The Study of Sound Field Reconstruction As an Inverse Problem *

Proceedings of the Institute of Acoustics

THE STUDY OF SOUND FIELD RECONSTRUCTION AS AN INVERSE PROBLEM *

F. M. Fazi ISVR, University of Southampton, U.K. P. A. Nelson ISVR, University of Southampton, U.K. R. Potthast University of Reading, U.K. J. Seo ETRI, Korea

1 INTRODUCTION

The problem of reconstructing a desired sound field using an array of transducers is a subject of large relevance in many branches of acoustics. In this paper, this problem is mathematically formulated as an integral equation of the first kind and represents therefore an inverse problem. It is shown that an exact solution to this problem can not be, in general, calculated but that an approximate solution can be computed, which minimizes the mean squares error between the desired and reconstructed sound field on the boundary of the reconstruction area. The robustness of the solution can be improved by applying a regularization scheme, as described in third section of this paper.

The sound field reconstruction system is assumed to be an ideally infinite distribution of monopole sources continuously arranged on a three dimensional surface S , that contains the region of space Ω over which the sound field reconstruction is attempted. This is represented in Figure 1. The sound field in that region can be described by the homogeneous Helmholtz equation, which means that no source of sound or diffracting objects are contained in Ω . It is also assumed that the information on the desired sound field is represented by the knowledge of the acoustic pressure p(,)x t on the boundary ∂Ω of the reconstruction area. This implies that either the original sound field was measured using an ideally infinite number of omnidirectional microphones continuously arranged over ∂Ω , or that p(,)x t was defined using an analytical model of the desired sound field.

Ω S0 x ∂Ω

Figure 1: Cross-section of the reconstruction volume Ω and of the 3D surface S

Some mathematical tolls and concepts such as operators, spaces of functions and orthogonal projections are used in this paper, and the reader is referred to [6], [7], [10], [11] and [12] for a more detailed explanation of these concepts.

* Much of the theoretical analysis presented here was first presented in Reference [14], at the 23rd IoA Conference on Reproduced Sound 2007

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2 FORMULATION OF THE INVERSE PROBLEM

Let the reconstruction area Ω ⊂ »3 be a region of space limited by a smooth, bounded and simply connected boundary ∂Ω . Assume that the acoustic pressure ψ (,)x t of a sound field is defined over this region, satisfying the homogeneous wave equation

1∂2ψ (x ,t ) ∇2ψ (,x t) − =0 x ∈Ω (1) c2 ∂ t 2 where c is the speed of sound, considered to be uniform over Ω , and the symbol Ω represents the closure of Ω , that is Ω = Ω ⊕ ∂Ω . For a monochromatic sound field with angular frequencyω , equation (1) can be reduced to the homogeneous Helmholtz equation

∇2ψ(x) +k 2 ψ ( x ) = 0 x ∈Ω (2) where k= ω / c is the wave number, and the harmonic time dependence e jω t is implicitly assumed. Let p(x ), x∈∂Ω be the continuous function that represents the value of ψ (,)x t on the boundary ∂Ω . Assume then that the loudspeaker array that is used for the reconstruction corresponds to an ideally continuous monopole source layer arranged over a smooth, bounded and simply connected surface S , as showed in Figure 1. It is also assumed that Ω is contained in S .

As it is shown in [14], if the wave number k in equation (2) is such that the homogeneous Dirichlet problem

∇2ψ(x) +k 2 ψ ( x ) = 0 x ∈Ω  (3) ψ (x )= 0 x∈∂Ω has only the trivial solution, then the same problem with inhomogeneous Dirichlet boundary conditions has a unique solution. This implies that, if the frequency in question is not one of the resonances of the pressure release cavity, the measurement and reconstruction effort can be limited to the boundary ∂Ω of the reconstruction area. Since the interior eigenvalues are discrete, we obtain reconstructability in the interior from the boundary values almost for all wave numbers and, thus, for the time-dependent fields.

Let G(y | x ) be the free field Green function solution to the free field inhomogeneous wave equation

∇2G(|)y x +k2 G (|) y x =δ ( y − x ) (4) y∈S, x ∈Ω and assume for simplicity that this function can be a good model, at a given frequency, of the electro-acoustic transfer function between each loudspeaker, represented by a point source located at y , and any point x in Ω . It is now possible to write an expression of the acoustic pressure ψˆ()x of the reconstructed sound field as the linear superposition of the infinite number of point sources arranged on S . That is

ψˆ (x)= G (y |)() x a y ds () y x∈Ω (5) ∫S where a()y is a complex function representing the driving signal (monopole strength) of each loudspeaker. In view of the uniqueness of the Dirichlet problem when the problem does not involve Vol. 30. Pt.2. 2008

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one of the resonance frequencies, the reconstructed sound field ψˆ()x equals the desired sound field ψ ()x if the acoustic pressure is correctly reconstructed on the boundary ∂Ω . That is to say, provided the loudspeaker driving function a()y is such that

p(x)= ( Ha )()x = G (|)() y x a y ds () y x∈∂Ω (6) ∫S

Equation (6) is a Fredholm integral equation of the first kind. This equation represents an inverse problem that is, in general, ill-posed. For the definition of an ill-posed problem the reader can refer to [7] or [11].

After having defined the inner product between two functions a()y and b()y as

a()()()()()y b y= a y b y ds y (7) ∫S the adjoint operator (H+ p )(y ) of (Ha )(x ) can be defined to be such that

(Ha )()()x p x= a ()( y H+ p )() y (8) which is a generalization of transposing a matrix. Under the proper assumptions of smoothness of the kernel G(y | x ) , the operator H defined in (6) is compact [8, p. 454]. As a consequence of the compactness of H , the adjoint operator H + is compact [8, p.416]. It can be observed that H + has the form [10]

(H+ p )()y = G (y | x )() p x ds () x y ∈ S (9) ∫∂Ω and can be understood as a “time reversed” acoustic propagation of an infinite distribution of monopole sources on ∂Ω to a point y ∈ S .

Let NH() be the null-space of the operator H , defined as the set of functions a()y such that

N( H )= { a (y ) : ( Ha )( x )= 0} (10)

The null space of an operator, as explained in [12], can be understood for the case under consideration as the set of loudspeaker driving functions for which, at the considered frequency, the reconstructed acoustic pressure profile pˆ( x )= 0 .

As it was shown in [14], it is possible to apply the singular value decomposition to the operator H and to express its action on a function a()y as

(Ha )(x )= ∑ µn an (y ) a ( y ) pn ( x ) . (11) n=1

where µn are the singular values of the operator and an ()y and pn ()x its singular functions, also called array modes [12]. It is also possible to represent the function p()x as

p()x = ∑ µn an ()()()()()y a y pn x+ Rp x (12) n=1

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The operator R is the orthogonal projection on the null-space of the adjoint operator NH()+ . The latter equation means that any acoustic pressure profile p()x defined over ∂Ω can be expressed as the sum of an acoustic pressure profile that can not be reconstructed by the monopole source distribution on S (the orthogonal projection (Rp )(x ) of p()x on NH()+ ) plus the linear superposition of different orthogonal modes pn ()x that can be reconstructed by the monopole source distribution on S .

3 SOLUTION TO THE INVERSE PROBLEM

As it has been shown, the solution to the inverse problem (6) does not always exist, meaning that the inverse problem is ill-posed. If the solution exists or if the choice is made of limiting the reconstruction to the component of the target pressure profile that is orthogonal to the null field of the adjoint operator NH()+ , then the solution can be expressed [14] as

N N 1 a()()()()()()()y= ∑ an y a y an y = ∑ pn x p x an y (13) n=1 n=1 µn

Even if a solution to the inverse problem (6) does not exist, that is if p()x has a nonzero orthogonal projection on NH()+ , the approximation of p()x expressed by

N 1 pˆ()x= G (|) y xp ()()()() x p x a y ds y (14) ∫S ∑ n n n=1 µn

Represents the pressure profile that can be generated by the continuous distribution of sources on S which minimizes the root mean square error

1 e=() p()()()()()x − pˆ x()p ˆ x − p x ds x 2 (15) (∫∂Ω ) as pˆ()x is the orthogonal projection of p()x on the subspace identified by the range of the operator H . If the minimization of the mean square error (15) is the target of the reconstruction, then (13) represents the best solution which can be achieved using a given continuous distribution of monopole sources. It is important to point out that the mean square error is defined over the boundary ∂Ω of the reconstruction area Ω and not on the reconstruction area itself.

Observing equation (13) it can be noticed that if the desired acoustic pressure profile p()x has a nonzero inner product pn ()()xp x with a mode pn ()x related to a very small singular value µn , then the related mode an ()y will become very large due to the inverse of µn in equation (13). This means that the reconstruction of some modes of the target pressure profile might require the loudspeaker array to generate a large amount of energy. Furthermore, if the acoustic pressure profile p()x is perturbed such that

δ p ()()()x=p x +δ pn x (16)

then the effect of this perturbation is amplified by a factor 1 µn in the loudspeaker driving function a()y , obtaining a perturbed solution

δ a ()y aδ a()()y= y + n . (17) µn Vol. 30. Pt.2. 2008

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In more detail, it holds that [7, p.91]

aδ ()()y y− a 1 = . (18) δ p ()()x− p x µn

This error amplification factor can become very large because of the roll-off of the singular values, which accumulate in general at 0 [14]. In a practical sense, this means that the presence of small singular values implies that a small error in the measurement of the desired acoustic pressure profile p()x or the inaccuracy in the positioning of the loudspeakers can have a devastating effect on the reconstructed sound field, The reconstructed field might then differ largely from that desired because of the presence of small errors in the data.

There are many different ways of regularizing the solution to the ill-posed problem (6) and to compute an approximate solution aˆ()y that does not generate an exact sound field reconstruction, but is more robust than the solution given by (13). One of the regularization methods is the spectral damping of the singular values of the integral operator H . It has been explained that it can be decided not to attempt the reconstruction of the component of the target pressure profile that do not belong to the range of the operator. In other words, the wise choice can be made not to attempt the reconstruction of what can not be reconstructed. In a similar way it can be chosen not to attempt the reconstruction of those modes which are related to very small singular values. This operation has the mathematical meaning of reducing the dimensions of the subspace defined by the range of the integral operator. In practice, with this operation it is possible to avoid the usage of the large amount of energy required for the reconstruction of the “critical” modes and it is also possible to avoid the dangerous error amplification shown by equation (18). An alternative regularization method is the Tikhonov regularization, which will not be discussed in this paper and the reader is invited to refer to [6] or [7].

4 CONCLUSIONS

The problem of sound field reconstruction has been expressed in the form of an inverse problem. The reconstruction system has been assumed to be constituted by a continuous distribution of monopole sources arranged on the surface of a closed, simply connected surface that constrains the reconstruction volume. The problem has been formulated as a Fredholm integral equation of the firs kind. The unknown of the problem is the function representing the driving signals of the monopole sources, while the kernel of the integral is the free filed Green function. It has been emphasized that, under appropriate conditions on the operating frequency, the accurate reconstruction of the target pressure profile on the boundary of the reconstruction volume implies the accurate reconstruction of the desired sound field over the entire reconstruction volume.The presented inverse problem has been proved to be ill-posed and therefore an exact solution can not, in general, be achieved for an arbitrary pressure profile. In more detail, the reconstruction must be limited to the component of the target pressure profile belonging to the subspace identified by the range of the integral operator can. It has also been shown that, because to the roll-off of the singular values of the operator, the reconstruction of the modes corresponding to small singular values can be critical in terms of the amount of energy involved and because of the amplification of errors. It has been explained that an approximate but more robust solution can be computed by applying a regularisation method. As an example of regularisation scheme, the spectral damping method has been briefly described.

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5 REFERENCES

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