The composition of a soundfield containing complex nearfield sources has interesting musical applications. A method is presented here using freefield expansions about exterior points of a harmonic multipole field. A derivation and verification are included. Binaural rendering is the natural method for displaying such soundfields. Some possible musical applications are discussed

Menzies, Dylan

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Holophonic synthesis of musical sources

Southampton (e-Prints Soton)

Holophonic SourcesHOLOPHONIC SYNTHESIS OF MUSICAL SOURCESDylan MenziesDe Montfort University, Leicester, UKdylan@dmu.ac.ukAbstractThe composition of a soundfield containing complex nearfieldsources has interesting musical applications. A method is pre-sented here using freefield expansions about exterior points of aharmonic multipole field. A derivation and verification are in-cluded. Binaural rendering is the natural method for display-ing such soundfields. Some possible musical applications are dis-cussed.INTRODUCTIONThe spatial content of sound is an important part of how it is perceived. Thishas been reflected in the production of music through the ages, though thespatial acoustics of auditoria and the way sound is projected from musicalsources. Through the development of the loudspeaker, and later digital sig-nal processing, the possibilities for generating and controlling spatial soundhas increased greatly. The traditions of electroacoustic and acousmatic mu-sic have explored these areas, and have sought to explicitly portray spatialqualities as musical attributes, both in recording and live performance. Inmainstream recording, there is an increasing interest in spatial sound, asstereo is being superseded with 5.1 systems. Here we are concerned, in par-ticular, with synthesizing complex virtual sources, with dynamic control overthe location and orientation of the sources. It should be possible to captureand manipulate the experience of listening to a rich source such as a violinin close proximity, rather than working with the current static methods ofrecording. Similarly, purely synthetic sources could be created, which havespatial variety associated with traditional instruments.Good results have been achieved for synthesizing distant sources, whichreach the listener as plane waves. Distance perception can be simulated usingdistance filtering and reverberation balance. It is also possible in low-orderAmbisonic systems, (Gerzon 1992; Gerzon 1985), to approximately synthe-size a diffuse source at varying distance, (Menzies 1999; Menzies 2002), whichcan be useful in a creative setting. Soundfield synthesis of an object withnon-uniform directivity has been considered in the farfield using sphericalharmonic representation, (Menzies 1999; Menzies 2002). With the devel-opment of high-order acoustic field construction, the simulation of nearfieldHolophonic Sources                              OBFigure 1: Overall scheme. O denotes the extended source object, and B thelistener.sources becomes feasible. This has been developed for a monopole sourcein the context of high-order Ambisonics by reconstructing a monopole fieldabout the listener, (Daniel 2003). Using the wavefield approach, (Berkhout,Vries, and Vogel 1993; Daniel, Nicol, and Moreau 2003), the directional prop-erties of objects have been encoded with filters that feed the speaker arraydirectly, (Caulkins, Corteel, and Warusfel 2004; Warusfel 2004).A localized source typically differs in two respects from simple monopolesource. The sound radiates from a region of non-zero width, and the di-rectivity of radiation is not uniform. Near to the object the soundfield willbe reactive, like a monopole’s, but possibly have a much more complex ge-ometry. We should fully expect this added richness to be exploited by theauditory system for its information content, and so to have perceptual signifi-cance. Although this does not appear to have been studied in detail, informallistening provides strong evidence of spatial perceptual variety among com-plex objects. The study of directional objects using the wavefield approachalso supports the hypothesis. For both practical and creative applications itwould be desirable to find a way to accurately represent a complex sourceand encode it into Ambisonic B-format. The conversion from source encod-ing to Ambisonic encoding depends on the location and orientation desiredof the source. From a single source encoding, that source can be renderedanywhere and in any orientation around the listener. Figure 1 illustratesthis scheme. The advantage of Ambisonic modularity is apparent here, inthat we seek a process that encodes into a format that is independent ofthe details of the rendering mechanism, whether it be a particular speakerarray or headphones. The wavefield approach lacks this intermediate stage,as well as proving less accurate for given order in some studies (Daniel, Nicol,and Moreau 2003). Binaural rendering of high-order Ambisonics, over head-phones, including the nearfield, has been considered, (Menzies and Al-Akaidi2007b).The article is organized as follows. First the source representation isdiscussed, followed by the main part, the development of a method to trans-form a source encoding, with knowledge of its position and orientation, intoan Ambisonic encoding. Some simulations are provided for verification andillustration. Finally we consider how the approach can be adapted for theAmbisonic encoding of reverberation depending on source and listener posi-tions and orientations. Owing to lack of space, this article has been shortenedHolophonic Sourcesconsiderably. Interested readers are referred to a forthcoming article, (Men-zies and Al-Akaidi 2007a).SOURCE REPRESENTATIONWe wish to use a representation which can encode any source to any desiredaccuracy, relates well to direct measurements of the field, and can be manipu-lated efficiently. The following possibilities suggest themselves. A source canbe modelled with several monopoles. This would be appropriate if it actuallyhas this structure, or because a rough and fast model is required. The sourcecan be positioned and orientated using standard cartesian transformations.For more accuracy we can attempt to use many monopoles distributed overthe source volume or surfaces. It is far from obvious how this would be donefor a general source. Such a representation contains considerable redundancysince it describes the structure of the object as well as the sound produced.The exterior harmonic expansionMultipoles in their original form consist of infinitesimal arrangements ofmonopole sources. A multipole of sufficient order can represent a the fieldaround a given extended object arbitrarily well. Although they are operatedon by simple cartesian operations, their infinitesimal nature does not lenditself to direct numerical manipulation. Also the relationship of multipoleparameters to the directionality of the field rapidly increase in complexitywith order. Closely related is the exterior expansion for the wave equation.This has basis functions in the frequency domain using spherical coordinates,hm(kr)Ymn(θ, δ), where hm(kr) are the spherical hankel functions of the sec-ond kind, (Morse and Ingard 1968). m is the multipole order of each function,and k = 2pi/λ is the wavenumber. The type of hankel function chosen givesan outward moving wave when associated with a positive frequency timepiece eiωt, the same convention used in (Daniel 2003).An infinitesimally defined multipole of order m can always be expressedexactly using an exterior expansion with terms up to orderm. For this reasonan exterior expansion is alternatively called an exterior multipole expansionor just a multipole, (Gumerov and Duraiswami 2005). Another term usedis singular expansion, since the center of the expansion has a singularity.The exterior expansion relates closely to the non-uniform directivity of asource, as discussed below, and our principal goal shall be to manipulate itto provide an Ambisonic source encoding. By multipole we shall mean anexterior expansion, unless otherwise stated.For convenience we define coefficients, Omn(k), by a general exterior ex-pansion,p(r, k) = k∑mi−m−1hm(kr)∑nYmn(θ, δ)Omn(k) , (1)so that in the farfield where hm(kr) tends to im+1e−ikr/kr, the field becomesHolophonic Sourcespfar =e−ikrr∑m,nYmn(θ, δ)Omn(k) . (2)The Omn(k) coefficients then directly express the non-uniform directivity inthis regime, where locally the field tends to an outward moving plane wave.The signals Omn(k) coincide with the O-format encoding used previously forAmbisonic synthesis, (Menzies 1999; Menzies 2002). The same name will beused here for the more general case described by (1). This is just a convenientconvention, in the same sense as B-format is defined. Nothing essentially newis added.Omn(k) can be readily calculated from measurements of the field on asphere at any radius r outside the source region. Applying an integral overthe sphere,∫dΩ Ymn(θ, δ) to (1) givesOmn(k) =im+1∫dΩ Ymn(θ, δ)p(r, k)4pikhm(kr). (3)Source approximation order and errorWe consider now the order to which a source is approximated, mmax. Wewish to minimize this subject to reconstruction error constraints. A sourcecan be arbitrarily small and still have power up to any multipole order, forexample using the explicit definition of infinitesimal multipoles. Howeverthis is unusual in a real acoustic source because opposed component sourcesare not usually found very close together. Here we just have room to give ageneral result for typical extended sources. For object radius r, the maximumorder required in the expansion, mmax ≈ kr.AMBISONIC ENCODING OF MULTIPOLESFreefield expansionHigh-order Ambisonics is founded on the interior expansion that we shallalso call the freefield expansion here, to emphasize that it is used to describea sourceless region around the listener. Eq. (4) is the version of the expan-sion using N3D harmonics, Ymn(θ, δ), and defines the B-format coefficients,Bmn(k), (Daniel 2003). The expansion converges quickly on any source-freefield, up to a given radius r. The typical order required to achieve ≈ 1%error for a regular freefield, such as a planewave, is mmax ≈ kr, (Gumerovand Duraiswami 2005; Ward and Abhayapala 2001).p(r, k) =∑mimjm(kr)∑nYmn(θ, δ)Bmn(k) (4)A freefield expansion is by definition a sourceless field. This means thatit can only be extended in radius as far as the nearest source, which howeverdoes not prevent us recreating the freefield around a listener due to a nearbysource.Holophonic SourcesrOrBrOBFigure 2: Vector notationMultipole to freefield coefficient transformationThe main task in this section is to find Bmn(k) in the presence of a multipoledescribed by Omn(k) at a given position. It would be desirable to find ageneralized closed form expression, as for the monopole case in (Daniel 2003).However, it is not very apparent how this could be done or even if it wouldbe the most practical method of calculation, so instead a more pragmaticapproach is adopted yielding eventually a manageable integral expression.To begin (4) and (1) are equated. The notation is modified according toFigure 2,∑mimjm(krB)∑nYmn(θB, δB)Bmn(k) = k∑mi−m−1hm(krO)∑nYmn(θO, δO)Omn(k)(5)To isolate Bmn(k) the operator∫dΩB Ym′n′(θB, δB) is applied, with rBa freely chosen constant, and θO, δO and rO are functions of the vector rB,yielding4piim′jm′(krB)Bm′n′(k) = k∑mi−m−1∑nOmn(k)∫dΩB Ym′n′(θB, δB)Ymn(θO, δO)hm(krO).(6)After some manipulations, described in (Menzies and Al-Akaidi 2007a),the following expression for Bmn(k) is found,Bmn(k) =∑n′Rmnn′(θ, φ)∑m′1rMmn′m′(kr)∑n′′Rm′n′n′′(θ,−φ) Om′n′′(k) (7)where the new 3-index coefficient matrix is defined,Mmnm′(k) =ki−m−m′−12jm(kB)∫+1−1dsBPˆmn(sB)Pˆm′n(sO)hm′(kO), (8)and,Pˆmn(sin δ) =√(2m+ 1)(m− |n|)!(m+ |n|)! Pm|n|(sin δ) . (9)Rmnn′ denotes a harmonic rotation, and (θ, φ) describes the location of thesource. Pmn(x) is the associated Legendre polynomial. kB = αk, kO =Holophonic Sourcesk√1 + α2 − 2αsB, sO = r(αsB − 1)/rO and sB = sin δB. Clearly this isdefined only for n < m and n < m′, so for a given source the numberof filters increases only linearly with B-format order required. The filtercoefficients are given in terms of one parameter, k. The actual filter actingin (8) is scaled in frequency by the radius r and there is a distance factor1/r. Choosing α = m/k gives good numerical performance.Validation and propertiesTo provide an immediate confidence test that the derived formulas are cor-rect, a random test 5th order multipole was constructed, shown in Figure 3,and compared with the 13th order freefield expansion calculated using thematrix (7), shown in Figure 4. The error contours in Figure 4 at 10% and 1%levels are for deviations from the original multipole shown in Figure 3. Theregion of agreement extends as far as the center of the original multipole,as expected, and indicates that the calculations described in this section arecorrect. Further tests, show that (7) agrees with the monopole case previ-ously considered. The general picture for higher orders is that with e−ikr/rfactored out, the response is always minimum phase. For small k the orderof the filter becomes m + m′, while for large k it is n. The location of thetransitional region increases linearly with m+m′ from k ≈ 2 for m+m′ = 1.For higher orders, the transitional region can be more complex. The transferfunctions grow large as k tends to zero. It can be shown that the func-tions can be limited without significantly affecting accuracy in the regions ofinterest.With rotations included, the number of filters required to synthesize agiven source is only linear in the maximum B-format order,mmax, formmax >m′, owing to zeros in Mmn′m′(kr), although the complexity of the filtersthemselves increases with m. As we saw earlier, mmax ≈ rk, where r is theradius of the 3-dimensional region constructed accurately. This demonstratesthe inherent efficiency of the method, despite the precision and complexityof reconstruction.Reverberation encoding and transformationRelated results have also been found for the generation and transformationof reverberation, (Menzies and Al-Akaidi 2007a). This provides an efficientmeans for a listener to experience a region of high quality reverberant soundfield, while being freely able to move and orientate their heads in that region.CONCLUSIONA method has been presented for encoding a general acoustic source, andtranscoding it to a high-order Ambisonic signal, dependent on source orien-tation, and position relative to the listener. The method also lends itself tothe direct measurement of real sources using an array of surrounding micro-phones, as well as more indirect synthetic processes. As well as a recordingand mixing tool, it can be seen as a compositional tool, and in the contextHolophonic Sources−3 −2 −1 0 1 2 3210−1−2Oz axiskz/2pikx/2piFigure 3: Cross-section of a field plot for a 5th order multipole, center at O.The cross-section is θ = 0. x, z are cartesian coordinates in length units.−3 −2 −1 0 1 2 3210−1−2Bz axiskz/2pikx/2piFigure 4: Cross-section of a field plot for a 13th order freefield expansion,center at B, of a multipole, center O. Error contours are shown at the 1%and 10% levels. The cross-section is θ = 0. x, z are cartesian coordinates inlength units.of progressive electronic music, a performance instrument. The approach isconsiderably more elaborate and costly than plane wave or monopole synthe-sis, however it is expected that in the context of complex sources displayedwith a high-quality rendering system, the efforts are worthwhile. Binauralheadphone reproduction is particularly attractive, because the encoding onlyneeds to be of sufficient order for a single listener rather than a listening area,Holophonic Sourcesso reducing computational costs. For instance for radius 0.2 m, up to 1500Hz, the required order, m = rk ≈ 6. In a speaker rendering environment thevalid listening region is necessarily fixed to accommodate multiple listeners.This places constraints on how nearfield sources can be arranged relative tothe listener, so for example it is impossible for a listener to experience nearsources directly on the left and the right sides while also having a large lis-tening area that can hold multiple listeners. Binaural reproduction does notsuffer this constraint, and so is the more natural method for nearfield ren-dering. A possible exception would be a small speaker array designed for oneperson. In the future we hope to investigate realizations of these methods.REFERENCESBerkhout, A., D. Vries, and P. Vogel (1993). Acoustic control by wave fieldsynthesis. J. Acoust. Soc. Am. 93, 2764–2778.Caulkins, T., E. Corteel, and O. Warusfel (2004). Synthesizing realisticsound sources in wfs installations. In Proc. DAFX04.Daniel, J. (2003). Spatial sound encoding including near field effect. InProc. AES 23nd International Conference.Daniel, J., R. Nicol, and S. Moreau (2003). Further investigations of highorder ambisonics and wavefield synthesis for holophonic sound imaging.Gerzon, M. (1985). Ambisonics in multichannel broadcasting and video.Journal of the Audio Engineering Society 33, 859–871.Gerzon, M. (1992). General metatheory of auditory localisation. In 92ndAES Convention Preprints.Gumerov, N. A. and R. Duraiswami (2005). Fast multipole methods for theHelmholtz equation in three dimensions. Elsevier Science.Menzies, D. (1999). New Performance Instruments for Electroacoustic Mu-sic. Ph. D. thesis, University of York, UK.Menzies, D. (2002). W-panning and O-format, tools for object spatialisa-tion. In Proc. AES 22nd International Conference.Menzies, D. and M. Al-Akaidi (2007a). Ambisonic synthesis of complexsources. Journal of the Audio Engineering Society , To appear.Menzies, D. and M. Al-Akaidi (2007b, March). Nearfield binaural synthe-sis. J. Acoust. Soc. Am. 121(3):1559-63.Morse, P. and K. Ingard (1968). Theoretical Acoustics. McGraw-Hill.Ward, D. B. and T. D. Abhayapala (2001, September). Reproduction of aplane-wave sound field using an array of loudspeakers. IEEE Transac-tions on Speech and Audio Processing 9(6), 697.Warusfel, O. (2004). Reproduction of sound source directivity for futureaudio applications. In Proc. International Congress on Acoustics.

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