Below 700Hz sound fields can be approximated well over a region of space that encloses the human head, using the acoustic pressure and gradient. With this representation convenient expressions are found for the resulting Interaural Time Difference (ITD) and Interaural Level Difference (ILD). This formulation facilitates the investigation of various head-related phenomena of natural and synthesised fields. As an example, perceived image direction is related to head direction and the sound field description. This result is then applied to a general amplitude panning system, and can be used to create images that are stable with respect to head direction

Fazi, Filippo M.

Menzies, Dylan

Southampton (e-Prints Soton)

Audio Engineering SocietyConvention PaperPresented at the 138th Convention2015 May 7–10 Warsaw, PolandThis Convention paper was selected based on a submitted abstract and 750-word precis that have been peer reviewedby at least two qualified anonymous reviewers. The complete manuscript was not peer reviewed. This conventionpaper has been reproduced from the author’s advance manuscript without editing, corrections, or consideration by theReview Board. The AES takes no responsibility for the contents. Additional papers may be obtained by sending requestand remittance to Audio Engineering Society, 60 East 42nd Street, New York, New York 10165-2520, USA; also seewww.aes.org. All rights reserved. Reproduction of this paper, or any portion thereof, is not permitted without directpermission from the Journal of the Audio Engineering Society.A theoretical analysis of sound localisation,with application to amplitude panningDylan Menzies1, Filippo Maria Fazi11 Institute of Sound and Vibration Research, University of Southampton, UKCorrespondence should be addressed to Dylan Menzies (d.menzies@soton.ac.uk)ABSTRACTBelow 700Hz sound fields can be approximated well over a region of space that encloses the human head,using the acoustic pressure and gradient. With this representation convenient expressions are found forthe resulting Interaural Time Difference (ITD) and Interaural Level Difference (ILD). This formulationfacilitates the investigation of various head-related phenomena of natural and synthesised fields. As anexample, perceived image direction is related to head direction and the sound field description. This resultis then applied to a general amplitude panning system, and can be used to create images that are stablewith respect to head direction.1. SOUND FIELD REPRESENTATIONA source free region of a sound field can be expandedas a series about any point in the region [5]. The firstorder approximation of the pressure P at a pointx, expanded about point x0 , can be given in thefrequency domain in terms of pressure and gradientbyP (x) ≈ P (x0) +∇P (x0) · (x− x0) (1)The approximation is good provided the wavelengthis considerably larger than the region, and at leastone of P (x0), ∇P (x0) is not very small comparedto higher order derivative terms. Below 700Hz atypical sound field region large enough to enclosethe human head satisfies these conditions. For nat-ural sound fields it would be very unusual for thefirst two terms to be highly suppressed relative tohigher terms, and we do not consider this possibil-ity here for synthetic fields. Further more at thesewavelengths the head is approximately acousticallytransparent, so the pressure signals at the ear posi-tions when the listener is present, that is the bin-aural signals, closely match the corresponding pres-sures of the incident field. Therefore to a first orderapproximation the binaural signals at the right andDylan Menzies AND Filippo Maria Fazi A theoretical analysis of sound localisationleft ears are given byPR = P + rH · ∇P (2)PL = P − rH · ∇P , (3)where P and∇P are the pressure and gradient at thecentral point between the ears and rH is the vectorfrom the centre to the right ear. The gradient isrelated to particle velocity V by Euler’s equation inthe frequency domain,∇P = −jkZ0V , (4)using the positive frequency convention p(x, t) =P (x)ejωt [8]. Z0 is the free-field impedance and kis the wavenumber. In the following discussion onlyrelative pressure phases are of interest, so it can beassumed without loss of generality that P is positivereal valued. V can be written as a sum of real andimaginary vectors,V = V< + jV= (5)Hence,PR = P + kZ0(rH · V= − jrH · V<) (6)PL = P − kZ0(rH · V= − jrH · V<) (7)The low frequency approximation condition can bewritten krH << 1.Fig 1 shows PR and PL in the complex plane whenonly V< is non-zero. There is then a phase differencebetween the binaural signals, but no amplitude dif-ference. The phase difference φRL is greatest whenthe velocity vector V< is along the inter-aural axis,since |rH ·V<| is then at maximum. As the head ro-tates PR and PL move inwards and outwards alongthe dashed line.Fig 2 shows the case where only V= is non-zero.There is no phase difference between the two bin-aural signals, however they do differ in amplitude.Figs 3,4 show more general examples where both V=and V< are non-zero. Fig 3 shows the case where V=and V< are in the same direction. ThenPR = P + kZ0rH · Vˆ (V= − V<j) (8)PL = P − kZ0rH · Vˆ (V= − V<j) , (9)PR = P − jkZ0rH ·VℜPL = P + jkZ0rH ·VℜφRLbb0ℑℜFig. 1: PR and PL in the complex plane for non-zeroV<PR = P + kZ0rH ·VℑPL = P − kZ0rH ·Vℑbb0Fig. 2: PR and PL in the complex plane for non-zeroV=PLPRb0bbℑℜFig. 3: PR and PL in the complex plane for alignedV< and V=PLPRbb0bℑℜFig. 4: PR and PL in the complex plane for non-aligned V< and V=AES 138th Convention, Warsaw, Poland, 2015 May 7–10Page 2 of 5Dylan Menzies AND Filippo Maria Fazi A theoretical analysis of sound localisationwhere the direction norm Vˆ = Vˆ< = Vˆ= , so PRand PL lie on a the same straight line for all headdirections determined by rH . Where V< and V= arein different directions then PR and PL move aroundan ellipse as the head rotates, as shown in Fig 4.2. LOCALIZATION CUESThere are two types of sound localisation cues oper-ating in the range below 700Hz, Interaural Time Dif-ference (ITD) and Interaural Level Difference (ILD)[1].For a harmonic field like that in (6,7) the ITD is alsocalled the Interaural Phase Delay, and is φRL/ω,where φRL = arg(PR/PL) is the Interaural PhaseDifference. For purely real V , i.e. V= = 0, P andV are in phase, and PR and PL are of equal ampli-tude, but differ in phase except when the listener ispointing in the direction V<. Thentan(φRL/2) =kZ0rH · V<P(10)For krH << 1 ITD is frequency independent [1],φRL/ω ≈ Z0rH · V /(cP ). For example for a planewave, P = Z0V .Interaural Level Difference (ILD) is the amplituderatio |PR/PL|. For purely imaginary V , i.e. V< = 0,then PR and PL are in phase or anti-phase, but differin amplitude except when the listener is pointing inthe direction V=. The ITD is then zero and the ILDis given by ∣∣∣∣PRPL∣∣∣∣ = ∣∣∣∣P + kZ0rH · V=P − kZ0rH · V=∣∣∣∣ (11)In general V= and V< are both non zero. If the vec-tors point in different directions then the cues areinconsistent for a single source. The difference be-tween the vectors could be used in constructing anobjective measure of image quality. ILD at low fre-quencies is only consistent with sources that are nearcompared to the ear separation 2r. Controlling bothV= and V< it may be possible to create consistentnear-field cues.V /P can be viewed as a local complex vector ad-mittance. For real values of V /P the incident fieldappears locally as a plane wave whose wavelength islonger or shorter than of a free-field plane wave ofthe same frequency, and the ITDs are correspond-ingly greater or smaller.V< and V= are directly related to the active andreactive intensity vectors Ia, Ir.V< =1|P |<(PV∗) =2|P |Ia (12)V= =1|P |=(PV∗) =2|P |Ir (13)The formula are valid for P and V both complexvalued, with V< and V= defined as before by firstrotating P and V in the complex plane so that P ispositive real valued.Ia is frequently used as a measure of localisation,however Ir is generally overlooked even though itcan have just as much impact. The use of Ir intandem with Ia should therefor be encouraged.3. IMAGE DIRECTIONSynthesized sound fields can produce virtual imageswhose direction varies significantly depending on thelistener’s head orientation. For example in Fig. 5an image is generated by feeding the same signal totwo loudspeakers. The image appears centrally ifthe listener faces it but if the listener faces away theimage direction is shifted towards the listener’s newdirection.b bFig. 5: Showing the image when two loudspeakerseach have the same signal, for different head direc-tions.This effect can cause unwanted perceptual instabil-ity. Pulkki investigated this for the case of panningbetween two loudspeakers [6], and created compen-sated panning functions by fitting to subjective mea-surements for particular angular loudspeaker sepa-rations. In the following, the off-median image di-rection is predicted from first principles using thesound field description.AES 138th Convention, Warsaw, Poland, 2015 May 7–10Page 3 of 5Dylan Menzies AND Filippo Maria Fazi A theoretical analysis of sound localisationAssuming V< 6= 0 and V= = 0, and discountingother factors such as high frequency spectral, visualand dynamic cues, a listener in the field will perceivean image in a direction based only on the currentITD. The key observation in this section is that wecan reasonably suppose that the listener processesthe ITD information under the assumption that itoriginates from a plane wave, because there is nocontrary cue, even though the field may be moregeneral as we consider here. The perceived directionshould therefor be that of a plane wave with sameITD as the incident field described by P and ∇P .From (10) the interaural phase difference φRL thensatisfiestan(φ/2) =kZ0rH · VP=kZ0rH · VIPI(14)where VI and PI describe a plane wave travellingfrom the perceived image direction. Since PI = Z0VIZ0rH · VP= rH · VˆI , (15)where VˆI is the normalised vector in the direction ofVI .And so,rˆH · (αVˆ + rˆI) = 0 (16)where the relative admittance α = Z0V/P , and thedirection norm to the image rˆI = −VˆI . Note thatrˆH could instead be replaced with the reverse vectorpointing to the left ear. (16) can also be writtenα cos θHV = cos θHI (17)where θHV and θHI are the angles between rˆH and−Vˆ , and rˆH and rˆI respectively.So given the head direction and the field descrip-tion, and provided α cos θHV = αrˆH · Vˆ < 1 thenthe image direction rˆI has possible values on a coneof confusion about rˆH with angle θHI . Otherwisethe most plausible value for rˆI is rˆH , as this givesthe closest ITD agreement. It is striking that rˆIis independent of the listener head dimensions, andany individual mapping between ITD and source di-rection. The relationships in (16,17) are illustratedby cross-section in Fig. 6. The cone of confusionextends out of the page.If the listener were facing in the direction −Vˆ thenthey would perceive an image in the median plane.rˆHrˆIrˆ′I−αVˆθHIθHVFig. 6: Spatial vector diagram showing the relation-ship between the possible image directions rˆI and rˆ′I ,inter-aural direction rˆH , and field description vector−αV . The circle has unit radius.For the case where all vectors all restricted to thehorizontal plane, the shift in perceived image direc-tion caused by a general listener head direction givenby rˆH , is θIV = θHV − θHI .If the field is static and the shift is non-zero, thena static image cannot be reproduced. However bytracking the head direction it is possible to modifythe field so that the image is static and stable. In thenext section the control of the field using panning isdescribed.For a field containing a mixture of frequencies, αmay be constant or vary over frequency. If it variesthen the image will be spread over a range of direc-tions.4. APPLICATION TO PANNINGAmplitude panning is a spatial audio reproductionmethod in which several loudspeakers are assumedto produce plane waves converging at the listenerin phase. The pressure and velocity at the listenerare given by the sum of the pressure and velocity ofthese waves. Soα = Z0|∑Vi|∑Pi=|∑PiVˆi|∑Pi=|∑ giVˆi|∑gi(18)where {gi} are gains applied to a source signal toprovide loudspeaker feeds. The Makita localisationAES 138th Convention, Warsaw, Poland, 2015 May 7–10Page 4 of 5Dylan Menzies AND Filippo Maria Fazi A theoretical analysis of sound localisationvector [2–4] is defined byrV =∑girˆi∑gi(19)where rˆi = −Vˆi are the direction norms to the loud-speakers. (16) can then be writtenrˆH · (rˆI − rV ) = 0 (20)rˆV = −Vˆ and α = rV = |rV |.According to ambisonic panning a plane wave is re-produced in a local region at the listener’s head,that is rV = 1, by using positive and negative gainsin loudspeakers that surround the listener, ideallyat regular intervals. If the listener moves, track-ing is required so that the sweetspot region can bemoved with the listener. Using the (20) the cues fora plane wave can reproduced using only loudspeak-ers within one half of the space around the listener.This could provide greater flexibility in some appli-cations. Tracking can also be used to achieve correctparallax according to target source distance, as dis-cussed in [7].5. CONCLUSIONIn the low frequency region ITD and ILD have a sim-ple representation in terms of the complex pressureand velocity components. This was used here to de-rive a compact formula relating image and head di-rections with the representation. The approach canbe used to find panning gains for a desired imagedirection given the listener’s head direction, givingsignificant improvement over panning based on thetangent law. Initial indication is that this agrees wellwith experiment. This and other applications of thepressure-velocity representation of ITD and ILD willbe further investigated in the future.6. ACKNOWLEDGEMENTSThis work was supported by the EPSRC ProgrammeGrant S3A: Future Spatial Audio for an ImmersiveListener Experience at Home (EP/L000539/1) andthe BBC as part of the BBC Audio Research Part-nership.7. REFERENCES[1] Jens Blauert. Spatial hearing, 1983.[2] M. A. Gerzon. Practical periphony: The repro-duction of full-sphere sound. In Proc. Audio En-gineering Society 65th Convention, 1980.[3] M. A. Gerzon. General metatheory of auditorylocalisation. In 92nd Audio Engineering SocietyConvention, Vienna, 1992.[4] Y Makita. On the directional localization ofsound in the stereophonic sound field. E.B.UReview, A(73):102–108, 1962.[5] P.M. Morse and K.U. Ingard. Theoretical Acous-tics. McGraw-Hill, 1968.[6] Ville Pulkki. Compensating displacement ofamplitude-panned virtual sources. In AudioEngineering Society Conference: 22nd Interna-tional Conference: Virtual, Synthetic, and En-tertainment Audio, Jun 2002.[7] M. Simon, D. Menzies, F.M. Fazi, T. de Campos,and A. Hilton. A listener position adaptive stereosystem for object based reproduction. In Proc.AES 138th Convention, Warsaw, May 2015.[8] E. Williams. Fourier Acoustics: sound radia-tion and nearfield acoustical holography. Else-vier, 1999.AES 138th Convention, Warsaw, Poland, 2015 May 7–10Page 5 of 5

A theoretical analysis of sound localisation, with application to amplitude panning

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