Sound field reconstruction in a room from spatiallydistributed measurements
Efren FERNANDEZ-GRANDE(1)(1) Technical University of Denmark (DTU), Department of Electrical Engineering, Denmark, efg@elektro.dtu.dk

Abstract
The full characterization of the sound field in a room from experimental data is a challenging problem, as
the direct measurement of the sound field in the volume requires considerable experimental effort. In thisstudy, the sound field in a room at low frequencies is characterized using a representation in wavenumber
domain that infers the modal structure of the sound field. The modal structure is used to synthesize frequency
response functions, to characterize the spatio-temporal properties of the wave field. The proposed method makes
it possible to infer frequency response functions between source-receiver pairs that have not been measured,
exploiting the representation of the sound field as a modal superposition, and thus reducing the dimensionalityof the reconstruction problem

Keywords: Acoustic holography, room acoustics, array processing, sound field analysis

1INTRODUCTION

In many applications, knowledge of the room impulse response functions throughout a room is valuable - e.g.,source localization [1], dereverberation [2], room equalization [3, 4], sound field analysis [5, 6], etc. However,measuring experimentally the spatio-temporal properties of the sound field in an entire room, i.e. measuring
impulse responses in the three-dimensional domain point by point, requires extensive experimental effort. The
alternative of interpolating and extrapolating the sound field - i.e. estimating room impulse response functions atpositions that have not been directly measured, is a powerful approach that has been explored in multiple studies[7, 8, 9, 10, 11, 12]. The rationale behind most approaches is to formulate a physical propagation model, often
based on elementary wave expansions (in frequency or time domain), onto which a limited set of measurements
is mapped. This constitutes an inverse acoustic problem [13], where the model is fitted to experimental data,
making it possible to predict or reconstruct the sound field anywhere in the domain [14, 15, 16]. These inverse
approaches typically do not require explicit knowledge of the geometry or boundary properties of the room, as
they are essentially concerned with the propagation of sound waves in the medium. Yet, prior information canbe incorporated into the problem, to promote meaningful solutions that lead to better predictions [17, 18, 19,20].
Several studies have addressed the reconstruction of the sound field in a room from a limited set of measurements [7, 8, 9, 10, 11, 12], typically at mid and low frequencies or over small apertures - due to the sampling
requirements. When considering the sound field in a room at low frequencies, the most distinct property of thesound field is its marked modal structure: i.e. the sound field results from the superposition of normal modes(eigenfunctions). When addressing sound field reconstruction in small or medium enclosures (low modal densities), this physical property can be exploited, i.e. [10, 21, 22, 23]. These studies propose to decompose thenormal modes into a sparse set of plane-wave functions, a case which is well-suited to rectangular enclosures,but is less suited to rooms of arbitrary shape. In the latter case, it is necessary to assume that an over-complete
set of basis functions can approximate the eigenfunctions sufficiently well.
In the present study we propose a method to reconstruct the sound field in a room, via incorporating the modal
structure of the sound field into the problem. The approach serves to predict the spatio-temporal properties of
the wave field at low frequencies over the volume of the room. As a result of introducing the normal modes
into the model, the method presented here enables to predict/synthesize frequency response functions between4961

any source-receiver pair in the room, including those that have not been measured.

2THEORYThe solution to the inhomogeneous wave equation in an enclosure, i.e. the Green’s function, has the well knownsolution [24]∞1 X Ψn (r0 )Ψn (r)G(r|r0 ) = −,(1)
V n=0 k2 − kn2 − jk/cτnwhere V is the room’s volume, Ψn are the normal modes (eigenfunctions) of the room, k is the wavenumber,kn the modal frequency of the mode of index n, and τn is its time constant. The sound pressure in the room isp(r) = jωρQG(r|r0 ), and Q is the volume velocity of the source. The normal modes can be expanded into anelementary wave basis∞XAnm ϕnm (r),(2)Ψn (r) =m

where ϕnm (r) are the elementary waves and Anm a set of unkown
coefficients.
 In the present study, planeT
propagating waves are chosen, ϕnm (r) = e−jknm ·r , where knm = k x,nm , ky,nm , kz,nm is the wavenumber vector ofmode n discretized into a basis of M plane waves. It should be noted that no assumption is made on thegeometry of the room or the shape of the normal modes. Assuming that all modes are excited (i.e. the source
is not placed on a nodal plane), the sound field in the room can be written asp(r) = −NM
−1 XjωρQ X  2 2k − kn − jk/(cτn )Anm ϕnm (r)Vnm(3)

where N is the number of normal modes considered, and M is the number of plane waves into which eachmode is discretized. As the modal frequencies are not available, and the modal shapes are unknown, thisproblem leads to a three-dimensional discretization of the wavenumber space or k-space, as shown in Fig. 1.
In the figure, two lattice sampling patterns are shown, one in concentric circles and another in a cubical lattice,where the wavenumber domain is sampled at equal intervals in the k x , ky and kz directions.

Figure 1. Sampling of wavenumber domain. Spherical lattice sampling (left). Cubical lattice sampling (right)
Problem (3) can be written in matrix form as p = Φb where p ∈ C M , b ∈ CN·M . This problem is highly underdetermined, given the multi-dimensional basis used to model the sound field, and the inversion is not trivial. We4962

Figure 2. Finite element model of the room, with the source placed near the origin, at (x, y, z) = (0.1, 0.1, 0.1) m(left). Distribution of measurement points in the volume of the room (right)
propose to make use of a synthesis prior in the solution [25, 26], which promotes solutions where the non-zeroplane wave functions are close to the driving frequency,b̃ = arg min||Φb − p||2 + λ||Lb||1(4)bThe matrix L = diag(w) is a diagonal matrix that contains the distance between the angular frequency of each
2 k. The estimation canplane wave and the modal frequency that is being excited, with elements wi = k2 − kknm,ibe further refined by normalizing the magnitude of the modes to match the measured sound pressure levels ina broadband sense, i.e. minimizing the functional arg minkp(rm , ω) − p̃(rm , ω, an )k where an is a scaling factor
anassociated to each mode, rm are the measurement positions and p̃ the reconstructed pressure. Once the eigenfunctions are estimated from (4), it is possible to predict the sound field at any point in the volume, through areconstruction matrix Φr that maps the estimated coefficients b̃ and the reconstruction pointsp̃r = Φr b̃

(5)with p̃r ∈ CR containing the predicted sound pressure at the reconstruction points.
A significant result derived from this model is the fact that based on the expansion of the normal modes intoplane waves (retaining the modal structure of the sound field), it becomes possible to predict the sound field in
the room at different source positions than the measured one. From Eqs. (2)-(5), the sound pressure field in theroom for any source at position r x can be synthesized asNjωρQ X Ψn (r x )Ψn (rr ).
p(rr ) = −V n=0 k2 − kn2 − jk/cτn
It is apparent that the proposed model enables to predict the sound pressure in the room for any combinationof source-receiver pairs, including those that have not been measured.

3RESULTS

A numerical study is presented to test the proposed methodology. A Finite Element Method (FEM) simulationof an approximately rectangular room of dimensions 3.3 × 3 × 4.5 m3 and reverberation time of T 60 = 2.2 seconds
is performed in COMSOL (Fig. 2 left). The sound pressure estimated from the FEM study is used as reference“true” data to assess the accuracy of the reconstruction method. The simulated measurement consists of 64
measurement points pseudo-randomly distributed throughout the volume of the room, with a minimum spacing4963

Figure 3. Reconstruction of the sound field in the room at 44 Hz (top row) and 78 Hz (bottom row). The
reconstructed field is shown in the left column, and the true reference sound field in the right column [dBSPL].
of 10 cm, and an average spacing of 75 cm (see Fig. 2 (right)). A wave basis is constructed consisting of
29000 plane waves, chosen to sample the wavenumber spectrum in a cubical lattice of dk=0.18 rad/m, andspanning the first 25 modes in the room. The reconstruction of the sound field is analyzed in the frequency
range between 30 Hz and 150 Hz, and the reconstruction domain is the entire volume of the room, discretizedin 43500 points.

Figure 4. Frequency response functions (averaged and min/max values in shaded area [dB SPL]) and MAC forthe frequency range between 30 and 135 Hz, when the room is driven with the source in the corner (x0 , y0 , z0 ) =(0.1, 0.1, 0.1) m.
Figure 2 shows the shape of the examined room, which is a model of one of the rooms at the DTU acoustics

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labs which is used for flanking transmission measurements. The figure also shows the measurement points
distributed throughout the volume.
Figure 3 shows the reconstructed and true sound pressure levels inside the volume of the room at 44 Hz (top),which is close to an axial mode, as well as the response at 78 Hz (bottom) which is between two modalfrequencies. The reconstructed results and the true reference sound field are shown for comparison. At 44 Hz
the pressure response is dominated by mode (1,0,0). In the case at 78 Hz, the resulting sound pressure is due tothe superposition of several modes - as expected due to the higher modal overlap, including the oblique (1,1,1)
mode. It is apparent from the figure that the estimation is fairly accurate in both cases, and the sound field in
the room is reconstructed correctly.
Figure 4 (right) shows the reconstructed frequency response functions in the room up to 135 Hz (averaged FRF
as well as max/min values in the shaded area), and the modal assurance criterion (MAC), defined as
MAC =kpH p̂k2,(pH p)(p̂H p̂)(6)which is used to quantify the spatial similarity of the modes (MAC= 1 indicates maximal similarity, and MAC=0 indicates maximal dissimilarity). The FRF’s and the MAC results show that the quantitative accuracy of themethod is good, particularly close to modal frequencies. This is in agreement with the fact that the estimation
problem (4) is solved at these frequencies.

Figure 5. Synthesized frequency response functions when the source is placed at position (x, y, z) = (1.5, 2, 1) m,
not too far from the center of the room. Reconstructed sound pressure levels [dB] at 87 Hz, over 869 points
in the room. Frequency response functions (averaged and min/max values in shaded area) and MAC for thefrequency range between 30 and 135 Hz.
It is perhaps of greater interest to consider the reconstruction of the sound field for source-receiver combinations

4965that have not been measured, i.e. if the presented model is used to synthesize or predict the frequency response
function for an arbitrary source position r x . Figure 5 (top) shows the predicted/reconstructed sound field if asource was placed at position (x x , y x , z x ) = (1.5, 2, 1) m, not too far from the center of the room. In this case thereconstruction is done over 869 points (the lower number of points in this test was chosen for computational
efficiency). The figure shows the reconstruction at 87 Hz, along with the reference true pressure, which wasestimated from a new FEM simulation with the source placed at (1.5, 2, 1) m. It is clear that the method predicts
well the actual sound field in the room. Figure 5 (bottom) shows the frequency response functions (averaged
as well as max-min values) as well as the MAC, for the reconstruction of the sound field in the case that thesource is placed in a new position. It is noticeable that several of the modes with nodal planes near the newsource position contribute less to the frequency response than in the previous case (source in the corner). Onaverage, the reconstructed responses and true reference response are fairly similar throughout the room and theMAC is higher at frequencies where a single mode dominates, or at higher modal densities, as the distance
between modes is smaller. The accuracy is somewhat lower than the case that the source is in the corner (Figs.
3 and 4), which is to be expected since the frequency responses are based on predicting the resulting soundfield if the source was placed in a different position. Nonetheless, the dominant spatial properties of the soundfield seem to be recovered correctly.

4CONCLUSIONS

A method is presented to reconstruct the sound field in a room at low frequencies. The sound field is modeledusing a wavenumber domain representation that infers the modal structure of the sound field. The method istested numerically at low frequencies, showing that the sound field in the volume of the room can be accurately
reconstructed. Due to incorporating the normal modes into the problem, it is possible to predict frequency
response functions between any source-receiver pair in the room, including those that have not been measured.
This result has interesting potential, as it leads to a model of the room that is independent of source position,
reducing the dimensionality of the reconstruction problem.

ACKNOWLEDGEMENTS
This work has been supported by the VILLUM Foundation, under grant number 19179 for the project ‘Largescale Acoustic Holography’. The author would also like to thank Samuel A. Verburg, Mélanie Nolan andRasmus E. Christiansen for valuable discussions, help with the simulation study and contributions to preceding
work .

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