ L

[dB

]

(a)

0 50 100 150 200 250 300 350

−10

−5

0

5

10

(φp − φs) [degrees]

20lo

g 10 |p

p|

|ps|

_

__

[dB

]

(b)

−20 dB

−5 dB

+6 dB

0 dB

+3 dB

Figure 2.2: Relative reduction in magnitude �L of the primary source strength pp (having a phase �p)due to a secondary source ps (having a phase �s) as a function of magnitude and phase differencesbetween them. Ideal situation, resulting in an infinite reduction, is obtained if pp=ps and �s=�p-180o.Figure (b) are equal reduction contours of the three-dimensional surface in (a).

22 State measurement and observation

‘‘Inside’’ a nonlinear system the superposition principle does not hold and at forehand we arenot able to give a maximum allowable magnitude and phase error. Still, the figures for linearsystems do already make it clear that our compensator needs to be very accurate, as it is notto be expected that larger errors are allowed in nonlinear systems.

2.4.1 State measurement and observation

Many of the techniques described thus far, require full state information. This can be done bymeasuring all states, which is hardly ever possible, by the use of an observer or bydifferentiation/integration of state/output measurement. The latter method suffers fromproblems with enormous gaining of noise when we use a differentiator, and integratorwind-up problems due to small DC offsets from sensor measurement.

A nonlinear observer can be designed using also the techniques described for control design.Generally we distinguish two approaches. The first one is the geometrical approach, also usedin feedback linearization, which attempts to find a coordinate transformation which linearizesthe error dynamics of the state estimation. It is hereby also possible to design an adaptiveobserver which performs two tasks: state estimation and parameter identification.

The second approach is based on the linearized model around some operating point. In spiteof the in theory only local convergence of these methods, they are widely used and give goodresults in practice. One of the most popular methods in state estimation of nonlinear systemsis the extended Kalman filter.

The Kalman filter is a recursive filter which uses a least-square estimate of the state-vector.The extended Kalman filter assumes that one can adapt the linear systems matrix (A) tochanging local linearizations by a system identification scheme. For a discrete systemdescription given by equation (C.1), the extended Kalman filter used as an observer is givenby a measurement update

x^(k� 1) � x^ u(k� 1)� K(k� 1)e(k� 1)

K(k� 1)� Pu(k� 1)HT(k� 1)�H(k� 1)Pu(k� 1)HT(k� 1)� R(k� 1)��1

P(k� 1)� { I � K(k� 1)H(k� 1)} Pu(k� 1)

(2.10)

and a time update

x^ u(k� 1) � f �x^(k),u(k)�

Pu(k� 1)� F(k)P(k)FT(k)� Q(k)(2.11)

where the error update, linearized system matrix and output matrix are respectively given by


e(k� 1) � y(k� 1)� h�x^u(k� 1),u(k� 1)�

F(k) ��f [x(k), u(k)]

�x(k)�x(k)�x^(k)

H(k� 1)� H�x^ u(k� 1),u(k� 1)� ��h[x(k� 1),u(k� 1)]

�x(k� 1)�x(k�1)�x^ u(k�1)

.

(2.12)

In case of a linear system disturbed by noise, optimal filtering in the maximum likelihoodsense is obtained when we choose Q(k) and R(k) as the covariance matrices of the system andmeasurement noise respectively. For nonlinear systems this optimality cannot be proved, butthey are frequently chosen similar to the linear case. For deterministic systems like given byequation (2.7), one sets Q(k)=0.

Next to these techniques we will introduce some other methods in this thesis to obtain stateinformation. The first one is by linear filtering of the input signal, and is used in chapter 4and 5. For low to medium driving levels of weakly nonlinear systems this method issufficient. Secondly we will use nonlinear filtering to perform nonlinear state predictionbased on the output signal of the compensator in chapter 6. Finally we can design a nonlinearstate observer in the same way as performed for linear systems. In this method we separate (ifpossible) the linear part from the nonlinear and design a linear state observer for the resultinglinear system. The nonlinear part is then used in the state estimation again. This method wasalso applied to design a nonlinear velocity observer for a robot arm recently [39].

2.5 Summary

Although many ways to describe a nonlinear system exist, as we have seen in section 2.2, wehave chosen to use parameterized modeling. This means that we use as much as possible ofthe available physical knowledge to build up nonlinear system models. Advantage of thisapproach is that the resulting compensators have parameters which are related to physicalcauses. This gives additional insight into the importance of one (nonlinear) parameter withrespect to another. Therefore application of nonlinear compensators may be used as an aid toperform improvements into the design of transducers, if we have decided that electricalcompensation is too expensive.Still, it is of interest to investigate wether other methods are suitable to form a basis of anonlinear compensator. Especially the block-oriented modeling techniques introduced insection 2.2.3, are thought to be very suitable for this task.

In the sequel of this thesis we assume that the nonlinearities can be described by smoothsingle valued functions, i.e. they are real-valued functions with continuous partial derivativesof any order. We exclude therefore effects like hysteresis.

From the wide field of nonlinear control several techniques are available to compensate

24 Summary

nonlinear behavior. Feedback linearization is a very suitable technique to perform this task.Disadvantage of this method its high sensitivity for parameter uncertainty. To overcome thisproblem several improvements of this methods are available, like robust, adaptive,fuzzy-logic or neural network based control. They will all result in more complex controllers,however. Therefore we will first focus on simple quite straightforward methods. From theresults with these compensators, we have better understanding of the problem and are capableto judge wether or not more complex methods are needed.

25

Chapter 3

The electrodynamic transducer

After a short review of the linear model of the electrodynamic loudspeaker, themajor nonlinear phenomena which cause nonlinear distortion in electrodynamictransducers and their loads are reviewed. The closed cabinet, the vented cabinetand the horn are the three most widely used acoustic configurations in practice. Ashort discussion of which nonlinear elements are of importance leads towards anonlinear model with three major nonlinearities for the first two load types. Inhorn loudspeakers the adiabatic behavior of the compressed air in thecompression driver and wave propagation in the horn are the major sources ofnonlinearity.

3.1 The electrodynamic loudspeaker

A schematic cross-sectional view of the electrodynamic loudspeaker is depicted in Figure 3.1.It consists of a voice coil which is situated inside a radial magnetic field of a permanentmagnet, which is driven by a current from an amplifier. The voice coil is mostly winded on apaper or aluminium cylinder which on its turn is connected to the cone. The cone is centeredby means of two flexible connections to the frame: a quite flexible rim and a quite stiff spider,where the latter prevents the voice coil from running out of the magnet. The dustcap isapplied to prevent dust and other dirt of coming between voice coil and air gaps.

ÉÉ

rim

conedustcap

spider

voice coil iron core

magnet

frame

ÉÉÉ

Figure 3.1: Schematic cross-sectional view of the electrodynamic loudspeaker

26 Electrodynamic loudspeaker

For low frequencies, where the wavelength is much larger than the dimensions of thetransducer, the cone behaves as a rigid (circular) piston, and the loudspeaker can be modeledin a relatively simple way using lumped elements. This impedance type analogy is easilyconstructed using the well-known analogous electrical network elements of mechanical andacoustical elements, see e.g. [8][87][42]. At higher frequencies this modeling is not valid asthe diaphragm behaves no longer as a rigid piston anymore. Even if the diaphragm is rigid,the flat piston model is an approximation. Although we can model this behavior by means ofa distributed lumped element model we will not consider it here [24][78].

The, widely used and accepted [8][11][42][87], electrical lumped element model of theelectrodynamic loudspeaker is given in Figure 3.2.

i

+ +

– –

F=–Bl i

Gy

m Cm=1/k RmRe Le

+

–

ue Zau � Blx.

x.

Figure 3.2: Electrical equivalent lumped element model of the voltage driven electrodynamicloudspeaker for low frequencies. Coupling between electrical and mechanical domain is performed bymeans of the gyrator Gy.

System parameters in Figure 3.2 are:ue driving voltage at the loudspeaker terminals [V].Re electrical resistance of the voice coil [�].Le inductance of the voice coil [H].i voice coil current [A].u voltage induced in voice coil (also known as motional voltage) [V].Bl force factor, product of B, the effective air gap flux density and l, effective length of

the voice coil wire inside magnetic field [Tm].F Lorentz force acting on the voice coil, for a current conducting voice coil element idl

this Lorentz force element is given by dF � idl � B, which, assuming a homogeneousmagnetic field distribution, simplifies towards a total force given by:

F � i �B� dl � B� l � i [N]. x

.velocity of voice coil [m/s].

Cm compliance of total resulting spring, i.e. of spider and rim. Equals 1/k where k is thestiffness of the same spring [m/N].

m mechanical moving mass [kg].Rm mechanical damping [Ns/m].Za(j�)=Ra(�) + jXa(�) complex acoustical load impedance (assuming a time dependency

v(t) � vme�j�t with � the angular frequency), transformed into its mechanicalequivalent, consisting of a front- and rear-side acoustical load impedance [Ns/m].

27The electrodynamic transducer

In this model we recognize two domains: the electrical and mechanical domain separated bythe gyrator. In fact, we have two other domains as well; the magnetic and acoustic. Themagnetic domain is omitted by transforming elements either to the electrical side (as for theself inductance) or to the mechanical side (as for the reluctance force, see section 3.3.1). Foracoustic elements this is done as well, but now to the mechanic domain. Both steps arefrequently performed in impedance like analogue system modeling and are done to simplifyanalysis.

In the mechanical domain we have a damped mass-spring system driven by the Lorentz forceacting on the voice coil. This system is terminated by a one-port: the acoustical loadimpedance1 Za transformed into its mechanical equivalent. This approach excludes thedescription of directional effects, but makes calculation of power radiated into the acousticaldomain with a loudspeaker in, for example, a closed cabinet relatively easy: the powerdissipated into Za. We will now shortly review the three most widely used acousticalterminations or loads, of the electrodynamic transducer as depicted in Figure 3.3(a)-(c):mounted in a closed- or vented-cabinet and the horn.

ÉÉ

É

ÉÉÉÉ

ÉÉ

ÉÉÉÉ

ÉÉ

(a) (b) (c)

Figure 3.3: The three acoustical configurations considered in this thesis. The electrodynamictransducer mounted in a closed cabinet (a), in a vented cabinet (b) and the horn (c).

3.1.1 Closed-cabinet

The most simple system is with the speaker mounted in a baffle or closed-cabinet (see Figure3.3(a)). Analysis and synthesis of loudspeaker designs inside closed-cabinets is a wellunderstood process [8][112].

______________1. For exact calculation of the sound radiation we have to solve the Helmholtz equation. In its integralform this equation comprises Green’s function and can only be solved analytically for some simple shapessuch as a piston in an infinite baffle or a pulsating sphere [87].

28 Closed–cabinet

Radiation impedance of the cone is approximated as if it behaves as a circular piston inside aninfinite baffle. This acoustical radiation impedance zrad(�) is given by [87]

zrad ��oco�a21� J1(2ka)

ka

� j��o

2�a4k3K1(2ka) (3.1)

with a the radius of the piston, �o the density of air, co the speed of sound in air, k=�/co thewavenumber and J1 and K1 Bessel functions approximated by the series

1�J1(2ka)

ka� k

2a22

� k4a4

223� k

6a6

22324� �� (3.2)

K1(2ka) �2�(2ka)3

3�

(2ka)5

325�

(2ka)7

3252

LINEARIZATION OF ELECTROACOUSTIC TRANSDUCERS · ‘‘vlak’’ zijn van de frekwentie responsie...

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