Inverse filtering of a loudspeaker and room acoustics using time-delay neural networks Po-Rong Chang, C. G. Lin, and Bao-Fuh Yeh Departmentof CommunicationEngineering,National Chiao-TungUniversity,Hsin-Chu, Taiwan

(Received30 July 1993;acceptedfor publication28 January 1994)

This paperpresentsthe designof a neural-based acousticcontrolusedfor the equalizationof the responseof a soundreproductionsystem.The systemusuallycan be modeledas a composite systemof a loudspeakerand an acousticsignal-transmission channel.Generally, an acoustic signalradiatedinsidea roomis linearlydistortedby wall reflections. However,in a loudspeaker, the nonlinearityin the suspension systemproducesa significantdistortionat low frequencies and the inhomogeneity in the flux densitycausesa nonlineardistortionat largeoutputsignals.Both the linear and nonlinear distortionsshould be reducedso that high fidelity sound can be reproduced.However, the traditional adaptiveequalizerwhich is only capableof dealingwith linear systemsor specificnonlinearsystemscannotcompensate thesenonlineardistortions.The time-delayfeedforwardneural network (TDNN) which has the capabilityto learn arbitrary nonlinearityand processthe temporal audio patternsare particularly recognizedas the best nonlinearinversefilter of the compositesystem.The performanceof a TDNN-based acoustic controller is verified by some simulation results. PACS numbers: 43.60.Gk, 43.38.Ar, 43.55.Me

INTRODUCTION

The objectiveof the sound reproductionsystemhas been assumedto be the "perfect" reproductionof the recorded signalsat the listener'sears, i.e., the signalsrecordedat two pointsin the recordedspaceare reproduced exactly at points in the listening space. Generally, the soundreproductionsystemis usedto achievethe perfect reproductionof the recordedaudio signalsat the listener's ears,i.e., the signalsrecordedat a point in the recording spaceare reproducedexactly at a point in the listening space.However,the originalaudio signalsare imperfectly reproducedat the earsof a listenerwhen thesesignalsare replayedvia loudspeakers in a listeningroom. The imperfectionsin the reproductionarisefrom two main sources: (i) the acousticsignalsradiatedinsidea room are linearly distortedby wall reflections,and (ii) in a loudspeaker,the suspension nonlinearityproducesa significantdistortionat low frequencies and the imhomogeneity in the flux density causesa nonlineardistortionat large output signals.In order to eliminate the above two undesired factors, it is

necessaryto introduceinversefiltersthat act on the inputs to the loudspeakers usedfor reproductionwhich will compensatefor both the loudspeakerresponseand the room response.Initial attemptsto designsuchinversefiltershas beenconsideredin designingthe filtersusedfor the equalization of the responseof the room acoustic signal-

versebut rather an approximateinverseof the acoustic system. The principal objective of such equalization schemeshasbeenassumedto be the productionof a "closestpossibleapproximation"to the exactreproductionof a recorded audio signal at a single point in the listening space.

An account 4 of workaimedat producing widespread effectiveness of the equalizationof low-frequencysound reproductionin automative interiors showsthat such an approachmay well be useful.Since the traditional equalization can only deal with the linear systemsor specific nonlinear systems,the suspensionnonlinearity of loudspeakerswill significantlydegradethe quality of reproduction at low frequenciesby usingsuchan equalization.For smallinput signals,the loudspeakers can be approximated as a linear system,and the transferbehavioris describedby a linear transfer response.However, the nonlineardistortions,i.e., harmonicsand intermodulation,increaserapidly when the input signalpowerbecomeslarger.This leadsto the nonlinearinversefiltersthat can equalizethe nonlinear distortionsof the loudspeakers. Most of them are basedon

theVolterraseries expansion. 5-7TheVolterraseries isboth a usefultool for analyzingweakly nonlinearsystemsand a basisfor synthesizingnonlinearfilterswith desiredparameters. Nevertheless, the realization of Volterra filters suffers

from its cumbersomerepresentationand computationalin-

efficiency. The emerging feedforward neuralnetworks 8-•ø transmission channel. NeelyandAllen• showed through have the capability to learn arbitrary nonlinearity and

computersimulationsthat the loudspeakerto microphone room impulseresponseis generallya nonminimumphase. This meansthat it is not possibleto realize the exact inverseof an acousticsystemthat has nonminimumphases.

Alternative approachs for therealization of theinverse 2'3 are on the basis of the conventionalleast-squareserror (LSE) methods. However, this inverse is not an exact in3400

J. Acoust.Soc. Am. 95 (6), June 1994

showgreat potentialfor nonlinearfilter application.Artificial neural networks are systemsthat use nonlinear computational elementsto model the neural behavior of the biologicalnervoussystems.The propertiesof neural networks include: massive parallelism, high computation rates,and easefor VLSI implementation.The neural-based inversefilters would be applied to the equalizationof the

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8

ListeningRoom

Power

Microphone Amplifier

Amplifier

Signal

exp [ j ((D/C) IRp-JrRrI] 4•rlR•,+Rrl

X exp(--jcot),

r Audio

•

Z Z p=l r----oo

(2)

whereRe denotes the eightpermutation vectors overthe positiveand negativesigns, Acoustic

Re= (x + x',y+ y',z+ z' ) , l