partial sound ﬁeld decomposition in multireference near-ﬁeld...

Partial sound field decomposition in multireference near-fieldacoustical holography by using optimally located virtualreferences

Yong-Joe Kima) and J. Stuart BoltonRay W. Herrick Laboratories, Purdue University, 140 South Intramural Drive, West Lafayette,Indiana 47907-2031

Hyu-Sang KwonAcoustics and Vibration Research Group, Korea Research Institute of Standards and Science, P.O. Box 102,Yusong, Taejon, 305-600, Korea

~Received 20 March 2003; revised 15 November 2003; accepted 24 November 2003!

It has been shown previously that the multiple reference and field signals recorded during a scanningacoustical holography measurement can be used to decompose the sound field radiated by acomposite sound source into mutually incoherent partial fields. To obtain physically meaningfulpartial fields, i.e., fields closely related to particular component sources, the reference microphonesshould be positioned as close as possible to the component physical sources that together comprisethe complete source. However, it is not always possible either to identify the optimal referencemicrophone locations prior to performing a holographic measurement, or to place referencemicrophones at those optimal locations, even if known, owing to physical constraints. Here,post-processing procedures are described that make it possible both to identify the optimal referencemicrophone locations and to place virtual references at those locationsafter performing aholographic measurement. The optimal reference microphone locations are defined to be those atwhich the MUSIC power is maximized in a three-dimensional space reconstructed by holographicprojection. The acoustic pressure signals at the locations thus identified can then be used as optimal‘‘virtual’’ reference signals. It is shown through an experiment and numerical simulation that theoptimal virtual reference signals can be successfully used to identify physically meaningful partialsound fields, particularly when used in conjunction with partial coherence decompositionprocedures. ©2004 Acoustical Society of America.@DOI: 10.1121/1.1642627#

PACS numbers: 43.60.Sx@EGW# Pages: 1641–1652

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I. INTRODUCTION

Near-field acoustical holography~NAH! is a useful toolfor visualizing noise sources and their associated sofields since it allows sound fields that are measured osurface to be reconstructed throughout a three-dimensispace.1,2 For example, after the sound pressure is measuon a surface of constant coordinate in plane, cylindrical,spherical coordinates, the pressure, particle velocity,sound intensity fields can be reconstructed on plane or ccentric surfaces, as appropriate, either closer to or farfrom the source.

For the NAH procedure to be successful in its simplform, the measured sound field must be spatially coherSuch a spatially coherent sound field can be obtainedcapturing the sound pressure on the entire measuremenface simultaneously. However, the latter measurementquires the use of many field microphones, perhaps hundor thousands, which may not always be practical. Whensound field generated by a single coherent source is staary, however, a relatively small number of scan microphocan be used in combination with a fixed-location referenmicrophone to sample data on portions of the hologram

a!Corresponding author. Phone:11-765-496-6008; fax:11-765-494-0787;electronic mail: [email protected]

J. Acoust. Soc. Am. 115 (4), April 2004 0001-4966/2004/115(4)/1

Downloaded 01 Dec 2012 to 165.91.74.118. Redistribution sub

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face sequentially. The reference spectral data can thencombined with the transfer functions linking the referencand the field points on the hologram surface to create a cplex pressure distribution on the latter surface that may tbe projected as desired.

A multireference procedure, referred to as the SpaTransformation of Sound Fields~STSF!, was introduced byHald to accommodate situations in which the sound fieldgenerated by a composite source comprising more thanmutually incoherent source.3 The latter procedure, describein terms of a reference cross-spectral matrix and a crospectral matrix linking the reference and field signals,based on using singular value decomposition~SVD! to sepa-rate the total sound field into a set of partial fields thatincoherent with each other. The partial fields can thenindependently projected and added together to yield the qdratic properties of the sound field on the reconstruction sfaces of interest. Note that in this procedure the charactetics of the partial fields depend on the reference microphlocations with respect to the component sources, and thuspartial field decomposition is not unique. As a result, tpartial fields determined in this way cannot usually be asciated with particular component sources.

Since the introduction of STSF, a number of investigtions have focused on separating the total sound field

1641641/12/$20.00 © 2004 Acoustical Society of America

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individual partial sound fields thatcan be associated withmeaningful physical sources. As an alternative to the Sprocedure, Hallman and Bolton4 suggested the application opartial coherence decomposition~PCD!, based on Gauselimination,5 to separate the partial fields. Kwon and Bolto6

compared the performance of the SVD and PCD methothey also introduced the use of the multiple signal classifition ~MUSIC! algorithm7 for the selection of the best refeences from among many candidates positioned arouncomposite sound source. It was shown that the referencecrophones should be positioned close to the individual phcal sources if physically meaningful partial sound fields wto be obtained. However, in practice, the locations ofmost prominent sources are sometimes not known prioperforming a holographic measurement. There are also cin which it is impossible to place references at the desilocations, even when those locations are known: forample, when a reference microphone placed close to a phcal source might itself induce flow noise or when the desilocations are otherwise inaccessible for practical reasons

Thus, to facilitate the decomposition of a sound fieinto physically meaningful components, it would be desable to be able to place ‘‘virtual’’ references at optimal loctions after performing a holographic measurement madeing a set of sufficient, but nonoptimally located reference

Nam and Kim introduced a virtual reference proceduin which virtual references are located at positions onsource surface where the amplitude of an acoustical propof interest, such as the pressure, particle velocity, or acintensity, is a maximum.8 The latter procedure was baseda vector representation9 of the sound field combined withknowledge of various point-to-point transfer functions, bomeasured and estimated by using holographic projectiNote that in their procedure it is required that the number‘‘real’’ and ‘‘virtual’’ references, first used to conduct thmeasurement and then to post-process the data, be theIn general, the number of virtual references should equalnumber of statistically independent sound sources. Howewhen a holographic measurement is performed, it maydesirable to use a number of real references larger thannumber of independent sound sources both to ensure thatotal sound field is captured completely and to allow treduction of noise effects in subsequent processing. Inlatter case, the virtual reference signals in excess of the nber of independent sources may not be properly determby using the procedure proposed by Nam and Kim, sithose virtual reference signals would be primarily relatedmeasurement noise. Through numerical simulations, Nand Kim demonstrated that a sound field can be decompinto physically meaningful partial fields when the virtual reerences are positioned appropriately on the source surand particularly when PCD is performed.

In the present article, the partial field decomposition pcedure based on the use of virtual references introduceNam and Kim is recast in a form that is based on the usesignal cross-spectral matrices and transfer functions withintent of developing a procedure that is compatible with cventional multireference NAH techniques.3 The procedurealso incorporates explicit relations between principal ref

1642 J. Acoust. Soc. Am., Vol. 115, No. 4, April 2004


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ence signals~i.e., noise-free singular values derived from treference cross-spectral matrix! and partial sound field signals on the hologram and reconstruction surfaces. Thusprocedure to be described here makes it possible to ularger number of real than virtual references, as would ually be the case in practice. In addition, a signal processalgorithm that can be used to calculate the MUSIC powanywhere within a three-dimensional sound field recostructed by NAH projection is introduced. An optimizatioprocedure combined with the latter algorithm can be usedhelp identify the optimal reference locations: i.e., the optimreference locations were identified by searching for thesitions in three-dimensional space at which the MUSpower was locally maximized. By using the proposed produre, it is possible to identify the optimal reference positioin terms of both the projection surface location and the lotion on the projection surface; in contrast, by simply mamizing the amplitude of a particular acoustical property, epressure, only the virtual reference location on any particuprojected surface can be identified. It is shown here, throboth experiment and numerical simulation, that the optimvirtual reference procedure can yield good estimates ofphysical partial sound fields. In particular, it is shown ththe use of a set of optimal virtual references positioned atmaximum MUSIC power locations results in a more accurestimation of the physically meaningful partial fields thwhen the decomposition is based on a set of virtual reences located on the source plane.

II. THEORY

A. Multireference NAH

When the total sound field generated by a composource comprising a finite number of uncorrelated physsources is completely sensed by a set of reference transers ~i.e., the number of references is equal to or larger ththe number of uncorrelated sources and each uncorrelsource is sensed by at least one reference transducer!, thefield signals on the hologram surface can be expressed infrequency domain as a linear combination of referencenals multiplied by appropriate acoustical transffunctions:6,10 i.e.,

F y1

y2

]

yN

G5F h11 h12 ¯ h1M

h21 h22 ¯ h2M

] ] � ]

hN1 hN2 ¯ hNM

GF r 1

r 2

]

r M

G , ~1a!

whereyi represents thei th field signal on the hologram surface (i 51,2,...,N, andN is the total number of measuremepoints on the hologram aperture!, r j denotes thej th referencesignal (j 51,2,...,M , M>1, and M is the total number ofreference microphones!, andhi j denotes the transfer functiobetween thej th reference andi th field signal. Equation~1a!can be expressed in compact vector-matrix form as6,10

y5Hyrr , ~1b!

wherey andr are theN by 1 field signal vector and theM by1 reference signal vector, respectively, andHyr is theN by M

Kim et al.: Virtual references in nearfield acoustical holography


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transfer function matrix that relates the reference and fisignals. When the holographic measurement is made usiscanning procedure in which data over portions of the hogram aperture are captured sequentially, it is conveniendescribe the sound field statistically. In the latter case, iconvenient to multiply Eq.~1b! by the Hermitian transposof the reference vector and then evaluate the expectatiothe result to give6,10

Syr5SryH 5E$yrH%5HyrE$rr H%5HyrSrr , ~2!

where E denotes the expectation operator, the superscridenotes the Hermitian transpose,Sry is the cross-spectral matrix between the real reference and field signals, andSrr isthe reference cross-spectral matrix. Here it is assumedthe number of incoherent sources isK<M : the number ofincoherent sources in any particular case can, in principledetermined by counting the number of significant singuvalues when the reference cross-spectral matrix is decposed by using SVD.11 In the latter case, the reference crosspectral matrix,Srr , can then be represented as the prodof a diagonal matrix whose elements are the singular valand a unitary matrix whose column vectors are the eigenvtors of Srr (Srr )

H. When the singular values associated wthe noise subspace are small enough to be ignored, theerence cross-spectral matrix can be approximated as cprising only the firstK singular values and eigenvectors: i.e

Srr 5VLVH, ~3!

where V is an M by K matrix whosemth column is thecolumn vector of the unitary matrix andL is the K by Kdiagonal matrix whosemth diagonal element is themth sin-gular value. By substituting Eq.~3! into Eq. ~2!, the transfermatrix relating the reference and field signals can bepressed as

Hyr5SryH Srr

215SryH VL21VH. ~4!

From Eq.~1b!, the cross-spectral matrix of the field signaon the hologram surface can be estimated by using theerence cross-spectral matrix in Eq.~3!: i.e.,

Syy5HyrSrr HyrH 5HyrVLVHHyr

H . ~5!

The cross-spectral matrix of field signals on the hologrsurface can then be decomposed to represent a set ofherent partial fields subject only to the condition thatSyy5YYH, where each column of the matrix,Y, represents apartial field vector: i.e.,

Y5HyrVL1/25HylL1/2, ~6!

whereL1/2 is the principal reference signal matrix andHyl isthe transfer matrix between the principal reference sigmatrix and the partial field matrix on the hologram surfa(Hyl5HyrV). The acoustical field on any reconstruction sface can then be expressed as

Y85Hy8yY5Hy8yHyrVL1/25Hy8yHylL1/25Hy8 lL

1/2,~7!

where Y8 is the partial field matrix on the reconstructiosurface andHy8 l represents the transfer matrix betweenprincipal reference signals and the partial field signals on

J. Acoust. Soc. Am., Vol. 115, No. 4, April 2004 K


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reconstruction surface. The matrixHy8y is composed of thetransfer functions that relate the field signals on the hologrand reconstruction surfaces, and it represents the NAHjection procedure, including the spatial Fourier transform~inthe planar case! and the propagation operation: the latter prcedure also incorporates the effects of discretization, wdowing, spatial filtering, zero-padding, etc.1–3 Note that, inpractice, the matrixHyr appearing in Eq.~7! should be cal-culated from the cross-spectral matrices,Syr ,(step) andSrr ,(step), measured during each individual scan, by usEqs. ~3! and ~4!, while V and L should be calculated fromthe reference cross-spectral matrix,Srr ,(avg), averaged overall the scans in order to suppress spatial noise causedsource nonstationarity~see Ref. 10 for details regardinsource nonstationarity compensation!. Finally note that eachcolumn of Hy8 l corresponds to the NAH projection of thcorresponding column ofHyl . The cross-spectral matrix othe field signals on a reconstruction surface can then beculated from Eq.~7! as

Sy8y85Y8Y8H5Hy8 lLHy8 lH . ~8!

B. Virtual reference procedure

Virtual references can, in principle, be placed anywhwithin the three-dimensional space covered by the NAH pjection. By making use of the partial field signal matricevaluated on the reconstruction surfaces, as in Eq.~7!, thevirtual reference signal matrix can be expressed as

X5F c1TY18

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18 l

c2THy

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]

ckTHy

k8 l

GL1/25HxlL1/2, ~9!

whereX is theK by K virtual reference signal matrix andHxlrepresents the transfer matrix between the principal referesignals and the virtual reference signals. In Eq.~9!, cm rep-resents theN by 1 reference selection vector: when themthvirtual reference is positioned at thei th field position on thereconstruction surface, all elements ofcm are zeros, excepfor the element at thei th row, which is itself unity. Note thatthe matrixYm8 in Eq. ~9! represents the partial field signamatrix on the reconstruction surface at which themth virtualreference is placed, and that the vector,cm, denotes themthvirtual reference location on themth reconstruction surfaceThus, the location of themth reconstruction surface in combination with the vector,cm , determines the location of thmth virtual reference in a three-dimensional space. Tcross-spectral matrices between the virtual reference signand between the virtual reference and the field signals onhologram surface, can then be obtained from Eqs.~6! and~9!: i.e.,

Sxx5XXH5HxlLHxlH , ~10!

and

Sxy5XYH5HxlLHylH , ~11!

1643im et al.: Virtual references in nearfield acoustical holography


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whereSxx is the virtual reference cross-spectral matrix aSxy is the cross-spectral matrix between the virtual refereand field signals on the hologram surface.

Recall that under ideal circumstances, the physicalerence microphones should be placed at the locationphysical, uncorrelated sources. As argued earlier, in pracit may not be possible to position real reference microphoat those points.

It has, however, been shown previously that the MUSalgorithm can be combined with the SVD procedure to guthe selection of the best ‘‘real’’ reference microphones froamong a large number of real microphones positionaround a noise source.6 Since the MUSIC algorithm can bused to find the locations of sources in a three-dimensiospace,7 the latter procedure has here been modified to idtify the optimal virtual reference locations: the optimal loctions are here identified as the points in a three-dimensiospace at which the MUSIC power, defined below, is mamized.

When the cross-spectral matrix of the field signals oreconstruction surface, i.e.,Sy8y8 in Eq. ~8!, is decomposedby using the SVD procedure (Sy8y85W(WH), the unitarymatrix, W, can be expressed in terms of the eigenvectorsSy8y8(Sy8y8)

H: i.e., W5@w1w2¯wN#, wherewn is the ntheigenvector associated with thenth singular value. Since thenumber of incoherent sources isK, the noise subspaceRnoise, can be defined in terms of the noise-related eigenvtors,wn (n5K11 to N!, as

Rnoise5 (n5K11

N

wnwnH . ~12!

The MUSIC power is then defined in terms ofRnoise as

PMUSIC51

uHRnoiseu, ~13!

whereu is the trial vector. Since the signal subspace spanby wn (n51 to K! is orthogonal to the noise subspace reresented byRnoise, the MUSIC power should be infinitewhenu5wn (n51 to K!.

Assume, for example, that a vector representing a soat thenth discrete point of the reconstruction surface canapproximated as the trial vector,

un5@0 ¯ 0 1 0 ¯ 0#T, ~14!

where thenth element ofun is unity and the otherN21elements are zeros. In that case, the MUSIC power assated with the trial vector,un , would be very large.

After calculating the MUSIC powers associated withpossible trial vectors,u5un (n51 to N!, a map of the MU-SIC power on a two-dimensional reconstruction surfacebe obtained by reshaping the MUSIC power vector intomatrix whose elements represent the MUSIC power apoint on a two-dimensional surface. A three-dimensioMUSIC power image can then be obtained by repeatinglatter operations on other projection surfaces in sequeSince the MUSIC power at source locations is large,optimal virtual reference locations are those where the MSIC power is maximized locally. The virtual reference s



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nals at those optimal locations can then be obtained from~9! since the reference locations that are identified by pforming the optimal search are expressed in terms ofreconstruction surface locations and the reference selecvectors that appear in the latter equation.

The trial vector defined in Eq.~14! closely represents thesound field generated by a monopole source. It is thus likthat the MUSIC power procedure as described here willcurately identify the locations of monopole-like sourceOther trial vectors may be more appropriate for identifyithe locations of more complicated sources, e.g., panel mobut here attention is limited to trial vectors of the type dfined in Eq.~14!.

Consider next two monopole sources that are spatiseparated but that are coherent: the MUSIC power criteas defined above would identify sources at both locatioHowever, in this case there is only one coherent soumechanism: i.e., the number of incoherent sources is smthan number of local MUSIC power maxima. In the lattcase, virtual references should first be placed at all loMUSIC power maxima. Based on the virtual reference crospectral matrix, Eq.~10!, the coherence between virtual reerencesi and j can be calculated: i.e.,

g i j2 5

uSi j u2

Sii •Si j, ~15!

whereSii andSj j are the autospectra of the virtual referenci and j, respectively, andSi j is the cross-spectrum betweethe virtual referencesi and j. If the coherence between thsignals at two candidate source locations is nearly unity,sources at those two locations are coherent. The virtualerence at the location at which the MUSIC power is smashould then be removed.

C. Partial sound field decomposition

Once the optimal virtual reference locations are idenfied, the virtual reference signals at those positions canobtained from Eq.~9!. In addition, the cross-spectral matrces between the virtual reference signals, and betweenvirtual reference and field signals on the hologram surfacan then be calculated from Eqs.~10! and ~11!. The cross-spectral matrix of the field signals on the hologram surfacan be expressed in terms of the latter cross-spectral mces: i.e.,

Syy5SxyH Sxx

21Sxy . ~16!

The cross-spectral matrix of field signals in Eq.~16! can bedecomposed into a partial field matrix,Yx , based on the‘‘virtual’’ reference signals subject only to the condition thSyy5Yx(Yx)

H: i.e.,

Yx5SxyH Sxx

21/2. ~17!

Since the decomposition in Eq.~17! is not unique~becausethe square root inverse of the reference cross-spectral mis not uniquely determined!, it is next desirable to selec‘‘good’’ partial fields from among all the possible ones.

The SVD and PCD procedures have been widely useperform the partial field decomposition expressed in E



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~17!.6 In the SVD method, the virtual reference crosspectral matrix is represented asSxx5USUH, whereU is theunitary matrix andS is the diagonal matrix of singular values. The partial fields can thus be represented as

Yx,SVD5SxyH US21/2. ~18!

The PCD procedure is based on the use of LU decomption, obtained from the Gauss elimination process,5 to sepa-rate the reference cross-spectral matrix into two matrici.e., Sxx5LDL H, where L is the lower triangular matrixwhose diagonal elements are unity andD is the diagonalmatrix. In the latter case the partial fields are written as

Yx,PCD5SxyH LH21

D21/2. ~19!

If the partial decomposition is based instead on the usereal reference signals, the cross-spectral matrices in Eqs.~18!and ~19! should be calculated using the real rather thanvirtual reference signals: i.e.,Srr andSry instead ofSxx andSxy . Note that when performing a partial coherence decoposition, the number of references should be equal tonumber of incoherent sources.

III. EXPERIMENT AND NUMERICAL SIMULATION

A planar acoustical holography experiment was pformed in an anechoic chamber to illustrate the optimal vtual reference procedure described above. Two loudspeahaving different frequency characteristics were driven bydependent white noise sources~see Fig. 1!.11 A horizontalline array of nine microphones spaced 0.051 m apartused to scan the hologram following the procedure indicain Fig. 1. The aperture thus comprised 32 subhologramsa total of 18 by 16 field points. The source surface~i.e., thez50 m plane! coincided with the front surfaces of the loudspeakers. The field measurements were made on a holosurface located atz50.05 m. During the scanning, five reerence microphones were fixed in front of the loudspeak



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as shown in Fig. 1. In each scan, the data record length512 points at a sampling rate of 4096 Hz and 20 linearerages were performed when estimating the various speduring the latter operations a 256 point overlap was useda Hanning window was applied to each record.

A numerical simulation involving three monopolsources was also performed to verify the performance ofoptimal virtual reference procedure described above. Inlatter case, monopole sources were located at each ofloudspeaker locations in thex-y plane as in Fig. 1, and a thirdone was located at (x,y)5(0.457 m,0.356 m): theirz loca-tions were chosen to be20.005 m,20.010 m, and20.015m, respectively: i.e., slightly behind the nominal sourplane atz50 m. References 1, 2 and 4 were placed atmonopole locations on the source plane~i.e., z50 m) whileRefs. 3 and 5 were placed at the locations of real Refs. 35, as shown in Fig. 1. The three monopole source strenwere represented by three independent random signals oproximately equal strength. The data obtained from the simlation was processed in the same way as the experimedata except that the number of virtual references was choto be three: the latter number could be determined by coing the significant singular values of the reference crospectral matrix.11

A source nonstationarity compensation was performbefore applying the NAH projection procedure to the hogram data in both the experimental and simulation case11

During the latter procedures, a spatial Tukey window with75% flat-top was applied to the hologram data to redutruncation effects and the hologram aperture size wastended to 36 by 32 by zero padding. No spatial filtering wapplied during the projections that were performed as parthe search for the optimal reference locations.

IV. RESULTS

The singular values of the measured reference crospectral matrix averaged over all scans are shown in Fig.



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FIG. 2. Spectra of the singular values of the reference cross-spectral m~experiment!.

FIG. 3. Music power on the source plane (z50 m) ~experiment!: ~a! from 0to 1600 Hz~maxima atn598 and 170!, and~b! mapped onto thex-y planeat 1200 Hz.



re-

The MUSIC power on the source plane~i.e., z50 m) isshown in Fig. 3~a! from 0 to 1600 Hz: a measurement poion thex-y plane is represented here by the vector index,n, onthex axis. In Fig. 3~b!, the MUSIC power vector at 1200 His plotted on thez50 plane: the vector was reshaped intomatrix whosei-, j th element represents thei-, j th measure-ment point on thex-y plane. The two loudspeaker locationcan easily be identified from the positions of the locmaxima of the MUSIC power: those locations correspondvector indices,n598 and n5170, the positions of loud-speakers 1 and 2, respectively. It can also be seen in Fig.~a!that there were distinct maxima atn598 and 170 at all fre-quencies except the very lowest.

The MUSIC power at 1200 Hz was then calculatedprojection planes fromz520.1 to 0.1 m, as shown in Fig4~a!. The MUSIC powers atn598 andn5170, i.e., at thetwo loudspeaker locations as identified from the resultsFig. 3, are plotted as a function ofz in Fig. 4~b!. Note thatthe two plots in Fig. 4~b! are cross-sectional views of Fig4~a! whenn598 andn5170. Local maxima of the MUSIC

rix

FIG. 4. Music power at 1200 Hz~experiment!: ~a! as a function ofn51 to288 andz520.1 to 0.1 m, and~b! plotted fromz520.1 to 0.1 m whenn5170 @MAX( PMUSIC)51.2976 at z520.0212 m] and whenn5170@MAX( PMUSIC)51.1351 atz520.0054 m].



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FIG. 5. Amplitudes of decomposed partial pressure fields on the hologram surface at 1200 Hz when the measurement was made using Re~experiment!: ~a! first field ~SVD!; ~b! second field~SVD!; ~c! first field ~PCD!; and ~d! second field~PCD!.

FIG. 6. Amplitudes of decomposed partial pressure fields on the hologram surface at 1200 Hz obtained using optimal virtual references when the mmentwas made by using Refs. 3 and 5~experiment!: ~a! first field ~SVD!; ~b! second field~SVD!; ~c! first field ~PCD!; and ~d! second field~PCD!.

1647J. Acoust. Soc. Am., Vol. 115, No. 4, April 2004 Kim et al.: Virtual references in nearfield acoustical holography

Downloaded 01 Dec 2012 to 165.91.74.118. Redistribution subject to ASA license or copyright; see http://asadl.org/terms

rement was

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FIG. 7. Amplitudes of decomposed partial pressure fields on the hologram surface at 1200 Hz obtained using the real references when the measumade by using Refs. 1 and 2~experiment!: ~a! first field ~SVD!; ~b! second field~SVD!; ~c! first field ~PCD!; and ~d! second field~PCD!.

FIG. 8. Amplitudes of decomposed partial pressure fields on the hologram surface at 1200 Hz obtained using optimal virtual references when the mmentwas made by using all references~experiment!: ~a! first field ~SVD!; ~b! second field~SVD!; ~c! first field ~PCD!; and ~d! second field~PCD!.

1648 J. Acoust. Soc. Am., Vol. 115, No. 4, April 2004 Kim et al.: Virtual references in nearfield acoustical holography


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power were found atz520.021 m for n598 and z520.005 m forn5170. These locations lie slightly behind thnominal source plane atz50 m, and these perhaps coincidwith the acoustical centers of the loudspeakers, which wcone shaped.

Since data on thex-y plane are available only at thdiscrete measurement points, an interpolation should, in pciple, be performed to obtain continuous data on the recstruction plane. To avoid the complexities introduced byinterpolation function, however, it is assumed here thavirtual reference may only be placed at a discrete measment point in thex-y plane. In the latter case the virtuareference locations on thex-y plane did not change withsmall variations of thez location @e.g., see Fig. 4~a!#, sincethex-y resolution, as determined by the distance betweendiscrete measurement points, was relatively coarse compwith the resolution in thez direction. Thus, the virtual references were first located on thex-y plane at the positions othe maxima of the MUSIC power on thez50 surface: e.g.,see Fig. 3~b!. The optimal reference locations were subsquently identified as those points at which the MUSIC powwas maximized as a function ofz at the corresponding location in thex-y plane: e.g., see Fig. 4~b!.

FIG. 9. Amplitudes of partial sound pressure fields atx50.356 m~obtainedfrom Figs. 7 and 8! as a function ofy: ~a! first partial pressure field, and~b!second partial pressure field.



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The measured data at 1200 Hz were used for the purpof comparing the characteristics of the partial fields decoposed by using either the SVD or PCD procedures combiwith either the set of real or optimal virtual references: tresulting partial pressure fields on the hologram surface~i.e.,at z50.05 m) are shown in Figs. 5–8. The partial pressfields decomposed by using the real and optimal virtual rerences are shown in Figs. 5 and 6, respectively, whenmeasurement was made by using real Refs. 3 and 5. Figushows the decomposed partial fields obtained by usingreal references when the measurement was made by uRefs. 1 and 2. The decomposed partial fields obtainedusing the optimal virtual references are shown in Fig. 8 whthe measurement was made by using all of the real reences.

When real Refs. 3 and 5 were used as the basis forpartial field decomposition, the decomposed partial fieldsnot represent the physical partial fields associated withloudspeakers, as may be seen in Fig. 5: i.e., both pafields contain contributions from both loudspeakers. Ho

FIG. 10. Music power at 1200 Hz~simulation!: ~a! as a function ofn51 to288 andz520.1 to 0.1 m, and~b! plotted fromz520.1 to 0.1 m whenn598 @MAX( PMUSIC)52.4484 at z520.0268 m], when n5170@MAX( PMUSIC)52.0323 at z520.0311 m], and when n5136@MAX( PMUSIC)51.7636 atz520.0353 m].



mal virtual
FIG. 11. Decomposed partial pressure fields on the hologram surface at 1200 Hz obtained by using the PCD procedure in combination with optireferences~simulation!: ~a! first field; ~b! second field;~c! third field; and~d! total field ~i.e., summed mean square pressures!.
FIG. 12. ‘‘Exact’’ partial pressure fields on the hologram surface at 1200 Hz~simulation!: ~a! first field; ~b! second field;~c! third field; and~d! total field ~i.e.,summed mean square pressures!.

1650 J. Acoust. Soc. Am., Vol. 115, No. 4, April 2004 Kim et al.: Virtual references in nearfield acoustical holography


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ever, the partial fields decomposed on the basis of a set eof ‘‘good’’ real references positioned close to the loudspeers~i.e., Refs. 1 and 2! or the optimal virtual references, arclosely associated with the physical sources. Note, howethat even in these cases there was a small amount of ‘‘leage’’ from one partial field to another: i.e., a faint trace of tsecond source is visible in the first partial field@see parts~a!and ~c! of Figs. 6–8#. By comparing Figs. 5~a!, 5~c!, 7~a!,and 7~c! for the first partial field or Figs. 5~b!, 5~d!, 7~b!, and7~d! for the second partial field, it can also be seen thatcharacteristics of the decomposed sound fields dependboth the reference locations and the decomposiprocedure.6

In order to compare the results of Figs. 7–8 quanttively, the amplitudes of the individual partial sound pressfields atx50.356 m are plotted as a function of they posi-tion in Fig. 9. In the first partial field, it can be seen that theis a contribution from the second loudspeaker aty50.457 m~i.e., the location of loudspeaker 2!: the leakagewas smaller when the sound fields were decomposed bying the PCD procedure rather than the SVD procedure,

FIG. 13. Error between the exact partial pressure fields and the papressure fields decomposed by using virtual references~simulation!: ~a!when the virtual references were placed on the source plane~PCD!, at thesource locations~PCD!, and at the maximum MUSIC power location~PCD!, and~b! when both PCD and SVD procedures were applied in cobination with virtual references placed at the maximum MUSIC powercations.



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when using the optimal virtual references rather than thereferences@see Fig. 9~a!#. That is, the smallest leakage ocurred when the PCD procedure was used in combinawith the optimal virtual references. Note that the pressamplitudes aty50.457 m in the second partial field appein the reverse order of the leakage amplitude in the fipartial field @see Fig. 9~b!#. That is, the sound pressure radated from the upper loudspeaker leaks into the first parfield which is primarily associated with the lower of the twloudspeakers, and is thus lost from the second partial fie

The results of the numerical simulation made using thmonopole sources are shown in Figs. 10–13. Figureshows the MUSIC power at 1200 Hz as the location ofprojection plane is varied fromz520.1 to 0.1 m. It can beseen in Fig. 10~a! that there are peaks of the MUSIC powat n598,n5136, andn5170: i.e., at thex-y locations of thethree sources. The MUSIC powers at those three locatare replotted in Fig. 10~b!: the z locations of the localmaxima were found to occur atz520.0268 m forn598,z520.0311 m forn5170, andz520.0353 m forn5136.Note that the locations identified according to the MUSpower criterion differ slightly from the real source locationthis discrepancy may result from small errors in estimatthe transfer matrix,Hy8y , representing the NAH projectionprocedure, owing to discretization, truncation, spatial wdowing, etc.

The partial pressure fields decomposed by usingPCD procedure in combination with the optimal virtual reerences~located according to the results of Fig. 10! areshown in Fig. 11. The corresponding ‘‘exact’’ partial pressufields obtained by turning on only one source at a timeshown in Fig. 12. It can be seen that each of the decompopartial pressure fields closely resembles the exact papressure fields.

To compare the decomposed partial pressure fieldsthe exact partial pressure fields on a quantitative basis,error, Ei j , between thei th decomposed partial field,pi ( i51 to 3! and thej th exact partial field,Pj ( j 51 to 3! wasdefined as

Ei j ~ f !51

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16 Uupi~ f ,xn ,ym!u22uPj~ f ,xn ,ym!u2

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wheref is frequency,xn is the location of thenth measure-ment point along thex axis, andym is the location of themthmeasurement point along they axis. The total error at thefrequency,f, is then defined as the sum of the three smallEi j ( f ) among all possible combinations ofi and j. In Fig.13~a!, the total errors in the cases when the set of virtreferences was placed first on the source plane, then amonopole locations, and finally at the maximum MUSpower locations are shown: in all cases the PCD procedwas used. The calculation based on the set of virtual reences placed at the maximum MUSIC power locatioyielded the least error: the error in the latter case is esmaller than when the virtual references were placed atknown source locations. In Fig. 13~b!, the partial fields de-composed by using the PCD procedure are comparedthose obtained by using the SVD procedure. At all frequ

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cies, the error associated with the PCD proceduresmaller than that resulting from the SVD procedure. Sinthe source levels were chosen here to be nearly the samesignificant singular values of the optimal virtual referencross-spectral matrix were close to each other, which mmake it difficult for the SVD procedure to reproduce tphysical partial fields: the latter effect does not affectPCD procedure.6

V. CONCLUSIONS

In this article, a post-processing procedure has beenscribed that makes it possible to identify virtual referensignals that can be used to identify physically meaningpartial fields after performing a holographic measuremebased on a nonoptimal, but sufficient, reference set. Thetimal virtual references were placed at the positions whthe MUSIC power was locally maximized. It was showthrough both experiment and numerical simulation thatoptimal virtual reference procedure results in partial fiethat are very similar to the ‘‘real’’ partial fields associatewith individual physical sources regardless of the locatioof the real references; however, the partial fields decompoby using the real reference signals directly were found tophysically meaningful only when the references were locavery close to the actual sources. It was also found that usthe set of virtual references located at the maximum MUSpower locations resulted in more accurate estimates ofphysically meaningful partial fields than were obtained usany other set of virtual references~e.g., on the source planor at the actual source locations! or even a set of ‘‘good’’ realreferences. It was also shown that the partial fields obtai



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using the PCD procedure suffered less leakage than thobtained using the SVD procedure. Thus, it was found tthe most meaningful partial fields were obtained whenPCD procedure was used in combination with optimallycated virtual references.

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