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By Francis RumseyStaff Technical Writer

Digital signal processing is usedincreasingly in loudspeakers tocompensate for a range of linear

and nonlinear distortion processes thattypically arise. DSP can also be used incrossover design and for controllingthe spatial radiation characteristics ofloudspeakers or loudspeaker arrays.This requires the detailed understand-ing and modeling of the acousticaldeficiencies and behavior of transduc-ers and cabinets. In this article wesummarize a number of papersdescribing recent research in this field;they were all presented at the AES32nd International Conference, held inDenmark in September 2007.

ACOUSTIC AND TRANSDUCERCONSIDERATIONS FOR DSPLOUDSPEAKERSMarshall Buck, in the paper “Acousticand Transducer Considerations forDSP Loudspeakers,” describes the fre-quency response of a loudspeaker asthe single most important parameterrelated to reproduction quality and theone most susceptible to treatmentusing DSP. He points out that by usingDSP a loudspeaker can be given apeak deviation in its frequencyresponse of less than 1 dB on axis. Buthow flat does it need to be before it isaudibly perfect? Referring to the workof Toole and Olive, he suggests that aresonant peak of less than 0.5 dB canin fact be audible if it is relativelybroad. There are a number of distor-tion processes that can affect theresponse by a number of decibelswhen designing a DSP loudspeaker.

Buck explains that the resonancesand band limitation of loudspeakerdrivers can be relatively easilyaddressed with DSP, but that the moreproblematic distortions arise fromcabinet resonances, thermal effects,directional variation, power compres-

sion, limiting, and protection. Hesuggests some approaches to trans-ducer selection and cabinet design thatmay help to overcome these problems.For example, the use of a metaldiaphragm can help to minimize theeffects of heat induced stiffnesschange. However, such a diaphragmcan have a very considerable peak inits response in the stop band (see Fig.1), which can be controlled by using adigital crossover with a high order (atleast 8th order). He also shows thatresponse smoothing in measurement(such as 1/20th octave) gives a goodidea of the octave-to-octave balance inthe frequency response and suggeststhat a flat response in this analysis maybe sufficient to deliver a perceptuallyperfect performance, making further

smoothing using DSP unnecessary. Inother words, flattening every minutedetail in a loudspeaker’s response maynot be needed for perceptually highsound quality. Buck concludes with theview that one may be tempted to useDSP to artificially flatten the responseof an otherwise mediocre loudspeaker,but this may be only part of thepicture, leaving other sources of erroruncontrolled.

MODELING LOUDSPEAKERNONLINEARITIESFinn Agerkvist, in his AES 32nd paperof this title, discusses the differentnonlinear processes that take place inloudspeakers and looks at differentways of simulating and compensatingfor them using signal processing. ➥

DSP in Loudspeakers

File: C:\MLS\ADATA\ZIPA\SEAS\012-01.FRQ 11-2-2000 11:18 AMTransfer Function Magnitude - dB SPL/volts

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Fig. 1. Response of magnesium cone woofer showing two different temperatures(solid and dotted lines), illustrating the peak in the response and the very smalldifference in temperature behavior (courtesy Buck).

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66 J. Audio Eng. Soc., Vol. 56, No. 1/2, 2008 January/February

He explains that it is generallyaccepted that three factors are of pri-mary importance in determining loud-speaker driver nonlinearity: the forcefactor, Bl (the magnetic field strengthtimes the length of the voice coil); thevoice coil inductance, Le; and the sus-pension compliance, Cm. Using a system developed by Klippel (seebelow), Agerkvist is able to plot thechanges in these parameters at differ-ent power levels, as shown in Fig. 2.In this example the largest change isobserved in the driver’s compliance,which increases overall with powerlevel but also changes shape to someextent. Agerkvist points out that it isdifficult to test nonlinearities such asthese because it requires that the loud-speaker is driven at very high powerlevels, which can run the risk ofdestroying the drive unit. For this rea-son it is desirable to have curve fittingtechniques that extrapolate well to lev-els that might be unsafe for physicaltesting.

One of the problems with high-orderpolynomial expansions for curvefitting in these cases is that they tend todeviate from the physical behavior ofreal loudspeakers, so it is important tohave a good knowledge of how loud-speakers really behave in order thatsuch oddities can be spotted. Oneexample is given, based on the knowl-edge that Bl is usually maximum whenthe cone is at rest and drops off ratherfast once the magnet gap is not filledwith the coil. For this reason it may bedesirable to fit the expansion to theinverse of the Bl curve or to use a setof localized Gaussian expansion func-tions that can be summed together.This also seems to work quite well forfitting the compliance curve. Forfitting the inductance curve one needsa function that has a negative slopearound zero and stable values forextreme positive and negative condi-tions, for example. One possibilityhere is to use the sigmoid function,because the standard polynomial

expansion does not seem to be particu-larly suitable.

It was found that one way ofmeasuring the compliance for thepurposes of modeling is to look at thedissipated power. This is because it isunderstood that any heating of thevoice coil will spread to the spider (theflexible device that holds the cone inits correct suspended place), and anyheating of this suspension is likely toaffect its flexibility. However, whenmodeling distortion profiles ofmodeled and real units, the simulationsdid not match perfectly and it wassuggested that other factors might alsobe important, such as the rms velocityand/or displacement of the suspension.The best prediction using power dissi-pation data was found at high powerlevels, where presumably the voicecoil heating had a more substantialeffect on the spider compliance, but atlower levels the prediction was lesssatisfactory.

In his paper “Optimal Design of

DSP in Loudspeakers

Fig. 2. Three main loudspeaker driver nonlinearities plotted with respect to different input power levels. A fourth, shown bottomright, is the derivative of the inductance. These plots are for a 5-inch woofer (courtesy Agerkvist).

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Loudspeakers with NonlinearControl,” Klippel discusses the practi-cal benefits of such technology. Thegeneral structure of an adaptive non-linear control system for a loudspeakeris shown in Fig. 3. A detector is usedto measure the loudspeaker parametersby looking at the voltage and current

characteristics of electrical signals atthe loudspeaker terminal. These arediagnosed and used to control a pre-processing system that treats the signalfed to the loudspeaker appropriately.Apparently it is possible to do thisdiagnosis using any ordinary audiosignal having sufficient bandwidth and

amplitude (for further discussion ofthis matter see a summary of the paperby Pedersen and Rubak, below).During the initial identification ofthose parameters the working range ofthe driver is determined. Klippel showsan interesting example (Fig. 4) inwhich the stiffness of a particular ➥

DSP in Loudspeakers

Fig. 3. Adaptive nonlinear control of a loudspeaker (Figs. 3–6 courtesy Klippel)

Fig. 4. Force factor Bl(x) and Stiffness Kms(x) versus voice coil displacement of loudspeaker under test

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5-inch automotive audio driver isasymmetrical: negative displacementproduces a higher restoring force thanpositive displacement. One of the diffi-culties of dealing with a phenomenonlike this using DSP for correction isthat the system may attempt to correctfor such asymmetry by producing aDC displacement of the voice coil,which moves it away from the Blmaximum thereby introducing inter-modulation distortion.

Klippel shows that voice coildisplacement is an interestingphenomenon to study, as illustrated inthe example in Fig. 5A. The voice coildisplacement is shown on the y axis

against the frequency of a single toneon the x axis for a number of equallyspaced voltage levels. However,although the voltages are equallyspaced, the displacement values arenot, suggesting that the loudspeakerbecomes less sensitive at higher signallevels and exhibits a certain dynamiccompression, as shown in Fig. 5B. Amost interesting point is that if oneobserves the probability distributionfunction (PDF) of voice coil displace-ment for music signals, as shown inFig. 6, the voice coil spends the major-ity of its time at zero and relativelylittle at the extremes; whereas for asine tone the opposite is true. He

suggests that the heating of the voicecoil in such a case for typical musicsignals is of little concern, whereas inthe case of sustained tones it can’t beneglected.

Generally there are two distortionsystems at work in parallel: a linearsystem that acts as a second- or higher-order high-pass filter obeying tradi-tional Thiele-Small parameters, and anonlinear system that comes into playmore at higher amplitudes of conedisplacement. In the latter case,Klippel explains, when the nonlinearcontrol feedback loop comes into playthere can be some unusual effectsbecause the distortion componentsaffect their own generation process. Heconcludes his paper by making theimportant point that one shouldconcentrate the design of the passivedriver in areas that cannot be corrected,such as vibration of the cone or mecha-nism, directivity, size/weight, cost ofmanufacture, and efficiency.

IDENTIFYING LINEARLOUDSPEAKER PARAMETERSIn their paper “Musical Transducer-Less Identification of Linear Loud-speaker Parameters” Pedersen andRubak explain that linear loudspeakerparameters such as resonance fre-quency, damping factor, and voice coilresistance can drift over time moresubstantially than nonlinear parame-ters such as force factor, compliance,and voice coil inductance as a functionof coil position. They drift as a result

DSP in Loudspeakers

A B

Fig. 5. (A) Positive and negative displacement values of an example driver with single tones varied in frequency and voltage.(B) Equivalent reduction in level of fundamental tone due to this compression effect.

Fig. 6. Probability distribution functions of cone displacement for music and a 50-Hz sine tone

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of temperature, humidity, aging, andproduction spread. In order to identifyboth linear and nonlinear parameters,a process known as system identifica-tion is employed, whereby the loud-speaker voltage and current are usedto measure the current state of thedriver. As music is often the main sig-nal used to drive the loudspeaker, theytried an approach that uses music asthe identification test signal. (Systemidentification in general is a process ofmathematical analysis that aims tomodel dynamic systems on the basisof measured input–output data.)

As shown in Fig. 7, an adaptive FIR(finite impulse response) filter is usedto match the loudspeaker parameters inorder that the least mean square of theerror signal is minimized. The authorsexplain that the length of the FIR filtershould be selected so that the trunca-tion error will be sufficiently low—thisalso determines the frequency resolu-tion of the filter. When greater accu-racy is needed, the frequency range ofthe identification process can belimited, for example to frequenciesonly a little above the bass resonancefrequency. Data to describe the loud-speaker can come either from ameasurement or a model. The authorsshow examples based on two Simulinkmodels, one linear and the othernonlinear, of a 6.5-inch bass/mid-rangeloudspeaker unit. First the filter wasoptimized using the linear model with

white noise as a test signal. A processof trial and error was used to choosethe best resolution for the process,leading to a sampling frequency of 4 kHz and a filter length of M=200.This was done to avoid the possibilityof one particular frequency componentof a musical signal having an undue

influence on the outcome. The esti-mates of driver resonant frequency andvoice coil resistance both turned out tobe very close to those of the realdriver, with an error of around onepercent or less, whereas the dampingfactor was slightly less well predicted,with an error of nearly six percent. Thelast parameter was believed to be inerror because of an over-simplifiedfitting function for the damping factor.Subsequently, the authors comparedthe results from their model usingwhite noise with those derived from amusic signal. As far as the resonancefrequency results were concerned, theresults were similar to those obtainedwith white noise but the damping-factor results had a higher spread atdifferent loudspeaker input levels.They suggest that the voice coil resis-tance is more difficult to estimate withmusic because it rarely has substantialenergy below 20 Hz. This can be dealtwith by employing a small DC offsetin the test signal, which can providethe necessary information.

In discussing the prediction andmodeling of nonlinear loudspeakerparameters toward the end of their

DSP in Loudspeakers

Fig. 7. General structure of an FIR system identification with a loudspeaker. Theerror signal: e(n) = d(n) – y(n)); d(n) is the desired response, loudspeaker current;y(n) is the output of the FIR filter (courtesy Pedersen and Rubak).

➥

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paper, the authors note that a simplemodel of compliance based only on afunction of the cone position seems tobe insufficient. Also they note thatsystem identification can sometimescome up with the wrong parameters,

particularly if the test signal is not suit-able for the algorithm concerned. Thedesign of such things needs to take intoaccount signal-to-noise ratio, clipping,and displacement of the loudspeaker.A system-identification model based

on an FIR filter was found to be effec-tive for identification of some loud-speaker parameters when one isdealing with small cone displacements.

DSP-BASED LOUDSPEAKERMANAGEMENT SYSTEMIn another AES 32nd Conferencepaper, Thaden et al. describe a loud-speaker management system based onsignal processing, comparing the rela-tive merits of IIR (infinite impulseresponse) and FIR filters. IIR filtersare like conventional analog filters,they explain, with their built-in non-minimum phase response and othertypical “analog” characteristics. How-ever, they are simple to implement andmost typical filter characteristics (suchas high-pass, low-pass with Butter-worth, Bessel, or Linkwitz-Rileyforms) can be built from IIR buildingblocks based on biquad filters, asshown in Fig. 8. Such filters can beused for both crossover design andequalization of loudspeakers, andhigher-order filters can be built bysimply cascading biquads. However,the authors also point out that thenoise and distortion performance ofIIR filters can be poorer than that ofFIR filters because of the feedbackloops involved, particularly whenusing fixed-point processing. Theauthors suggest that 48-point data andarithmetic paths provide a gooddynamic range, whereas it is only nec-essary to use 24-bit precision for thefilter coefficients. They found that 48-bit integer processing (fixed point)produced considerably less noise anddistortion than 32-bit floating pointprocessing for medium and high signallevels.

The advantages of FIR filters lie inthe possibility for creating filters with amuch wider range of phase characteris-tics, as well as the chance of makingone FIR filter that can replace a wholechain of IIR filters. While it is indeedpossible to combine both types of filterin one design, it is apparently morecommon to employ a one-FIR-filter-does-it-all approach. In this way onecan design an FIR filter that deals withthe equalization and crossover functionstogether. One of the advantages ofusing linear-phase FIR filters forcrossovers is that they can be designed

DSP in Loudspeakers

Fig. 8. Topology of a biquad IIR filter in direct form I, with optional error feedback(EFB). The graph shows the self-generated noise of such a filter when using single-and double-precision (48 bit) processing (Figs. 8 and 9 courtesy Thaden et al.).

Fig. 9. Simplified principle of complex EQ FIR filter coefficient generation for a mid-range loudspeaker. (1) Measured loudspeaker response is (2) inverted to form acompensation response, which is combined (3) with the crossover filter characteristicto lead to a target response for the FIR filter. An inverse FFT process (4) is used tocreate a time domain response that can be turned into a suitable FIR filter.

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to avoid any of the problematic cancel-lations and dips in the frequencyresponse that can otherwise arise in thecrossover range. One problem with FIRfilter design is that the processingpower required to implement complexfilters across the entire audio frequencyrange is considerable. This is becausethe length and number of coefficients ofan FIR filter that are needed to get good

frequency selectivity are proportional tothe frequency. For this reason differentbands tend to be operated at differentsampling frequencies, resulting in amultirate filter design. One of the sideeffects of multirate filtering is thatdifferent frequency bands end uphaving different group delays; also thesignal-to-noise ratio becomes worse asthe sampling rate drops. The typical

process for generating the appropriatecoefficients for equalizing and crossingover a typical loudspeaker are shown inFig. 9.

DIGITAL CROSSOVER FILTERSIn two AES 32nd papers dealing withthe application of linear digitalcrossover filters to pair-wise symmet-ric multiway loudspeakers, Horbachand Keele consider both the control ofoff-axis frequency response andbeamwidth or polar pattern of theloudspeaker. They start by pointingout that it is desirable to have a loud-speaker with a uniform and smoothoff-axis response, as this is a featurethat is widely accepted as giving riseto high-quality sound. There are vari-ous ways of attempting to achieve this,including the use of a large number ofdrive units in a long curved array, butthese are not always practical andhave various drawbacks. Quite com-monly engineers will modify theparameters of multiway crossovers inan attempt to “voice” a loudspeakerfor optimum perceived sound quality.An alternative, proposed by Horbachand Keele, is to use a DSP-basedcrossover with a loudspeaker arraythat has a single central tweeter andpairs of lower-frequency unitsarranged on either side, as shown inFig. 10. The idea is that only one ortwo pairs of speakers are operating ata time, although the single centraltweeter operates on its own, whichallows the method to be applied toboth equally and nonequally spaceddrivers. The aim is to specify acrossover response shape that forces aflat response at an arbitrary off-axisangle. The authors found that if this isdone the response at other off-axisangles is also reasonably flat.

The bandpass crossover filter theauthors arrived at through their processof development, considered by exam-ining primarily the far-field responseof the loudspeaker, tends to have ashape like that shown in Fig. 11. It hasan unusual pointed-top shape that rollsoff very steeply to either side of a so-called critical frequency at which onlyone pair of drivers is energized. Thespacing of the speakers and thecrossover frequencies are related sothat the spacing is a constant

DSP in Loudspeakers

Fig. 10. Conventional, left, and pairwise symmetric three-way loudspeaker, right,(Figs. 10 and 11 courtesy Horbach and Keele)

Fig. 11. Approximation of a crossover filter using an FIR filter ➥

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distance apart in terms of the acousticwavelength at the critical frequency ofthe crossover. The filter heavily attenu-ates the signal at frequencies aboveand below the critical frequencies ofadjacent drivers. Between the criticalfrequencies only two pairs of speakersoperate.

One of the beneficial side effects ofthis design is that the vertical polarpattern is also maintained relativelyconsistently over the frequency rangesof different drivers, so that the patternwhen only one pair of sources is activeis relatively similar to that which arisesbetween the critical frequencies whenmore than one pair is active. It issuggested that the spacings of thedrivers at their critical frequenciesshould be in the range of 0.4 to 0.6 ofone wavelength in order to achieve this.A spacing of 0.5 wavelength is nearoptimum and gives a polar pattern withno side lobes and a vertical beam widthof around 84 degrees. The authors foundthat simplifying the design technique, soas to restrict the flattened off-axisresponse to the 6-dB-down levelcompared with on axis, defined thebeamwidth, and that forcing the 6-dB-down off-axis frequency response to beflat forced the beamwidth to be constantin the same frequency region.

SUMMARYBased on the summary of these papers,it can be seen that there is considerablepotential these days to compensate forboth linear and nonlinear characteristicsof loudspeakers by using digital signalprocessing. One might be able toemploy drive units previously consid-ered unusable or inappropriate and usesignal processing to make the resultingsound more acceptable. The relativecosts of improving drive units and ofsignal processing are beginning to reachthe point where it may be more eco-nomical to employ signal processingthan to improve the mechanics oracoustics of the physical loudspeaker,although there are still things that can-not be adequately dealt with using sig-nal processing. Such techniques maylead to the possibility for more compacttransducers that have previouslyunheard of sonic performance, althoughthe laws of physics naturally still limitwhat can be achieved in this way.

DSP in Loudspeakers

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DSP forLOUDSPEAKERS

DSP for Loudspeakers is now more relevant than ever. This iseffectively demonstrated by these 21 scientific papers, whichdescribe how digital signal processing really can make thedifference in loudspeaker design and usage. The proceedingshold 235 pages of state-of-the-art technology in thisimportant research field.

Also available as adownloadable PDF (11.3MB).

Purchase online atwww.aes.org/publications/conf.cfm

2007 September 21– 23

Conference Chair: Jan Abildgaard Pedersen

Hillerød, Copenhagen, Denmark

THE PROCEEDINGS

OF THE AES 32nd

INTERNATIONAL

CONFERENCE

For more informationemail Donna Vivero atDMV@aes.org orcall +1 212 661 8528, ext. 42

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