Computational sound field in a virtual environment via field data in an arbitrary real environment

It is useful to compute sound field of a source in a virtual environment which is different from the measurement environment. For example, some properties of sound source, such as directivity index and frequency response curve, are required to be measured in an anechoic room or free space, but both of them cannot be always accessible. Consequently, it will be useful to compute sound field of a source in free space when sound field of the source is not measured in the free space. In the aforementioned example, the free space is a virtual environment. Based on reciprocity theorem and modal expansion, a method to predict sound field of a source in a virtual environment is given in this paper when the scattering effect of the source can be neglected. Reciprocity theorem builds the relationship between measured sound field and predicted sound field, which plays an important role in the method. Green’s function in the virtual environment is needed in the method. To restrict measurement points on an enclosed surface, the Green’s function is expanded by a set of modes. A simulation is given to examine the validity of the method.


Introduction
Virtual acoustic reconstruction (VAR) which mainly focuses on computer-aided simulation of building acoustics is a well-established tool for the study of the acoustic behavior of such spaces and is often combined with virtual acoustic representation (auralisation) to provide direct aural impression of their response to speech or music signals [1][2][3][4]. To compute sound field stimulated by a source in a virtual environment by VAR, the properties of the source, such as directivity and frequency response, must be known. However the properties of a sound source cannot be measured directly in an ordinary room, and the sound field stimulated by the source with unknown properties in a virtual environment cannot computed by VAR. When the virtual environment is equal to the real environment, sound field reconstruction methods, such as near-field acoustical holography (NAH) [5][6][7][8][9][10][11][12][13][14][15][16], inverse boundary element method (IBEM) [17][18][19], and least-squares method (LSM) [20,21], can be used to compute sound field stimulated by a certain source and the properties of the source do not need to be known in advance. However, when a virtual environment is different from the real environment, sound field reconstruction is not a suitable method to compute sound field stimulated by a certain source in a virtual environment.
In this paper, a method called extended virtual acoustic reconstruction (EVAR) is proposed which maps sound field in an arbitrary real environment to that in a virtual environment without knowing properties of the excitation source in advance when scattering effect of the source can be neglected. EVAR is derived based mainly on reciprocity theorem [22,23]. Use SFR and SFV as the acronyms for sound field in the real environment and that in a virtual environment respectively. To obtain SFV, the Green's function in the virtual environment should be known. However one cannot reconstruct SFV with the Green's function directly based on reciprocity theorem when SFR is not accessible in the whole domain. This problem can be overcome by expanding the Green's function with a set of modes which satisfy homogeneous Helmholtz equation, and the measurement points are restricted on an enclosed surface which contains the sound source. Both pressure and its gradient need to be measured on the measurement surface in EVAR. A simulation is given to examine the validity of this method. This paper is organized as follows. Section II presents the EVAR using reciprocity theorem, which is examined by a simulation in section III. A conclusion is given in section IV.

Theory of EVAR
In this paper, the sound field is considered in frequency domain which is related to time domain by Fourier transform. Let 1 ( ) ϕ r be the SFR which satisfies both the boundary conditions in the real environment and inhomogeneous Helmholtz equation where / k c ω = is wave number, ω is angular frequency, c is sound speed of medium and ( ) S r is source term whose support set is limited in a finite domain D , see Fig.1.
It is well known that 2 ( ) ϕ r can be represented by the The integral in Eq.
(12) builds a mapping between two sound fields satisfying different boundary conditions and is the central part of EVAR. In the next section, a simulation is given to examine EVAR.

Simulations
To illuminate EVAR more clearly and to examine its validity, a simulation is given in this part and the simulation is restricted in the two-dimensional space. Supposing there are six identical point sound sources whose working frequency is 1500 Hz f = in a rectangle cavity. The medium in the cavity is homogeneous and isotropic with sound speed 1500 m/s c = . Length of the rectangle cavity is 6.3 a λ = and width is 5.
is the sound wavelength in the medium. Acoustic boundary conditions of the rectangle cavity are perfectly rigid. The Cartesian coordinate system is shown in Fig.2 Fig.2(b) gives the distribution of the absolute value of SFR 1 ( ) ϕ r in the cavity. The measurement surface D ∂ is consistent with the rigid boundary of the rectangle cavity. Place the same point sources in a two-dimensional ideal waveguide with H height and build the same Cartesian coordinate system as figure 2(a), see Figure  3(a). SFV 2 ( ) ϕ r should satisfy the following boundary conditions in the two-dimensional ideal waveguide case.  (13) where ( , ) x y = r . The Green's function in a twodimensional ideal waveguide has the following form [24]   1 ( ) ϕ r , and 1 ( ) ϕ r is shown in Figure 3(d  Figure   3 (16) which is plotted in Fig.3(c). As Fig.3(c) shows, 2 ( ) x ∆Φ decreases along x because the influence of evanescent waves decreases along x . Through this simulation, the validity of EVAR is verified.

Conclusions
A method called EVAR is proposed in this paper, which can reconstruct SFV by SFR. To make sure that SFR 1 ( ) ϕ r and its gradient are only measured on an enclosed surface D ∂ in the method, the Green's function 0 ( , ) t G r r in the virtual environment is expanded by a set of complete functions { ( )} u α r which satisfy homogeneous Helmholtz equation when applying EVAR. A simulation is given to verify the validity of EVAR. Because there are not any requirements on the boundary conditions in the real environment and in the virtual environment in the EVAR, the method proposed in this paper maybe useful in dereverberation, home theatre, virtual reality, sound source calibration etc.