A WAVENUMBER-FITTING EXTRAPOLATION METHOD FOR FFT-BASED NEAR-FIELDACOUSTIC HOLOGRAPHY USING MICROPHONE ARRAY

Benxu Liu, Bremananth Ramachandran, Andy W. H. Khong

School of Electrical & Electronic Engineering,Nanyang Technological University, Singapore,

Email: liub0008@e.ntu.edu.sg, {bremananth, andykhong}@ntu.edu.sg

ABSTRACT

Near-field acoustic holography (NAH) is an important acoustic vi-sualization technique which can be implemented efficiently throughthe fast Fourier transform (FFT). However, when the FFT is appliedon a small measurement aperture via a microphone array, significantspectral leakage occurs. Windowing is often employed to reducethis leakage but at the expense of degradation in reconstructionperformance due to the corruption of measured data, especiallywhen the source is located around the edge of the sensor array. Awavenumber-fitting method is proposed to extrapolate the measure-ment aperture effectively. This is achieved by exploiting the waveequation using measurement aperture as the boundary condition.The quality of NAH source reconstruction is shown to be improvedsignificantly using the proposed approach.

Index Terms— Extrapolation, microphone array, near-fieldacoustic holography, wavenumber-fitting.

1. INTRODUCTION

Near-field acoustic holography (NAH) is an important tool for soundradiation visualization and acoustic source localization [1]. Con-ventional NAH first measures the acoustic field on a measurementaperture using an array of microphones after which the fast Fouriertransform (FFT) is used to decompose the measurement field intoa spectrum of plane propagating and evanescent waves. These twotypes of waves, with predetermined characteristics, are then backpropagated to an assumed source aperture after which the inverseFFT is utilized to compute the pressure distribution on this aperturefor visualization or localization. The coordinate system of the NAHis presented in Fig. 1, where the acoustic propagation is in the z-axiswhile zs, zh and zd denote, respectively, positions of the source,measurement apertures and the distance between them.

The existence of evanescent waves, with amplitude decaying ex-ponentially from the source to measurement aperture, renders NAHan ill-posed problem [2]. Any errors due to spectral leakage will beamplified exponentially during back propagation. One way to reducesuch spectral leakage is to employ a large measurement apertureconstructed using a large number of microphones [1]. For practi-cal applications, however, this is often prohibitive due to constraintspertaining to the array size as well as computational complexity.

One of the most popular solution is to extend the aperture syn-thetically via zero-padding [1]. This idea stems from the assump-tion that any unmeasured acoustic field is treated as zero. It is how-

This work is supported by the Singapore National Research Foun-dation Interactive Digital Media R & D Program, under research grantNRF2008IDM-IDM004-010.

Measurementaperture

Sourceaperture

Microphonearray

Back propagation distance

-axis

-axis -axisx

y

z

sz hz

dz

Fig. 1. A coordination system of NAH.

ever important to note that zero-padding does not increase the spec-tral resolution and hence does not reduce spectral leakage. A one-dimensional (1-D) linear prediction (LP) algorithm has been pro-posed to estimate the acoustic pressure outside the measurementaperture [3]. Apart from the challenging task of determining theoptimal predication order, one of the main disadvantages is that thealgorithm utilizes only neighboring acoustic pressures lying on oneof the axes. The statistically optimized NAH (SONAH) [4] has alsobeen proposed to remove spectral leakage by avoiding the use ofFFT. Although arbitrary placement of the microphones is allowed,significant computational load is required due to the computation ofa continuous spectrum. Extrapolation of head-related transfer func-tion data from limited measurements has also been proposed in [5].

In this work, we exploit the computational efficiency of FFT andhence seek to extend, synthetically, the finite measurement apertureeffectively. We utilize the solution of the homogenous Helmholtzwave equation and propose a wavenumber-fitting method to estimatethe unmeasured acoustic field in a least-norm sense. This allows usto extrapolate the measurement aperture which in turn reduces thespectral leakage due to FFT operation in NAH.

2. ILL-POSED PROBLEM AND FINITE APERTURE

Conventional planar NAH propagates the acoustic field in the trans-formed wavenumber domain. Defining F and F−1 as the two-dimensional (2-D) Fourier and inverse Fourier transform, the processcan be expressed as [1]

P (kx, ky, zh) = F{p�(x, y, zh)}, (1)

p(x, y, zs) = F−1{P (kx, ky, zh)G(kx, ky, zs − zh)}, (2)

where p�(x, y, zh) and p(x, y, zs) denote, respectively, the acousticpressure on the finite measurement and reconstructed source aper-tures. As shown in (1), the measured acoustic field is first trans-formed to the Fourier domain described by the angular spectrumP (kx, ky, zh) where kx and ky are the directional wavenumbers

145978-1-4577-0539-7/11/$26.00 ©2011 IEEE ICASSP 2011

along the x and y axes respectively. Defining j =√−1, the spec-

trum P (kx, ky, zh) is in turn back propagated, as shown in (2), viathe propagating factor

G(kx, ky, zs − zh) = e−j|zs−zh|kz , (3)

kz =

{ √k2 − k2

x − k2y k2

x + k2y ≤ k2

j√

k2x + k2

y − k2 k2x + k2

y > k2 , (4)

where acoustic wavenumber k = 2π/λ and λ is the wavelength.Note that when k2

x + k2y > k2, G(kx, ky, zs − zh) increases with

kx and ky , which corresponds to the back propagation of evanescentwave [2]. This implies that, using (2), any errors in P (kx, ky, zh)will be amplified significantly rendering the NAH an ill-posed prob-lem. Addressing this ill-posed problem requires the finite measure-ment aperture p�(x, y, zh) to be large enough so that the acousticfield outside this aperture is negligible. This requires a measurementaperture that is significantly larger than the source, which is oftenimpractical.

The spectral leakage problem can be explained from the finitemeasurement aperture which can be expressed by

p�(x, y, zh) = p(x, y, zh)Π(x/Lx)Π(y/Ly), (5)

Π(x/Lx) =

⎧⎨⎩

1 |x| < Lx/20.5 |x| = Lx/20 |x| > Lx/2

, (6)

where Π(y/Ly) is defined similar to (6) and p(x, y, zh) is the acous-tic field of an infinitely large measurement aperture, while Lx andLy are the size of the aperture along the x and y axes, respec-tively. Since multiplication in (5) corresponds to convolution in thespectral domain, P (kx, ky, zh) is equivalent to the convolution be-tween F{p(x, y, zh)} with the spectrum of the windowing functionsF{Π(x/Lx)} and F{Π(y/Ly)} where [2]

F{Π(x/Lx)} =Lx sin(kxLx/2)

kxLx/2= Lxsinc(kxLx/2). (7)

We therefore note that the main lobe of the sinc function increasesin width and reduces in amplitude when Lx and/or Ly reduces. Thisresults in spectral leakage which, in turn, leads to poor acoustic re-construction. Although applying windows such as Tukey [6] will re-duce this leakage, the tapering edge of the window often corrupts thealready insufficient measured acoustic field which in turn introduceshigher reconstruction errors. This error becomes more significantwhen the source is located near the edge. In addition, if size of themeasurement aperture is too small, windowing alone is insufficientto suppress the leakage since the acoustic field outside the measure-ment aperture cannot be treated as zero.

3. WAVENUMBER-FITTING FOR EXTRAPOLATION

Unlike existing methods described above, and in order to extrapolatethe acoustic field outside the measurement aperture, we utilize thewave propagation model. For clarity of presentation, we define rmas the (x, y, z) coordinates of the mth microphone, m = 1, . . . ,M ,where M is the total number of microphones while p�(rm) is themeasured acoustic pressure of the mth microphone. Wave prop-agation in the source-free field is governed by the homogenousHelmholtz equation and the measurement aperture forms the bound-ary condition. Based on this, our extrapolation problem can beformulated as

∇2p(r) + k2p(r) = 0, (8)

p(rm) = p�(rm), m = 1, 2, . . . ,M,

where p(r) is the acoustic pressure at location r.The solution to this partial differential equation can be con-

structed using a linear combination of elementary wave components(EWCs) [2]

φk(r) = e−j(kxx+kyy+kzz), (9)

where each EWC has zero initial phase and is fully determined by thedirectional wavenumbers kx, ky , and kz . Similar to the zero-paddingtechniques, the extrapolated area and the measurement aperture areon the same plane and this implies that (9) can be simplified into

φk(r) = e−j(kxx+kyy). (10)

Using this EWC, the solution to the extrapolation problem can bedescribed as [7]

p(r) ≈N∑

n=1

anφkn(r), (11)

where N is the chosen number of EWCs while the approximationimplies that the boundary condition is incomplete due to insufficientdata measurement. The solutions are therefore solved in a least-norm sense. The problem of extrapolation is therefore translated tothe choice of φkn(r) as well as the determination of coefficients an.

We note that the FFT can be a solution of (11) via the decom-position of the measured pressures into wave components definedin (9). This solution corresponds to evaluating an uniformly aroundthe unit circle in the k-space with ky and kx being the ordinateand abscissa axes respectively. However, direct application of suchEWCs for extrapolating the finite aperture is undesirable since doingso will introduce the wrap-around problem due to the periodic im-ages of the FFT. To address this wrap-around problem, continuousEWCs have been utilized by the SONAH algorithm [4] to recon-struct the source field which, as a consequence, require a significantcomputational load since a large number of coefficients have to besolved.

To achieve a good compromise between accuracy and compu-tational load, a discrete spectrum is preferred. Selection of suit-able wavenumber components is therefore important for the reduc-tion of wrap-around errors. We first note that the resolution of thewavenumber spectrum is determined by the size of the extrapolatedaperture. It is also important to note that the highest wavenumber de-termines the distance between the neighboring periodic images in thespectral domain. Therefore, these two factors determine the choiceof EWCs that are suitable in estimating the acoustic field outside themeasurement aperture.

For clarity, we consider a 1-D aperture since this approach canbe extended directly to the 2-D case. For this 1-D case, we assumethat we have an array of length L with a uniform microphone sep-aration of Δd. In order to reduce the spectral leakage, we extendthe aperture length to 2L and reduce Δd by a factor of four syn-thetically. This is to increase the highest sampling frequency whichin turn allows us to separate the periodic spectra. This results in awavenumber spectrum with the highest wavenumber kmax and reso-lution �k to be

kmax =2π

�d/4, �k =

2π

2L. (12)

With these conditions satisfied, and similar to the case of FFT, themost direct way to discretize the continuous spectrum in the range(0, kmax) is by uniform sampling via

kn = (n− 1)�k, n = 1, 2, . . . , N , N =kmax

�k. (13)

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It is however important to note that this uniform sampling will inher-ently result in a periodic acoustic field. This periodicity is undesir-able because the acoustic field outside the measurement aperture isoften not a periodic image of the acoustic field inside this aperture. Inorder to avoid this periodicity, we propose to sample the continuouswavenumber spectrum non-uniformly. Two different non-uniformsampling techniques, piecewise and log sampling are proposed. It isalso worthwhile to note that, for fair performance evaluation, kmax

and number of samples N should be the same for different spectralsampling methods.

The coefficients an will have to be determined once the wavenum-ber components are chosen. Determination of an can be achievedvia substituting p�(rm) into (11). With M number of microphones,we formulate M linear equations and re-write (11) in matrix notationgiven by

Φa = p, (14)

Φ =

⎡⎢⎣

φk1(r1) φk2(r1) · · · φkN(r1)

φk1(r2) φk2(r2) · · · φkN(r2)

· · · · · · · · · · · ·φk1(rM ) φk2(rM ) · · · φkN

(rM )

⎤⎥⎦ ,

p = [p(r1), p(r2), · · · , p(rM )]T ,

a = [a1, a2, · · · , aN ]T .

Since M < N , the system of equations in (14) is under-determined and we propose to incorporate the Tikhonov regulariza-tion [8] such that

a = ΦH(ΦΦH + εI)−1p (15)

is solved in a least-norm sense. The regularization parameter ε en-sures good conditioning of the matrix ΦΦH . With this a, the ex-trapolated acoustic pressure can be computed by

p(r) =N∑

n=1

anφkn(r). (16)

4. SIMULATION RESULTS

We evaluate the performance of our algorithm in the context ofwave field visualization using monochromatic sources at 1200 Hzwhere the impulse responses are generated for an enclosed envi-ronment with the reverberation time of 300 ms using the methodof images. The speed of sound is 350 m/s giving a wavelengthλ = 350/1200 = 0.2917 m. A finite aperture of 13 × 13 micro-phones with separation of 4 cm is deployed. We extrapolate thisfinite aperture using both zero-padding [1] and 1-D LP [3] as well asour proposed wavenumber-fitting method. We then backpropagatethe extrapolated acoustic field to the source aperture.

For performance evaluation, extrapolation error (XTPE) is com-puted before any back-propagation by comparing the extrapolatedacoustic field with that of the reference acoustic field obtained usinga large measurement aperture with 76×76 microphones. The XTPEis computed using

η =

√√√√∑Mm=1(pex(rm)− pre(rm))2∑

Mm=1 pre(rm)2

, (17)

where pex(rm) and pre(rm) are the extrapolated and reference

acoustic pressure at location rm while M is the number of extrap-olated locations. We also evaluate the reconstruction error (RCSE)

-500 0 500-500

0

500

kx

Ky

0 20 400

10

20

30

40

EWC kx index

EWC

ky in

dex

-500 0 500-500

0

500

kx

Ky

0 20 400

10

20

30

40

EWC kx index

EWC

ky in

dex

-500 0 500-500

0

500

kx

Ky

0 20 400

10

20

30

40

EWC kx index

EWC

ky in

dex

(a) (b) (c)

Fig. 2. EWCs sampling using (a) uniform, (b) non-uniform piece-wise and (c) log methods with their respective spectra.

by comparing, after back propagation, the reconstructed source fieldusing extrapolated acoustic field with that of a reference sourcefield back propagated using the large measurement aperture. TheRCSE is computed similar to (17). Defining Ns as the total numberof independent simulation trials, we compute the mean percentageimprovement of η over such trials given by

β =1

Ns

Ns∑i=1

(ηother(i)− ηwnf(i)

ηother(i)

), (18)

where ηother(i) and ηwnf(i) denote XTPE/RCSE of other algorithmsand our proposed wavenumber-fitting method for the ith simulationrun, respectively.

4.1. Uniform and Non-uniform sampling of EWCs

As described in Section 3, EWCs can be sampled uniformly ornon-uniformly. Coefficients a in (11) are the amplitudes of differ-ent EWCs and thus they can be viewed as the spectra of mea-sured acoustic field. In our case, the highest wavenumber iskmax = 8 × 2 π/λ = 172.34 m−1 and 48 × 48 EWCs aresampled within the range (0, kmax). Figure 2 illustrates differentEWCs sampling methods and their corresponding a. The uniformsampling and its computed spectrum is shown in Fig. 2(a) wherewe can see that for all the EWCs sampled, only some discrete areashave non-zero amplitude and these non-zero areas are uniformlydistributed. This discrete spectrum will generate a periodic acous-tic field after extrapolation, which is undesirable. The same effectalso occurs for the piecewise non-uniform sampling as shown inFig. 2(b). Our piecewise method samples the middle portion thatis three times denser than that of the edges. We therefore note,from the computed spectrum that the non-zero interval in the middleportion of the spectrum is nearly three times that of the edge portion.This spectrum is still discrete and thus is suitable for our applicationeither. On the other hand, a non-uniform log sampling, with differ-ent intervals between any two sampled wavenumbers, reduces theperiodic extrapolation effect as illustrated in Fig. 2(c) where we notethat the spectrum is more continuous than the other two methods.Our experiments also show that log sampling of the EWCs producesthe best estimation in extrapolating acoustic field as compared toother sampling methods. The following experimental results arebased on log sampling of EWCs.

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5 10 15 20

5

10

15

20

Microphone index in xMic

roph

one

inde

x in

y

5 10 15 20

5

10

15

20

Microphone index in xMic

roph

one

inde

x in

y5 10 15 20

5

10

15

20

Microphone index in xMic

roph

one

inde

x in

y

5 10 15 20

5

10

15

20

Microphone index in xMic

roph

one

inde

x in

y−0.5

0

0.5

1

1.5

(d)

(b)(a)

(c)

Fig. 3. Illustrations of (a) original acoustic field, and small aperturemeasurement extrapolated using (b) zero-padding, (c) 1-D LP and(d) proposed wavenumber-fitting methods.

5 6 7 8 9 10 11 1210

20

30

40

50

60

70

Impr

ovem

ent (

%)

LP orders

Extrapolation Improvement

Reconstruction Improvement

Fig. 4. Improvement of wavenumber-fitting over 1-D LP for differ-ent orders.

4.2. Extrapolation Error

In our experiment, the small measurement aperture is extended us-ing different extrapolation methods. The results are shown in Fig. 3with the source of interest located near the edge of the measure-ment aperture. An area inside the white rectangle in each subplot ofFig. 3 represents the small aperture of 13 × 13 microphones whilethe outside region signifies the extrapolated area. Figure 3(a) illus-trates the original acoustic field inside and outside the small aper-ture. Figure 3(b) depicts the small aperture which has been extendedusing zero-padding. The extrapolated acoustic field using 1-D LPmethod is shown in Fig. 3(c) and the extrapolated acoustic field us-ing proposed wavenumber-fitting method is presented in Fig. 3(d).From Figs. 3(b)-(d), we observe that our proposed method outper-forms other two methods by reconstructing the energy distribution inthe extrapolated area. We performed 100 simulation runs with fourpoint sources that are randomly positioned. Figure 4 shows that ourwavenumber-fitting method achieves an improvement between 16%to 24% in terms of extrapolation accuracy compared to different 1-DLP orders.

4.3. Reconstruction Error

After extrapolation, the FFT-based NAH is used to reconstruct thesource aperture. Figure 5 illustrates the RCSE of different extrap-olation methods for 100 simulation runs where, as before for eachrun, the positions of the sources is varied randomly. We can seethat zero-padding introduced significant RCSE due to spectral leak-age brought about by the small measurement aperture. The 1-D LPmethod and the wavenumber-fitting method both attenuate these er-rors while the latter can achieve lower reconstruction error. Resultsshow that the proposed method produces approximately 87% recon-

0 10 20 30 40 50 60 70 80 90 100

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Index for different simulation run

Rec

onst

ruct

ion

Erro

r

1D LP extrapolation with order=8

Zeropadding

Proposed wavenumber-fitting extrapolation

Fig. 5. Reconstruction errors using different methods.

struction improvement over zero-padding. In addition, Fig. 4 showsthat the improvement over 1-D LP for various LP orders range be-tween 50% to 60%. We note that the reconstruction improvement de-creases modestly with a higher LP order, since a higher order impliesthat more measured pressures are used for the extrapolation processin 1-D LP. However, unlike our proposed method, since 1-D LP doesnot take into account wave propagation, it is therefore expected thatour proposed algorithm outperforms the 1-D LP algorithm.

5. CONCLUSION

A robust wavenumber-fitting extrapolation method was proposed toimprove the performance of conventional FFT-based NAH on smallaperture. Wave propagation theory was utilized to extrapolate thesmall measurement aperture. In addition, a log sampling techniquewas employed to discretize the wavenumber spectrum which, inturn, prevents the extrapolated acoustic field to be periodic. It hasbeen shown through simulations that the proposed wavenumber-fitting algorithm has a better performance in terms of extrapolationand reconstruction errors compared to zero-padding and 1-D LPalgorithms.

6. REFERENCES

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[2] E. G. Williams, Fourier acoustics: sound radiation andnearfield acoustical holography, Academic Press, c1999.

[3] R. Scholte, I. Lopez, N. B. Roozen, and H. Nijmeijer, “Trun-cated aperture extrapolation for Fourier-based near-field acous-tic holography by means of border-padding,” J. Acoust. Soc.Am., vol. 125, no. 6, pp. 3844–3854, 2009.

[4] J. Hald, “Basic theory and properties of statistically optimizednear-field acoustical holography,” J. Acoust. Soc. Am., vol. 125,no. 4, pp. 2105–2120, 2009.

[5] R. Duraiswami, D. N. Zotkin, and N. A. Gumerov, “Interpo-lation and range extrapolation of HRTFS,” Prof. of IEEE Int.Conf. on Acoustic, Speech, and Signal Processing, vol. 4, pp.iv–45 – iv–48 vol.4, May 2004.

[6] F. J. Harris, “On the use of windows for harmonic analysis withthe discrete Fourier transform,” Proc. of IEEE, vol. 66, no. 1,pp. 51 – 83, Jan 1978.

[7] Z. Wang, Helmholtz Equation-Least Squares (HELS) Methodfor Inverse Acoustic Radiation Problems, Ph.D. thesis, WayneState University, 1995.

[8] P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems:Numerical Aspects of Linear Inversion, SIAM, c1998.

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