Deliberate Nonlinear Design and Control of Loudspeakers Overview

Deliberate Nonlinear Design andControl of LoudspeakersAndrew BrightNokia CorporationHelsinki, FinlandSymposium “Nonlinear Compensation of Loudspeakers”Technical University of DenmarkDec. 2, 20041 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company ConfidentialOverview• Deliberate nonlinear design: the tradeoffs• Controller: Architecture & general considerations• Loudspeaker model: Alternatives & affects on controller• Constructing a nonlinear compensation controller from a loudspeakermodel• Tuning the controller• Optimal design with nonlinear control2 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential

2 years in psychotherapy training and delivery, and a bit of management consultancy.One can’t just do nothing with all that. While ‘retirement’ seems to preclude activeselling, I do want to offer training, coaching and mentoring, for practitioners, buyersand users of qualitative market research. I’m delighted that a well-known ad agencyhas already taken up the offer. If you’re interested, please contact me. (Ravereviews available on request…)And, incidentally, “will travel” is true: UK, most of Europe, certainly; North Americaand Australia, if generosity is on offer. I promise to charge less than the average HighStreet solicitor.3. Evangelise (a little) for Better Quality Qualitative ResearchFor decades, I’ve had a bee in my proverbial bonnet (the only sort I wear). Given thenear-universal wish to see high-quality qualitative research, why on earth doacademic and commercial-world ‘quallies’ have such profound distrust of eachother? Why don’t market research quallies consult the collective memory, stop reinventingthe wheel, and ask themselves if their data are true and reliable? Whydon’t academic quallies stop obsessing about what exactly they should do, and startasking themselves “what does it all mean?” (Ooh – that’s better.)It started with a paper I wrote for the Market Research Society (MRS), back in 1987,which has recently been placed on the UK Open Resource Network: ‘WhatDetermines Quality in Qualitative Research?’, downloadable fromhttp://bit.ly/QnEr3U . (One very serious commentator at the time failed to noticethe tongue-in-cheek conclusions, so I’d better admit their existence…)There were some intervening publications on this theme, but let’s fast-forward to myMarket Research ‘Valedictory Lecture’ (2011): ‘Market Segmentation, QualitativeResearch and Conversations Across the Garden Wall’. You can view it athttp://bit.ly/wYwgxl . The ‘garden wall’ is the one that has been built betweenacademic and commercial world researchers.*I really do want to tell more people about this, and suggest ways to aim higher, andbetter. If you’d like a guest lecture, let me know. I’ll probably do it for expensesonly.There’s one more ‘public’ lecture that needs to be aired. Of late, I’ve begun to feelthat I really should answer that original 1987 question – at least partially. As soon asI find a suitable forum, I’ll deliver it. The provisional title is ‘Towards a Formula forQuality in Qualitative Research’.4. Be Charitable (waive the fee: make it free)Geoffrey Roughton attended the first-ever MRS Conference in 1957 as a Client Execfor Television Audience Measurement. As he has now reached such seniority that hewould be well entitled to contemplate the research world while sitting cross-leggedon a mountain top, I’m awfully impressed that he has recently, with MRS support,launched ‘Research Aid’: an initiative to bring market research advice to smallcharities, which would find it impossible to commission their own, full-price ad hoc

Change in basic parameters vs. coilheightAssumptions:• Coil lengthproportional toheight• ConstantresonancefrequencyV AS increases withdecreasing height5 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company ConfidentialSmall-signal sensitivity vs. coil height6 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential

Overview• Deliberate nonlinear design: the tradeoffs• Controller: Architecture & general considerations• Loudspeaker model: Alternatives & affects on controller• Constructing a nonlinear compensation controller from a loudspeakermodel• Tuning the controller• Optimal design with nonlinear control7 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company ConfidentialGeneral considerations for a controllerAudioinput signalSignalProcessorPowerAmplifierLoudspeaker+-processedaudio signalpoweraudio signalacousticsound field• Many audio products require DSP (MP3, dig. cellular telephony)• Analogue:+Can be designed from physical model (s-space; differential eqs.)–Drift problems–Not (easily) programmable• Digital:+Programmable+Cheap (free?) HW – if already required– Algorithms very different from physical model8 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential

Feedback linearisationDigital controller design: from plant model• Continuous time model• “Digitised” with numeric integrators• Model built from traditional LPM• Difficult parameter updating• Discrete time model• Easy digital controller design? How to build model? (Ldspk. LPM is in continuous-time)11 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company ConfidentialOverview• Deliberate nonlinear design: the tradeoffs• Controller: Architecture & general considerations• Loudspeaker model: Alternatives & affects on controller• Constructing a nonlinear compensation controller from a loudspeakermodel• Tuning the controller• Optimal design with nonlinear control12 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential

Lumped-Parameter Nonlinear LoudspeakerModel• Combination of electrical andmechanical LPM models• Acoustic dynamics treatedas mechanical-equivalents• Parameter variation withdisplacement producesnonlinearity• Chief culprits:• φ(x): transductioncoefficient• k(x): suspension stiffness• Nonlinear differential equationsmay be analysed by numericalintegration, other methods.power amplifierφ 0to voicecoilk dc dm d+f c+x d( x ( t))dic( t)dxd( t)dLebd dxd( t)vc( t)= Rebic( t)+ Leb( xd( t)) + φ( xd( t)) + ic(t)dtdt dx dt2d xd( t)dxd( t)1 2 dLeb( ) ( ) ( )( xd( t))ic( t)φ xd( t)= mtxd( t)+ c k x ( t)x ( t)i ( t)2 t + t d d −2 cdt dtdx ( t)dZ rm13 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential• Linear modelElectricaladmittance(blocked)v c (s)+1(s)-R ebSystem dynamics viewMechanicalreceptance(open-circuit)Mechanoacoustictransduction,radiationx d (s) p(sρ s 2i cf c(s))10S φ d s0m s2 + c s + k4 πtttφ0s14 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential

Adding nonlinear components• Added as zero-memory nonlinear systems to linear modelf c(s)v c(s)i c(s)p(s)+ 1 x+d(s)1ρφ( x)0S s 2dR mts2 + cts + k4tπeb- -x·k1( x)φ( x) sforce-factor non-uniformitysuspension stiffness nonuniformity15 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company ConfidentialDSP implementation of model• Linear filters needed for:• Mechanical receptance• Differentiation• Other components are “zero-memory”f c(s)v c(s)i c(s)p(s)+ 1 x (s)+dρφ( x)0S s 2Linear filterdR eb- -φ( x)Linearfilterx·k1( x)4π16 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential

FIR filter Mechanical Receptance• FIR inherently stable –attractive for adaptive filtering• Loudspeaker’s mechanicaldynamics are ‘resonant’• Inefficiently modelled byall-zero filterExample electricaladmittance impulseresponse17 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company ConfidentialIIR Filter for Mechanical Receptance• Resonant behaviour modelled with low-order IIR filter• Same structure can be used to approximate differentiation• Nonlinear-elements are zero-memory systems: straightforward digitalimplementation• Forms basis for feedback linearisation-based design of nonlinearcontrollerφ( x)v n c[ ]+ 1/R ebic[ n] + f c[ n ] ws [] nx nΣ ΣΣΣ- -z -1k1( x)Σz -1a 2Σz -1b dt· 1a dt· 1σ xφ( x)a 1ud[ n]b dt·0Σ18 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company ConfidentialΣd·est[ ]

Overview• Deliberate nonlinear design: the tradeoffs• Controller: Architecture & general considerations• Loudspeaker model: Alternatives & affects on controller• Constructing a nonlinear compensation controller from a loudspeakermodel• Tuning the controller• Optimal design with nonlinear control19 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company ConfidentialInverted digital model• Algebraic inversion of voltage & displacement described by model• Basis for nonlinear compensation algorithm• Describes voltage that would have created some specifieddisplacement1/φ()xrp[ n]1/σ xR ebΣ ΣΣz -1z -1a 1·mΣk x 1( )φ()xrlin[ n]a 2·mr n p[ -1]Σz -1 b dt·0Σud·e[ n]20 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidentiala dt·1b dt·1

Complete nonlinear compensatorPre-filteringNonlinear compensator-10-20-30-40-50-60-70100 1000 Fs/250403020100-10100 1000 Fs/221 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company ConfidentialOverview• Deliberate nonlinear design: the tradeoffs• Controller: Architecture & general considerations• Loudspeaker model: Alternatives & affects on controller• Constructing a nonlinear compensation controller from a loudspeakermodel• Tuning the controller• Optimal design with nonlinear control22 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential

Parametric drift and uncertainty• Loudspeaker characteristics change with:• Manufacturing tolerances• Temperature variationsParameter nameSymbolTemperature VariationCoefficientManufacturing toleranceDC-resistance R eb 0.004·R eb ·0°C ±10%Suspension damping c d -0.05 ±10Suspension stiffness k d (none) ±30%Transduction coefficient φ 0 -0.005 n/a• Parameter drift causes mistuning of feedforward controller‣ Fatal problem for pure feedforward control• Controller must be tuned to changes in the loudspeaker23 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company ConfidentialTracking Changes in the Loudspeaker• System identification: Tracking changes using adaptive filteringut ()Poweramplifieri cshunt res. v cPlantLoudspeakerv cAdaptive filter(plant model)-Σε[ n]ic( t)ud( t)pt ()• Identified parameters are updated in pre-processor• Measurement of electrical current, mechanical vibration, or acousticpressure can be measured24 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential

Overview• Deliberate nonlinear design: the tradeoffs• Controller: Architecture & general considerations• Loudspeaker model: Alternatives & affects on controller• Constructing a nonlinear compensation controller from a loudspeakermodel• Tuning the controller• Optimal design with nonlinear control25 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company ConfidentialSimulation of effective sensitivity• Effective sensitivity calculated as a function of coil height calculated1S eff =Ccp1mc• C c is additional amplifier output required for distortioncompensationPk( u)C c =Pk( w)( s)v ( s)x~026 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential

486Hz27 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential825Hz28 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential

1074Hz29 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential3082Hz30 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential

Measurement• Modified voice-coil height samplesprepared for measurement• 1.2mm (standard height)• 0.8mm• 0.4mm• 0.2mmVoice Coil Height1.2 mmPlastic insert0.8 mmOriginal Coil HeightCoil Height = 2 × Magnet GapPlastic insertCoil Height = Magnet Gap0.4 mmPlastic insertCoil Height = ½ × Magnet Gap0.2 mm31 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company ConfidentialLinear FRF of shortened-height V-Csamps.• Shorter heights providehigher sensitivity• 0.2mm shows ~10dBhigher voltage sensitivity,+4dB power efficiency32 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential

Compensation of distortionMeasurements from 0.2mm height coil• Broken: No control• Sold: With control33 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company ConfidentialConclusions• Deliberate introduction of nonlinearity can increase loudspeakersensitivity• Nonlinearities can be compensated by digital processing• Optimal design point not fully clear34 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential