pre-printThis paper presents a novel equalizer for nonlinear distortions in direct-radiator loudspeakers in a closed cabinet by constructing an exact inverse of an electro-mechanical model of the loudspeaker. This exact inverse compensates for distortions introduced by excursion and current-dependent nonlinearities. The equalizer compensates for the nonlinearities in the force factor, voice coil inductance, eddy currents and the stiffness of the loudspeaker. Simulation results demonstrating substantial reduction in the harmonic distortions at the output of the loudspeaker are included in this paper

Mathews, V. John

Payal, Shreyas Srikanth.

English

The University of Utah: J. Willard Marriott Digital Library

EQUALIZATION OF EXCURSION AND CURRENT-DEPENDENT NONLINEARITIES INLOUDSPEAKERSShreyas Srikanth Payal , V. John MathewsDept. of Electrical and Computer EngineeringUniversity of Utah,Salt Lake City, UT 84112, USAAjay Iyer, Russ Lambert, Jeffrey HutchingsHarman International10653 South River Front Pkwy, Suite 300South Jordan, UT 84095, USAABSTRACTThis paper presents a novel equalizer for nonlinear distortionsin direct-radiator loudspeakers in a closed cabinet by con-structing an exact inverse of an electro-mechanical model ofthe loudspeaker. This exact inverse compensates for distor-tions introduced by excursion and current-dependent nonlin-earities. The equalizer compensates for the nonlinearities inthe force factor, voice coil inductance, eddy currents and thestiffness of the loudspeaker. Simulation results demonstratingsubstantial reduction in the harmonic distortions at the outputof the loudspeaker are included in this paper.Index Terms— equalizers, nonlinear distortion, nonlin-ear systems, loudspeakers.1. INTRODUCTIONDirect-radiator loudspeakers are used in many applicationssuch as entertainment devices, mobile phones, tablet comput-ers, laptop computers and desktop speakers. They are inex-pensive and usually have a good linear response over a wideband of frequencies. However, when they are driven withlarge amplitude inputs, nonlinear distortions degrade the au-dio quality. Pre-processing of the input signal using a digitalcorrector for the distortions can mitigate these nonlinear ef-fects and reproduce high-ﬁdelity audio. This paper presentsa pre-equalizer for direct-radiator loudspeakers that compen-sates for distortions introduced into audio signals by the non-linearities in the force factor, the voice coil inductance, thestiffness of the loudspeaker diaphragm and eddy currents.These nonlinear effects of the loudspeaker depend on itsgeometric construction and the materials used in the voicecoil, the diaphragm and the enclosure [1]. Figure 1 displaysa schematic diagram of the loudspeaker, explicitly showinghow the electrical and mechanical components are coupledtogether. The ﬁgure also shows the primary components inthe loudspeaker that exhibit nonlinear behavior.A lumped parameter electro-mechanical model of theloudspeaker is shown in Figure 2. The suspension stiffness email: shreyas.payal@utah.eduFig. 1. Diagrammatic representation of a direct-radiator loud-speaker (adapted from [1, 2] ).Kms(x) arises from the material properties in the surroundand the spider of the loudspeaker. The force factor nonlin-earity, Bl(x), describes the frequency-independent electro-mechanical coupling in the loudspeaker. This is modeled as agyrator. The inductance of the voice coil, L0(x, i), is a func-tion of its displacement x(t) and the current in the voice coil,i(t). To model eddy currents in the pole pieces and the plates,a parallel combination of an inductance L2 and a resistanceR2 is included [1, 3, 4]. These inductances L0 and L2 alsoinduce a reluctance force in the moving components of thespeaker. The electrical voice coil windings have a resistanceRvc. The electrical components are modeled via Kirchoff’svoltage law (KVL) [1, 5]. A force equation is written forthe mechanical components to include the mechanical loadresistance (like friction) Rms that affects the diaphragm andMms, the moving mass of the diaphragm. In this paper,we construct an equalizer for the direct-radiator loudspeakerusing the model in Figure 2. Pre-equalization of the inputaudio using this equalizer provides substantial reduction inharmonic distortion at the output of the loudspeaker.The rest of this paper is organized as follows. Section2 provides an overview of the previous work on nonlinearequalization in loudspeakers. Section 3 describes a simpli-ﬁed loudspeaker model and a formal derivation of an inverseto this loudspeaker model. Evaluation results are discussed inSection 4. Conclusions and future work follow in Section 5.2014 IEEE International Conference on Acoustic, Speech and Signal Processing (ICASSP)978-1-4799-2893-4/14/$31.00 ©2014 IEEE 6697+−u(t)Rvc L0(x, i)L2(x, i2)i2(t)R2(x, i− i2) i(t) v(t)+−i2(t)2dL0(x, i)dx1Kms(x)+−i22(t)2dL2(x, i2)dxRmsMms+−Bl(x)v(t)+−Bl(x)i(t)ELECTRICAL MECHANICALFig. 2. Lumped-parameter representation of a loudspeaker.2. PRIOR WORKNonlinear equalizers were based on truncated Volterra ﬁl-ters in [6, 7] and on Volterra-Wiener-Hammerstein models in[8]1. Gao and Snelgrove [9] introduced an adaptive truncatedVolterra system equalizer for loudspeakers. The mirror ﬁlter[10] was obtained by “reﬂecting” the order of nonlinearitiesin a lumped parameter model of the speaker.We model the loudspeaker using the electrical and me-chanical components of the loudspeaker, and then implementa pre-inverse of this model that pre-distorts the input to thespeaker model. Our equalizer is obtained by inverting all op-erations in the loudspeaker model. Unlike prior work [10, 11,12, 13, 14], our model and the equalizer includes effects ofcurrent-dependent nonlinearities as well as the distortions re-sulting from eddy currents. Unlike the approaches using trun-cated Volterra ﬁlters, our method is an exact inverse. Also,this equalizer can be implemented using one differentiator.This is an advantage over the mirror ﬁlter in real-time appli-cations.3. LOUDSPEAKER EQUALIZER3.1. Loudspeaker modelConsider the equivalent representation in Figure 2. Let thevoltage u(t) denote the input stimulus to the system. Then,the voice coil current i(t) is related to the parameters of thesystem through the KVL as,u(t)−Bl(x)v(t) = i(t)Rvc + ddt(L0(x, i)i(t))+ddt(L2(x, i2)i2(t)) (1)where, i2(t) is the current in the para-inductance, Bl(x)v(t)represents the back-emf induced by the force factor and v(t)is the velocity of the speaker diaphragm. Using KVL in theL2-R2 loop, we getddt(L2(x, i2)i2(t)) = (i(t)− i2(t))R2(x, i− i2) (2)Similarly, for the mechanical components in the lumped1The authors thank the anonymous reviewers for this reference.+ Voice Coil Bl(x)i(t)+ Suspensionfin(t)Bl(x)∫ t0dtPconstu(t) u1(t)+i2(t)2dL0(x, i)dx+i22(t)2dL2(x, i2)dxp(t)v(t)−Bl(x)v(t)Fig. 3. Block diagram for the loudspeaker model.+ HBl(x)∫dtPconstu(t) u1(t) p(t)v(t)−Bl(x)v(t)Fig. 4. Simpliﬁed model of a loudspeaker.parameter circuit, the equation for the force on moving di-aphragm is given by,Mmsdv(t)dt=i2(t)2dL0(x, i)dx−Kms(x)∫ t0v(τ)dτ+Bl(x)i(t)−Rmsv(t) + i22(t)2dL2(x, i2)dx(3)An equivalent block diagram for the loudspeaker can be de-rived as in Figure 3. by combining (1), (2) and (3). The forcesignal input fin(t) to the suspension block is given by,fin(t) = Bl(x)i(t)+i2(t)2dL0(x, i)dx+i22(t)2dL2(x, i2)dx(4)This force moves the diaphragm modulating the air pressurein the region around the speaker. The resulting pressure signalp(t) is modeled as,p(t) = Pconstdv(t)dt(5)where, Pconst =ρπr2spkr2πd, d is the distance at which the pres-sure is measured, πr2spkr represents the effective area of thediaphragm, rspkr represents the radius of the diaphragm andρ corresponds to the air density.Figure 3 can be further simpliﬁed as shown in Figure 4 bycombining a cascade of the voice coil, the force factor non-linearity and the suspension system into an equivalent systemH. The operations in H require the signals x(t) and v(t).3.2. Exact pre-equalizationWe assume that the input signal pin(t) to the equalizer is thepressure waveform that we wish to reproduce at some dis-tance from the loudspeaker. The diaphragm velocity wave-form vp(t) can be estimated using (5), and the displacement6698∫dtPconstBl(xp)H−1 +pin(t) vp(t)pin(t)Bl(xp)vp(t)+u1(t) uout(t)Fig. 5. Loudspeaker pre-inverse.signal xp(t) can be obtained from the velocity waveform. Fig-ure 5 shows a block diagram of the pre-inverse of the model inFigure 4. To see this, we start with an expression for uout(t)in Figure 5.uout(t) = H−1[pin(t)] +Bl(xp)vp(t) (6)This output voltage uout(t) from the pre-equalizer is appliedto the loudspeaker model to generate an output y(t). UsingFigure 4, this is formulated as,y(t) = H [uout(t)−Bl(xy)vy(t)] (7)xy(t), vy(t) were derived from y(t). Substituting for uout(t)using (6) in (7),y(t) = H[H−1[pin(t)] +Bl(xp)vp(t)−Bl(xy)vy(t)](8)Thus, y(t) = pin(t) for all t can be proved by mathematicalinduction and causality of the system.For system H−1, a pressure input signal pin(t) is con-verted to a voltage output u1(t). System H−1 is implementedby reversing the order of operations in H, namely suspensionfollowed by nonlinear scaling, followed by the voice coil, i.e.,(3), (2) and (1). Using the estimated velocity waveform, thedisplacement signal and the acceleration waveform, the cur-rent in the voice coil ip(t) is estimated as,ip(t) =1Bl(xp)(Kms(xp)xp(t)−i2p(t)2dL0(xp, ip)dxMmsdvp(t)dt+Rmsvp(t)−i22p(t)2dL2(xp, i2p)dx)(9)where i2p(t) is the estimated para-inductance current. Thenonlinearities in H−1 are obtained using signals ip(t) andxp(t). Using (9), we may estimate the output voltage signalof system H−1, u1(t), asu1(t) =ddt(L0(xp, ip)ip(t)) +ddt(L2(xp, i2p)i2p(t))ip(t)Rvc (10)where the voltage across the para-inductance L2 is computedrecursively as,ddt(L2(xp, i2p)i2p(t))=(ip(t)− i2p(t))R2(xp, ip − i2p)(11)102 103 104708090Frequency (Hz.)SPL(dB)CascadeWooferFig. 6. Comparison of linear responses.4. PERFORMANCE EVALUATIONAn explicit Runge-Kutta (RK) algorithm [15] of order 4 wasused to perform integration operations in the model. Anapproximate differentiator was constructed as the inverseof this RK-integrator. All signals involved were sampledat 48 kHz. The parameters of the loudspeaker model wereobtained through actual characterization of a 3-inch wooferusing the Klippel Analyzer at JBL Pro, Northridge, CA. Themodel assumed that the nonlinearities L0(x, i), L2(x, i2) andR2(x, i− i2) were separable, i.e.,L0(x, i) = L0(x)F0(i), (12)L2(x, i2) = L2(x)F2(i2), (13)R2(x, i− i2) = R2(x)G2(i− i2) (14)The coefﬁcients for F0(i), F2(i2) and G2(i − i2) were as-sumed to be equal [16]. The excursion-dependent termsL0(x), L2(x) and R2(x) were related as in [1, 16].Figure 6 compares the linear responses of the woofer anda cascade comprising of this shaping ﬁlter, the equalizer andwoofer. The shaping ﬁlter used had a linear response that wassimilar to that of the loudspeaker. Minor differences in theseresponses were observed in Figure 6.Second, third and fourth order harmonic distortions weremeasured for the forward loudspeaker model and the cascadeof the shaping ﬁlter, equalizer and the loudspeaker model us-ing sinusoidal waveforms of frequencies in the range 24 Hz.and 20 kHz and amplitude 8V. The sound pressure level (SPL)for the n-th harmonic of a sine waveform with frequency ω0,was computed as the ratio of the power at frequency nω0 tothe reference sound pressure level, 20 μPa.Figure 7 displays the measured harmonic distortions forthe woofer and the equalized woofer. A substantial reductionin the harmonic distortion was observed because of equaliza-tion using our pre-equalizer. Ideally, the distortions associ-ated with the equalized woofer should be zero. These non-ideal results in the ﬁgure were attributed to computation andmeasurement errors.6699102 103−30−1501530456075Frequency (Hz.)SPL(dB)(a) 2nd order harmonic distortion102 103−30−15015304560Frequency (Hz.)SPL(dB)(b) 3rd order harmonic distortion102 103−30−150153045Frequency (Hz.)SPL(dB)(c) 4th order harmonic distortionFig. 7. Harmonic distortion (HD) responses for the cascaded system (bold lines) and for the nonlinear woofer (dashed lines).102 103−30−1501530456075Frequency (Hz.)SPL(dB)(a) 2nd order harmonic distortion102 103−30−15015304560Frequency (Hz.)SPL(dB)(b) 3rd order harmonic distortion102 103−30−150153045Frequency (Hz.)SPL(dB)(c) 4th order harmonic distortionFig. 8. Harmonic distortion (HD) for the cascaded system when the nonlinear parameters in the woofer are perturbed using arandom error with α = 0.1. Black bold lines : mean HD response over 100 iterations. Blue dash-dot lines : mean HD response+ σ (standard deviation) of the HD responses. Red dashed lines : Response of the nonlinear woofer.In order to test the system for robustness against inaccu-racies in the loudspeaker model, we introduced a mismatchbetween equalizer and loudspeaker model parameters. Thiscould also be used to simulate aging in the loudspeaker.Speciﬁcally, the parameters of the loudspeaker model wereperturbed by multiplicative noise in the formwp = w(1 + κ) (15)wherew is any coefﬁcient of the original model and κ is a uni-formly distributed random variable in the range [-α, α]. Theequalizer used the original parameters w while the wooferused the perturbed parameters wp.Figure 8 displays the harmonic distortions for orders two,three and four for this case of model mismatch with α = 0.1.We can see that even with this level of mismatch, the sys-tem on average is able to reduce the harmonic distortion inthe output of the equalizer by 10-15 dB at most frequencies.Accurate modeling of the loudspeaker is key to additional,substantial mitigation of the distortions.5. CONCLUDING REMARKSA novel pre-equalizer structure for nonlinear equalization ofloudspeakers was presented in this paper. This structure wasable to compensate for nonlinear effects due to excursionand current-dependent nonlinearities and those due to eddycurrents. Performance evaluation involving perturbed coefﬁ-cients of the model indicates that our approach is reasonablyrobust to inaccuracies in the loudspeaker model. The stabil-ity of this inverse can be demonstrated but was not includedin this paper because of space limitations. Additional workon parameter estimation for loudspeaker models, real-timeimplementation and adaptive equalization of time-varyingnonlinearities is underway at this time.ACKNOWLEDGMENTSThis work done at the University of Utah was supported inpart by a research contract from Harman International. Theauthors thank Alex Voishvillo, Felix Kochendo¨rfer and DougButton at JBL Pro, Northridge, CA. for loudspeaker measure-ments, their insights and many discussions.67006. REFERENCES[1] W. Klippel, “Tutorial: Loudspeaker nonlinearities-causes, parameters, symptoms,” J. Audio Eng. 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Equalization of excursion and current-dependent nonlinearities in loudspeakers

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Equalization of excursion and current-dependent nonlinearities in loudspeakers

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