2010 Sound field extrapolation: Inverse problems, virtual microphone arrays and spatial filters

Sound field extrapolation: Inverse problems,virtual microphone arrays and spatial filtersPhilippe-Aubert Gauthier1, Cedric Camier1, Yann Pasco1, Eric Chambatte1, and Alain Berry1

1GAUS, Universite de Sherbrooke, Sherbrooke, J1K 2R1, Canada and CIRMMT, McGill University, Montreal, H3A 1E3, Canada

Correspondence should be addressed to Philippe-Aubert Gauthier (philippe aubert [email protected])

ABSTRACT

Sound field extrapolation is useful for measurement, description and characterization of sound environmentsand sound fields that must be reproduced using spatial sound system such as Wave Field Synthesis, Am-bisonics, etc. In this paper, two methods are compared: inverse problems and virtual microphone arrayswith filtering in the cylindrical harmonics domain. The goal was to define and identify methods that couldaccommodate to various non-uniform sensor arrays (i.e. non array-specific methods) and that are less sensi-tive to measurement noise. According to the results presented in this paper, it seems that the method basedon inverse problem with Tikhonov regularization is less sensitive to measurement noise.

1. INTRODUCTION

In recent years, a strong and wider interest for acoustictransducer arrays and multichannel signal processing isobserved. Applications range from: acoustic imaging,sound source localization and separation using nearfieldacoustical holography [1], beamforming [2], subspacemethods [2], time-reversal and other algorithms [2] to:spatial sound reproduction using Wave Field Synthesis(WFS) [3, 4], Ambisonics [5], multichannel Surroundsound [6], etc. These applications motivate a plethora ofresearch works both for enhanced spatial sound record-ing and spatial sound reproduction. This paper deals withspatial sound recording for subsequent spatial sound re-production. More precisely, it belongs to a larger projectof sound environment reproduction in aircraft cabins andcockpits.

1.1. Microphone array and acoustic imagingMost of the microphone arrays applications developedfor acoustic imaging aim at the experimental visualiza-tion and characterization of noise sources. Redesign orcontrol can then be undertaken to reduce the noise ra-diated by the given object or machine. For those ap-plications, a microphone array is used to measure (on adiscrete surface) the sound field radiated by the source.Post-processing then involves: a) sound field extrapola-tion outside the measurement grid up to the sound source[1] or b) direction-dependent mapping of the incoming

sound level on the array [2]. This type of applications iscommon for noise abatement purposes.

Applications such as sound source localization, identifi-cation and separation are based on algorithms which caneither state: a) angular direction from which the targetsound comes from (parameter estimation [2]) or b) out-put a signal which is caused by a single target source ina noisy or reverberating environment (source separationor waveform estimation [2]). This second type of appli-cations is mostly sought after for human-machine inter-faces and communication purposes. Large microphonearray technologies and research are also gaining atten-tion for audio applications [7].1.2. Multichannel sound reproductionOn the loudspeaker counterpart of array processing, partof the applications are related to sound field reproduc-tion. In this paper, we are specifically concerned by mi-crophone array measurements for spatial sound field re-production. The sound field reproduction applicationscan be further differentiated in terms of their targets. Themost straightforward being a sound pressure field: thetarget is then sound pressure as function of spatial coor-dinates. Other methods may involve spatial targets suchas sound intensity, sound diffuseness [8], sound contrastmaximization, sound power minimization [9], randompressure fields on planar surfaces [10] or even psychoa-coustics metrics. For all these sound field reproductionmethods, the target is either measured in situ or synthe-

AES 40TH INTERNATIONAL CONFERENCE, Tokyo, Japan, October 8101

Gauthier et al. Sound field extrapolation

sized from a theoretical or virtual definition.1.3. Sound field measurement and descriptionFor spatial sound and sound field reproduction usingloudspeaker arrays, there is therefore a need for on-sitesound field measurement, characterization and descrip-tion. These spatial measurements and characterizationsof the sound environment are subsequently used as tar-gets at the reproduction stage. In this paper, we searchfor a sound field extrapolation method that allows laterpost-processing and sound field characterization for sub-sequent sound field reproduction using loudspeaker orvibration source arrays in closed space (i.e. listeningrooms or mock-ups). The main objective of the presentedmethods is to achieve the largest extrapolation region sothat any relevant spatial parameters (sound pressure field,spatial coherence, intensity maps, direct and diffuse en-ergy densities as function of spatial coordinates, etc.) canbe computed and characterized over a large region froma single measurement array. Another requirement wasthe resulting extrapolation description should be generalenough: plane wave spectrum, helical wave spectrum,etc [1]. Furthermore, we were interested by the defini-tion of a method that could accommodate various mi-crophone array configurations, regular or not, and thatcan address measurement noise issues. Two methods arecompared: a) regularized inverse problems and b) cylin-drical harmonics filtering at virtual sensor array. The sec-ond method is a new method (different from the mostcommon inverse problem regularization methods [11])which is compared to the first to evaluate its potential.For illustration purpose, numerical results are shown fora specific and simple array. However, the reader shouldnote that the method can be applied to any array geome-try.1.4. Paper outlineAfter the introductory Sec. 1, Sec. 2 presents the classicalinverse problem formulation adapted to sound field mea-surements and extrapolation from a region centered onthe recording area. Typical sound field spectra and cylin-drical harmonics spectra are analyzed in Sec. 3 wherethe nature of the second method is evoked. In Sec. 4, it isshown how the inverse problem formulation can functionas a spatial resampling algorithm on the basis of virtualmicrophone arrays. In Sec. 4, classical spatial transformmethods are finally applied to virtual sensor arrays. Sec-tion 5 presents the comparison between the two meth-ods. Concluding remarks, future research avenues andupcoming practical developments are presented in Sec. 6.

2. INVERSE PROBLEM

In this section, the inverse problem method is definedand illustrated for a microphone array. With this ap-proach, one is looking for an optimal source distributionthat best fits the measured sound field at the microphonearray [11]. This optimal source distribution is then usedto extrapolate the measured sound field outside the areaof the array. The inverse problem, which differs from ex-trapolation methods based on spatial transform (such asfor classical Nearfield Acoustic Holography (NAH) [1],or in-plane sound field extrapolation [7]), is especiallywell-suited to irregular microphone arrays or particularconfigurations. Although inverse problems are used fornon-conformal nearfield holography, these applicationsinvolve sound extrapolation in a direction perpendicularto the array. In our case, we are concerned by in-planeextrapolation.

In this paper, an URA (Uniform Rectangular Array) con-figuration was selected for illustration purposes since it isthe configuration which is the easiest to build. The selec-tion of this configuration was also influenced by the factthat a similar array is currently built in our laboratory forpractical applications. Moreover, as the URA covers anentire spatial region, such type of two-dimensional arraymight more easily cope with a real 3D sound field or withthe proximity of a sound source or a diffracting object ina confined space such as a vehicle cabin.

2.1. Inverse problem formulation

A typical geometrical model of URA is depicted inFig. 1. The symbol and geometrical convention holds forany microphone array, either 2D or 3D. Any field point isdescribed by a position vector x= [x1,x2,x3] in rectangu-lar coordinates. Using cylindrical coordinates, this pointis described by [ ,r,x3]. A real microphone is describedby a position vector x(m)m . The complete real microphoneset is described by a vector set x(m)1 ,x

(m)2 , ,x(m)M for an

array made of M microphones. Using this URA, a com-plex sound pressure field is measured for an angular fre-quency [rad/s] and stored in a M-component vectorp( ,x(m)m ) [Pa]. In case of real measurements, the com-plex pressure field is obtained by a Fourier transform ofthe measured microphone signals, assuming stationarysound fields or short-duration frames. The smallest dis-tance between two microphones is denoted x(m). The

AES 40TH INTERNATIONAL CONFERENCE, Tokyo, Japan, October 810Page 2 of 10


x1

x2x3

b b b b bb

b b b b bb

b b b b bb

b b b b bb

b b b b bb

x(m)m

x

Impinging wave

r

x3

Fig. 1: A microphone array. Microphones are marked bysmall filled circles. x is any field point location, x(m)m isthe m-th microphone location. The region covered by theURA is shown by the rectangle.

following principles hold up to the spatial aliasing fre-quency falias = c/2x(m) [Hz], where c = 343 m/s is thesound velocity.

To extrapolate the measured sound field p( ,x(m)m ) to anyfield point x outside or inside the URA region, an inverseproblem is posed. Lets first assume a set of L planewaves: uniformly covering a 4pi steradians solid anglerange for the 3D case and uniformly covering a 2pi ra-dians angle range for the 2D (in-plane) case. Note thatselection of the plane waves propagation directions canbe different or adapted to the problem at hand, dependingon a priori information. The pressure produced by thisset of L plane waves at the sensor array is given by

p( ,x(m)m ) = Z( ,x(m)m , (l)l , (l)l )S( , (l)l , (l)l ) (1)where L-component vector S is the plane waves complexamplitudes, (l)l (from the set

(l)1 ,

(l)2 , , (l)L ) the l-th

plane wave azimuth and (l)l the l-th plane wave eleva-tion (zero for the 2D case). The matrix Z is the transfermatrix (dimension: ML) from plane wave amplitudesto resulting pressures at the microphones. The transfermatrix is defined by its ml-th element

Zml( ,x(m)m , (l)l , (l)l ) = e jk

(l)l x

(m)m (2)

where k(l)l = k cos((l)l )cos( (l)l )e1 +

k sin( (l)l )cos( (l)l )e2 + k sin(l)e3, ei being a unitvector along xi. k [rad/m] is the wave number and k(l)l isthe wave-number vector (k(l)l = kn

(l)l , with n

(l)l being the

unit vector pointing in the l-th propagation direction).The selected time convention is e jt , with time t [sec],and imaginary number j =1.The inverse problem is posed so that p( ,x(m)m ) matchesp( ,x(m)m ). From Eq. (1)

p( ,x(m)m ) = ZS( , (l)l , (l)l ) (3)and the solution is

S( , (l)l , (l)l ) = Z+p( ,x(m)m ) (4)where superscript + denotes pseudo-inversion [12].Pseudo-inversion is used since the matrix Z is generallynot a square matrix. The solution S( , (l)l , (l)l ) thengives the plane waves amplitudes which will reproducethe measured sound field at the array. The sound fieldcan then be extrapolated for any point x outside or insidethe region covered by the URA using

p( ,x) =L

l=1

Sl( , (l)l , (l)l )e jk(l)l x (5)

with Sl obtained from Eq. (4). This is the basis of soundfield extrapolation using inverse problem. Stated simply,one should understand that the previous equations and in-verse problem aim at computing a set of L virtual planarsources that optimally (in the least-mean-square sense)recreate p, an approximation of the measured sound fieldp. A great advantage of inverse problem over spatialtransform method [7] is that the inverse problem ap-proach can be readily applied to any non-uniform or uni-form microphone array. Indeed, no particular assump-tions are made about x(m)m to obtain Eqs. (4) and (5).Moreover, although classical NAH applications usinguniform rectangular array take great advantage of 2Dspatial transform, it is solely for forward and backwardsound field extrapolation along the array perpendicularaxis (x3 in Fig. 1). It is important to note that, althougha set of plane waves was used in the inverse problemformulation, the operation represented by Eq. (4) doesnot correspond to a 2D or 3D Fourier transform of thesound field as used in NAH [1] or sound field descrip-tion as reported by Poletti [13]. In our case, we aremostly interested by in-plane (x1,x2) sound field extrap-olation for subsequent in-plane sound field reproduction.This is a simple situation which prevents the use of the2D Fourier transform as this spatial transform assumes a



Fig. 2: (a) and (b): Real and imaginary parts of the impinging sound field. Microphones are shown as small filledcircles. (c) and (d): Real and imaginary parts of the extrapolated sound field by inverse problem. The isocontour linesof the resulting local quadratic extrapolation error are shown for 0.1% (white line) and 5% (black line).

repetitive spatial signal along the in-plane axes (x1,x2).This is not the case for a real sound field impingingon an array. Finally, the selection of a plane wave setas a solution basis for the inverse problem is not arbi-trary. Indeed, other general orthogonal solutions of theHelmholtz equation such as cylindrical or spherical har-monics are not bounded at the origin of the coordinatessystem which corresponds to the array center [1, 7] . Inthat case, the inverse problem based on these other or-thogonal solutions of the wave equation is extremely ill-conditioned and p may even not approach p. This hasbeen verified by the authors.

An example of sound field extrapolation using the in-verse problem approach is shown in Fig. 2 for an URAof 64 microphones covering a square with a side lengthof 0.77 m. The microphone separation distance is 11 cm.Theoretically, this URA could then measure a sound fieldwithout spatial aliasing up to 1559 Hz. The harmonicsound field measured at 500 Hz is shown in Fig. 2(a)and (b). The sound field impinging on the URA is aplane wave with a propagation angle of 45 deg. Thecorresponding extrapolated sound field obtained by theinverse problem solution is shown in Fig. 2(c) and (d).The inverse problem included a set of L = 128 planewaves covering 2pi radians. The inverse problem methodclearly performs sound field extrapolation in the mea-surement plane. The effective extrapolation region cov-ers (defined by a local quadratic error less than 0.1%)a circular region of roughly 2 m radius which is largerthan the URA. It should be kept in mind that the size ofeffective extrapolation region varies with frequency.

2.2. Inverse problem condition number

Although the example shown in Fig. 2 seems promisingfor sound field extrapolation, the condition number ofZ is very large: 3.5 1016 The condition number is theratio of the largest and smallest (greater than zero) singu-lar values: = max/min where the singular values areobtained by singular value decomposition [12].As long as measurement noise is absent, the pseudo-inverse solution (Eq. (4)) to the inverse problem gives aconsistent result. However, any noise or input variabilityhas the potential to make the solution S excessively large.Indeed, any variation p on the measured sound field (in-cluding noise) will imply a variation S on the solutionS that is bounded by the condition number [11, 14]

||S||/||S|| (Z)||p||/||p|| (6)

where ||...|| denotes the Euclidean norm. This high-lights the amplification potential of the condition num-ber. However, even if S becomes very large, the cor-responding extrapolated sound field may be able to ap-proach the measured sound field at the sensor array(p( ,x(m)m ) p( ,x(m)m )). However this solution mightnot provide a convenient sound field extrapolation out-side the array region. An example is shown in Fig. 3 fora case similar to the one shown above with an additional0.1% of random and spatially incoherent noise. Fig. 3 il-lustrates that the extrapolated sound field is only valid inthe array region. At first glance, this suggests that inverseproblem method for sound field extrapolation might not



Fig. 3: Real and imaginary parts of the extrapolatedsound field by inverse problem with 0.1% of noise (60dB SNR). Sound pressure is clipped at values larger than100.

be appropriate, except, possibly, as a spatial resamplingalgorithm for the region covered by the array.Note that for the two cases reported in Secs. 2.1 and 2.2,the numerical pseudo-inversion only involves the dis-carding of singular values smaller than 9 1012. Thistype of regularization (truncated singular value decom-position) [15] simply prevents instability caused by nu-merical machine limited precision.2.3. Inverse problem regularizationTo circumvent this noise sensitivity problem, regulariza-tion is introduced. We first recast the problem as an errorminimization task using a quadratic cost function

J = eHe+ SHS (7)where e is the error column vector defined by e = p p.In Eq. (7), is the regularization parameter. Combin-ing this error vector definition with Eq. (3), it is straight-forward to show that the cost function is a Hermitianquadratic function of S. The optimal S(opt) that will min-imize this cost function is [16, 17]

S(opt)L1 =[

ZH

ZHZ+ I]

LMpM1 (8)

where I is the LL identity matrix. The purpose of theregularization parameter is simple: it populates the maindiagonal of the matrix to be inverted. This filters any toosmall singular values which have the potential to amplifynoise. This is Tikhonov regularization [15].The choice of the regularization parameter is not trivial.Methods such as ordinary cross-validation and general-ized cross-validation are defined for this unique purpose

in acoustics [18]. Methods for choosing the regulariza-tion parameter are, by themselves, a research topic. Onthe one hands, while the selection of a small penaliza-tion parameter may stabilize the inversion (p may ap-proach p very closely in the microphone array region),it is again not guaranteed that optimal solution will giveany significant results outside the sensor array. It mightagain diverge at large distance from the array. On theother hands, selecting a large penalization parameter maystrongly stabilize the inversion but also reduce the spatialresolution of the extrapolated sound field (including thearray region). Examples of sound field extrapolation us-ing inverse problem method with varying regularizationparameter are shown in Fig. 8(a) to (f). Sec. 5 explainsthese results.

One way to select the optimal regularization parameteris based on the evaluation of the L-curve [15] which isa representation of the quadratic error eHe as functionof the solution quadratic amplitude S(opt)HS(opt) for awide range of . However, this selection implies a se-rious amount of computation time as the problem mustbe solved for an entire distribution of . It would be in-teresting to benefit from a more physical approach thatwould simplify the regularization process. In the follow-ing section, the typical cylindrical harmonics spectra ofextrapolated sound field with and without noise will pro-vide some hints on how could that be done. Moreover,this original method will be illustrated.

3. CYLINDRICAL HARMONICS SPECTRA

To analyze typical sound field spectra, cylindrical coor-dinates [ ,r,x3] are used. The definition of cylindricalharmonics (orthogonal functions of the Helmholtz equa-tion expressed in cylindrical coordinates) are recalledfor the 2D case. The sound pressure field of diverging(P(1)k ) and converging (P

(2)k ) cylindrical harmonics (CH)

are given by multiplication of Hankel and complex expo-nential functions

P(1)k (r, ,) = H(1)k (kr)e

jk , (9)

P(2)k (r, ,) = H(2)k (kr)e

jk . (10)

Any two-dimensional sound field can be represented by



Fig. 4: Magnitude of the pressure and velocity angularspectra for a circular sensor array centered at the originwith radius R = 0.3 m on which impinges a broadbandplane wave with the propagation direction = 45 deg.

a linear combination of CH [1, 7]:

p(r, ,) =

k=

M(1)(k ,)H(1)k (kr)e

jk +

k=

M(2)(k ,)H(2)k (kr)e

jk . (11)

The terms M(1)(k ,) and M(2)(k ,) are the complexCH coefficients, creating a CH spectrum.The M(1) and M(2) are directly computed from soundpressure and normal sound pressure gradient Fouriertransforms (denoted P(R,k ,) and Vn(R,k ,), re-spectively) over a continuous circular aperture of radiusR [7]

M(1) =H (2)k (kR)PH

(2)k (kR) jcVn

H(1)k (kR)H(2)k (kR)H

(2)k (kR)H

(1)k (kR)

, (12)

M(2) =H (1)k (kR)PH

(1)k (kR) jcVn

H(2)k (kR)H(1)k (kR)H

(1)k (kR)H

(2)k (kR)

(13)

where is the angular frequency [rad/s], k the wavenum-ber (k = /c) [rad/m], k the circumferential wave in-dex (k = ...2,1,0,1, ...), H(i)k is the i-type Hankel

function of order k , H(i)k is the spatial derivative of the

Hankel function and is the air density [kg/m3] (1.18kg/m3). P(R,k ,) and Vn(R,k ,) are the sound pres-sure and particle normal velocity in the circumferentialwave index and frequency domain.

For a circular continuous aperture, the sound pres-sure and particle normal velocity spectra are defined asFourier series expansion coefficients

P(R,k ,) =1

2pi

2pi0

p(R, ,)e jk d (14)

Vn(R,k ,) =1

2pi

2pi0

vn(R, ,)e jk d (15)

where p(R, ,) and vn(R, ,) are the complex-valuedsound pressure and particle normal velocity along a cir-cular sensing aperture (radius R). The discrete ver-sions of these two transforms simply involve replacingthe integration by a proper summation. For the methodpresented in this paper, it is assumed that the circularaperture is a virtual microphone array. The pressurep(R, ,) and velocity vn(R, ,) at this virtual arrayare computed from Eq. (5) once the inverse problem issolved and used as a non-uniform resampling algorithm(as evoked in Sec. 2).For practical cases, Eq. (11) should not be used for ex-trapolation since Hankel functions are not bounded atr = 0 and the summation over k may be numericallyunstable [7]. To circumvent this problem, one must firsttransform the CH coefficients in another set of divergingand converging plane waves amplitudes [7]

S(1)l ( ,(l)l ) =

k=

(1/2pi) jk M(1)(k ,)e jk (l)l ,

(16)S(2)l ( ,

(l)l ) =

k=

(1/2pi) jk M(2)(k ,)e jk (l)l .

(17)The extrapolated sound field is then computed withoutnumerical difficulty using

p( ,x) =L

l=1

(S(1)l e

jk(l)l x +S(2)l ejk(l)l x

) (l) (18)

where (l) is the angular separation [rad] of the planewaves propagation directions.



20 10 0 10 201020

1010

100

|P(k )

|

Magnitude of pressure angular spectra P(k) at 500 Hz

20 10 0 10 201020

1010

100

k = M/2...,2,1,0,1,...M/2

|V n(k

)|

Magnitude of velocity angular spectra Vn(k) at 500 Hz

k =

kR

k =

kR

No noise0.1% noise0.001% noise

Fig. 5: Magnitude of sound pressure and velocity an-gular spectra at 500 Hz without noise and with 0.1% ofnoise. Additional spectra with noise (103%) are alsoshown as dotted lines. The arrows indicate the decreaseof these curves for a decreasing noise level.

3.1. CH spectra: Numerical examplesThe hypothesis which underlies the method described inthe remainder of this paper relies on a simple observationof the typical sound field spectra and corresponding CHspectra. It will be shown how a very specific region ofthe sound field spectra is affected by the inverse problemsensitivity to noise. This observation helps in the defini-tion of the second method based on CH filtering.

The noise-free sound-field spectra are shown in Fig. 4 fora virtual circular aperture of radius R = 0.3 m on whichimpinges a broadband plane wave with the propagationdirection = 45 deg. The virtual sensor array is made of100 microphones arranged in a double circle (inner andouter) for the discrete evaluation of pressure and veloc-ity. Note that most of the spectras significant energy iscontained in the |k | kR region, here called the pass-band region. This spectra region corresponds to prop-agative waves. For |k | > kR, the corresponding wavesare cylindrical evanescent-like waves [1].A slice of these spectra are shown in Fig. 5, with and

without noise. For the case without noise, all the spectracomponents are relevant sources of information about theoriginal sound field. Indeed, even the part of the spec-tra which belongs to the evanescent-like region bringsinformation about the original sound field. Therefore,that information should be conserved except if noise ispresent. The spectra for the same impinging plane wavewith additional noise at the original microphone array(URA shown in Fig. 2) used for sound field extrapola-tion to the virtual circular sensor array are also shownin Fig. 5. (For that case, the spectra are obtained us-ing Eqs. (14) and (15) where sound pressure and particlenormal velocity are computed using the inverse problemextrapolated sound field evaluated at the virtual circu-lar aperture.) Clearly, outside the passband region, thesignal-to-noise ratio is drastically reduced when noise ispresent. Once the sound field spectra are used to com-pute the CH spectra using Eqs. (12) and (13), the noiseoutside the passband region can be drastically amplifiedby the Hankel functions included in both the numeratorand denominator of Eqs. (12) and (13).Resulting CH spectra are shown in Fig. 6. Onecan clearly see how noise affects the computed CHspectra and consequently the extrapolated sound field.Two important features of these CH spectra should benoted: a) the passband region is not affected by noiseand b) the peaks in the CH spectra with noise all occurat |k |= 14 (for the shown frequency) (thin dashed line),i.e. the peaks positions do not depend on noise level. Theregion between these peaks and |k |= kR will be definedas the transition band. The peaks in the CH spectra arecaused by the amplifying Hankel functions in Eqs. (12)and (13). An example of the terms of Eq. (12) for the re-ported case is shown in Fig. 7 where one can see the noiseamplification potential of |H(2)|, |H (2)| and Eq. (12)sdenominator. The Hankel functions H(2) and H (2) inthe numerator boost the angular spectra with increasingk while the denominator becomes suddenly large for agiven k which seems to define the end of the transitionband. Any noise present in the transition band will beamplified (see Fig. 6).

4. VIRTUAL MICROPHONE ARRAYS

To benefit from the observations mentioned above, theinverse problem will be used as a non-uniform spatial re-sampling method. This resampling allows the use of vir-tual microphone arrays of any size and of varying geom-



20 10 0 10 201020

1010

100

|M(1)

(k )| a

nd |M

(2)(k

)|

CH spectra at 500 Hz

k=M/2...,2,1,0,1,...M/2

k =

kR

k =

kR

|M(1)| (noise) |M(2)| (noise) |M(1)| |M(2)|

Fig. 6: CH spectra at 500 Hz for the case withoutand with 0.1% of noise. Additional spectra with noise(103%) are also shown as dotted lines. Arrows indicatea decreasing noise level. Spectras peaks are circled

etry as function of frequency. In this paper, the inverse-problem extrapolated sound field described by Eqs. (4)and (5) will be used to compute the sound field over a cir-cular virtual array of microphones described by locationvectors x(v)v (from a set x(v)1 , ,x(v)V ). If it is assumedthat the virtual array is located inside the region coveredby the real array, the resampling by inverse problem canbe based on pseudo-inversion without Tikhonov regular-ization. Indeed, for this method based on virtual arrays,we are not concerned by the inverse problem reproducedsound field outside the URA. The virtual array is madecircular so that the CH transforms mentioned in Sec. 3can then be applied.

Using a virtual array that fits the circular array as usedin Fig. 4, the CH coefficients are computed (Eqs. (12)and (13)), then transformed in plane waves amplitudes(Eqs. (16) and (17)) and finally converted to the extrapo-lated sound field (Eq. (18)). Note that for the case with-out noise, conversion from inverse problem plane wavessolution to the CH spectra using spatial resampling andtransform gives an extrapolated sound field very similarto the one obtained by the inverse problem method. Ifnoise is present, the sound field extrapolation by CH isless efficient and it is expected that a low-pass CH filtermight remove the noise peaks in the CH spectra. Accord-ingly, a CH filter is designed so that conversion from theCH coefficients M(1) and M(2) involve a zeroing of theCH coefficients which corresponds to a k above kR but

20 10 0 10 20500

250

0

250

500

k=M/2,...,1,0,1,...M/2

Gai

ns [d

B ref

1]

Terms gains for the evaluation of M(1) at 500Hz

k =

kR

k =

kR

k =

14

k =

14

|H(2)k

||H(2)k

|

|1/Den||H(2)/Den||H(2)k

/Den|

Fig. 7: Gains at 500 Hz of the various terms in Eq. (12).The maximums, which cause the peaks if Fig. 6, aremarked by small circles.

within the transition region. The zeroing starts from acut-off k . Examples of sound field extrapolation us-ing virtual sensor array and CH filtering are shown inFig. 8(g) to (l). These results are explained in Sec. 5.

5. COMPARISON OF THE TWO METHODS

Figure 8 shows sound field extrapolation examples forthe two methods where both the regularization parame-ter and the cut-off k are varied within an range of in-terest. This range was selected to reach almost divergingand spatially-limited sound field extrapolation. Althoughthe virtual array with CH filtering method seems to re-duce the diverging part of the extrapolated sound field(see the diverging case in Fig. 3), the size of the effectiveextrapolation region diminishes greatly as the cut-off kis reduced. Interestingly, the effect of the regularizationparameter in the regularized inverse problem is different.Indeed, the effective extrapolation region is much less re-duced as the regularization parameter is increased (andas the diverging part of the extrapolated sound field is de-creased). In fact, the size of the effective extrapolationseems to be less affected by the regularization parame-ter than the CH cut-off k . This suggests that soundfield extrapolation on the basis of inverse problem withTikhonov regularization might be more efficient than thesecond method based on virtual sensor array and CH fil-tering. Moreover, the inverse problem method involvesless processing than the virtual sensor method which in-



Fig. 8: Comparisons of the extrapolated sound fields by the inverse regularized problem and CH filtering with virtualarrays with 0.1% noise at the URA. (a) and (b): Inverse problem with = 1e9. (c) and (d): Inverse problem with = 1e7. (e) and (f): Inverse problem with = 1e5. (g) and (h): Virtual array and a cut-off at k = 9. (i) and (j):Virtual array and a cut-off at k = 7. (k) and (l): Virtual array and a cut-off at k = 5.

volves two additional transforms. Finally, the inverseproblem method directly outputs a set of plane waves asa sound field descriptor. This is also a great advantage,beside its simplicity, of sound field extrapolation methodbased on regularized inverse problem.

6. CONCLUSION

This paper has investigated the properties of two soundfield extrapolation methods that may be applied to uni-form rectangular array or non-uniform array. The firstmethod based on inverse problem theory is sensitive tonoise. The introduction of Tikhonov regularization in

the inverse problem can however circumvent this noisesensitivity and provide efficient sound field extrapola-tion. The second method based on virtual sensor arrayand CH filtering was inspired by the fact that measure-ment noise manifests itself in a very specific region ofthe CH spectra. The performance of this second methodis however less interesting. Indeed, the reduction of thediverging part of the extrapolated sound field by CH fil-tering also involved a strong reduction of the effectiveregion of the extrapolation. On the basis of these obser-vations, it seems that inverse problem extrapolation is abetter method for in-plane sound field extrapolation.



Future research could be done to evaluate the perfor-mance of the inverse problem method for the 3D cases,i.e. 3D sound field measured using a 2D array or a 2Darray with variations in the vertical direction. A morein-depth study of inverse problem properties for soundfield extrapolation should be done. Recent works by theauthors in that direction is devoted to a new promisingregularization method for the inverse problem, this willbe the topic of a future paper.

Once sound field extrapolation is achieved, it is possibleto move to the next step: characterization of the soundfield or sound environment for subsequent reproduction.Beside the definition and uses of classical targets, manyother types of target could come from a multidisciplinarydescription of the sound environment that must be spa-tially reproduced.

7. ACKNOWLEDGMENT

This work is part of a project involving: Consortium forResearch and Innovation in Aerospace in Quebec, Bom-bardier Aeronautique, CAE and McGill University, sup-ported by a Natural Sciences and Engineering ResearchCouncil of Canada grant.

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2010 Sound field extrapolation: Inverse problems, virtual microphone arrays and spatial filters