A wavenumber approach to characterizing the diffuse field conditions in reverberation rooms
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1 PROCEEDINGS of the 22 nd International Congress on Acoustics Isotropy and Diffuseness in Room Acoustics: Paper ICA A wavenumber approach to characterizing the diffuse field conditions in reverberation rooms Mélanie Nolan (a), Efren Fernandez-Grande (b), Jonas Brunskog (c), Antoine Richard (d) Cheol-Ho Jeong (e) Acoustic Technology, Department of Electrical Engineering, Technical University of Denmark, Denmark (a) melnola@elektro.dtu.dk (b) efg@elektro.dtu.dk (c) jbr@elektro.dtu.dk (d) apar@elektro.dtu.dk (e) chj@elektro.dtu.dk Abstract This study proposes a wavenumber analysis method for evaluating the diffuse field conditions in a reverberant space. The wavenumber (or angular) spectrum, which results from expanding an arbitrary sound field into a plane-wave basis, is used to characterize the spatial properties of the observed sound field. Subsequently, the obtained angular spectrum is expanded into a series of spherical harmonics, and the multipole moments from this spherical expansion are used to characterize the wave field, in terms of both isotropic conditions and phase distribution. The paper examines the validity of the method and investigates how the results relate to the existing theory for the diffuse sound field in a reverberation room, based on Waterhouse s random wave model. The analytical framework is presented, and the proposed methodology evaluated numerically based on simulated measurements using a spherical microphone array. Keywords: Diffusion, Isotropy, Wavenumber transform.
2 A wavenumber approach to characterizing the diffuse field conditions in reverberation rooms 1 Introduction Reverberation chambers constitute the primary instrument for a number of standardized measurements; some of the most important being sound power determination, sound absorption and sound transmission loss measurements. A central concept in most theories about reverberation room measurements is that of a diffuse sound field, although various and equivocal definitions can be found in the literature [1]. The following two formulations [2] seem to have gained acceptance and are adopted here: (i) in a diffuse sound field there is equal probability of energy flow in all directions; (ii) a diffuse sound field comprises a infinite number of plane propagating waves with random phase relations, arriving from uniformly distributed directions. In reference [2] Jacobsen shows that the condition stated in definition (i) is achieved provided that the sound field corresponds to definition (ii). Consequently, definition (ii) is used as a basis for describing the diffuse sound field conditions in this study. Formulation (ii) is fundamental in the stochastic random wave theory originally developed by Waterhouse [3], although this model is only suitable from a conceptual point of view. Hence, most of the work on statistical room acoustics has been devoted to theory, as there is no direct metric for any of these concepts and consequently, no practical metric for determining quantitatively the diffusion in a given room. Methods have been proposed for evaluating the degree of diffusion, such as cross-correlation observations between pressure measurements at neighboring positions [4]. Properties of the field correlation functions are of particular interest as they essentially determine the statistics of the variables of interest. In [5], Jacobsen et al. presented a new and efficient method of determining spatial correlation functions in a room, suitable to other quantities than the sound pressure. In this regard, noteworthy and perhaps overlooked is the work by Ebeling [6] who interpreted the cross-correlation function derived by Cook et al. [4] in the spatial frequency domain. Subsequently, he proposed a multipole expansion of the spatial correlation function leading to a measure for spatial diffusivity. However, as his interest is concentrated on the spatial correlation function of the sound pressure and thus, on the mean square pressure, the phase distribution of the actual sound field is disregarded. This, at most, leads to a measure for isotropy of the wave field. The present study proposes a wavenumber analysis method for evaluating the diffuse field conditions in a non-anechoic enclosure, in terms of both isotropic conditions and phase distribution. The analytical framework is far simpler than the theory developed in [6], in that it considers the actual pressure field directly. The corresponding wavenumber (or angular) spectrum is evaluated, based on which the diffusivity is characterized via the coefficients of its spherical harmonic expansion. The paper examines the validity of the method and investigates how the results relate to the existing theory for the diffuse sound field in a reverberation room, based on Waterhouse s random wave model [3]. The proposed methodology is evaluated numerically based on simulated measurements using a spherical microphone array. 2
3 2 Theoretical background 2.1 Brief summary of the theory The proposed method consists of expressing an arbitrary sound field on an array of sensors as the superposition of a set of plane waves: ψ 1 (r 1 ) ψ 2 (r 1 ) ψ n (r 1 ) c 2 p(r) = [ ] [ ] = Wc, (1) ψ 1 (r m ) ψ 2 (r m ) ψ n (r m ) c n where p(r) is the measured sound pressure, c is a complex coefficient vector and W is a matrix containing the plane wave functions e jk r. Pure-tone sound fields are considered in this paper. The sound pressure measured at each of the M transducers is expressed as c 1 n p(r m ) = P(k i )e jk i r m = c i e jk i r m, (2) i=1 where P(k i ) = c i = P(k i ) e jφ(p(k i )) corresponds to the wavenumber (or angular) spectrum with P(k i ) and φ(p(k i )) its magnitude and phase, respectively. A spherical coordinate system r m = (r, θ m, φ m ) = (r, Ω m ) is considered where the direction of the waves is given by k i = (k, Ω i ). Subsequently, the magnitude of the obtained complex angular spectrum is expanded into a series of spherical harmonics: n i=1 n P(k, Ω) = A mn (k)y m n (Ω). (3) n=0 m= n Since the spherical harmonics are orthonormal, the complex coefficients A mn (k) of the expansion can be calculated from 2π π A mn (k) = P(k, Ω) Y m n (Ω)dΩ. (4) 0 0 A wave field is termed isotropic if the wavenumber vectors of the incident plane waves are uniformly distributed over all angles of incidence (corresponding to a sinusoidal distribution of the polar angles and a uniform distribution of the azimuth angles) [2]. In this case, the magnitude of the angular spectrum is constant over the entire solid angle. Consequently, all the energy of the angular spectrum P(k, Ω) resides on the monopole moment A 00 (k) of its spherical harmonic expansion in equation (3). This will not be the case if the contributing waves 3
4 cover just a partial section of the solid angle, as the higher-order moments would characterize the wave field. The energy of the n-th order moment is given by n=0 n m= n A mn (k) 2, so that the relative energy of the monopole contribution can now be expressed as A 00 (k) 2 ι(k) = A mn (k) 2. (5) n=0 n m= n This quantity is here suggested as a potential isotropy indicator and will be denoted by ι(k) in the following. The measure ranges between zero and one and equals unity in the case where the flow of acoustic energy is equal in all directions. Conversely, it approaches zero if the incident waves propagate in a single direction. Similarly, the phase of the obtained complex angular spectrum φ(p(k, Ω)) in P(k, Ω) = P(k, Ω) e jφ(p(k,ω)) is expanded into a series of spherical harmonics: n φ(p(k, Ω)) = B mn (k)y m n (Ω), (6) n=0 m= n where the complex coefficients B mn (k) of the expansion can be calculated from 2π π B mn (k) m = φ(p(k, Ω))Y n (Ω)dΩ. (7) 0 0 It is interesting to note that equations (4) and (7) correspond to spherical Fourier transforms [7]. A diffuse sound field, besides isotropy, is characterized by the phase of the incident waves being random and uniformly distributed between 0 and 2π [2]. In this case, the corresponding phase of the angular spectrum φ(p(k, Ω)) follows such distribution. Consequently, it is expected that higher-order moments from the spherical harmonic expansion in equation (6) are necessary to describe the phase of the angular spectrum (that is, B mn (k) 0 for (m, n) (0,0)). This will not be the case if the contributing waves are of equal phase, as only the monopole moment would characterize the wave field. 2.2 Simulation study A simulation study is presented to validate the theoretical considerations introduced in section 2.1. In this section, we consider the ideal wavenumber spectrum; that is, as if the wavenumber spectrum P(k, Ω) was estimated perfectly (hence disregarding numerical errors). The complex coefficients A mn and B mn from the spherical harmonic expansions in equations (3) and (6) are calculated from the theoretical wavenumber spectrum, discretized into 1000 points. Three reference test cases are considered, where the sound field is modeled as: (a) a single propagating plane wave; (b) a perfectly isotropic wave field, where all propagating plane waves 4
5 have the same phase; (c) a perfectly isotropic wave field, where the phase of the propagating plane waves is random, with values between 0 and 2π. Figure 1 shows the magnitude and phase of the theoretical wavenumber spectrum, along with the first five moments (i.e. up to n=4) from their respective spherical harmonic expansions. The moments are displayed in terms of their normalized energy ( (1 N) n m= n A mn 2 and (1 N) n m= n B mn 2, respectively), where the normalization N corresponds to the number of complex coefficients contributing to each moment. Figure 1(a) shows the case of a single propagating plane wave. It is apparent that all moments have equal normalized energy and that the isotropy indicator in equation (5) approaches zero. Indeed, the magnitude and phase of such a spectrum (see figure 1(a)) correspond to a Dirac delta function over the sphere: δ(cos θ cos θ ) δ(φ φ ), the spherical Fourier coefficients of which are simply the spherical harmonics [7] A mn = B mn = [Y m n (θ, φ )]. As an illustration of the equality in the normalized energies, we consider the dipole moment of the spherical harmonic expansion of the magnitude of the wavenumber spectrum: A m1 2 = 1 3 ( A A A 11 2 ) = 1 3 ( Y 1 1 (θ, φ ) Y 1 (θ, φ ) Y 1 (θ, φ ) 2 ) m= = 1 3 ( 3 8π sin θ e iφ + 3 4π cos θ + 3 8π sin θ e iφ ). Basic calculus shows that 1 1 A 3 m= 1 m1 2 = 1 = Y 4π 0 0 (θ, φ ) 2 = A Analogous derivations hold for the normalized energies of the higher-order moments. Figure 1(b) displays the case of an isotropic sound field (that is, uniform distribution of the directions of propagation) with waves of equal phase. It can be seen that the energy of both the magnitude and phase of the wavenumber spectrum resides on the monopole moment. Analytically, the magnitude and phase of such a spectrum correspond to a constant function over the sphere (see figure 1(b)), the spherical Fourier coefficients of which are proportional to δ n0 δ m0 [7]. Consequently, the magnitude and phase of the wavenumber spectrum are represented using the zero-th order spherical harmonic only. Figure 1(c) illustrates the case of a diffuse sound field (where the term diffuse is here understood as in formulation (ii); that is, uniform distribution of the directions of propagation and random phase relations). As the sound field is isotropic, the energy of the magnitude of the wavenumber spectrum naturally resides on the monopole moment. In this case however, the higher-order moments of the spherical harmonic expansion in equation (6) are needed to describe the phase of the angular spectrum (i.e. B mn (k) 0 for (m, n) (0,0)). Consequently, comparison of results from figures 1(b) and 1(c) confirms the well-established idea [2] that an isotropic sound field is not necessarily diffuse (although any diffuse sound field is isotropic). 5
6 (a) (b) (c) Figure 1: Normalized energy of the first moments from the spherical harmonic expansions of the magnitude and phase of the theoretical wavenumber spectrum resulting from (a) a single propagating plane wave; (b) a perfectly isotropic wave field, where all propagating plane waves have same phase; (c) a perfectly isotropic wave field, where the phase of the propagating plane waves is random, with values between 0 and 2π. Frequency: 500 Hz. 6
7 3 Numerical results A numerical study is conducted to examine the validity of the proposed methodology based on simulated measurements using a spherical microphone array. Spherical microphone arrays are widely used for the analysis and reconstruction of complex sound fields [8,9,10], and are particularly well suited for applications in non-anechoic enclosures, such as reverberation rooms [11], where the sound waves impinge on the array from multiple directions. The array used for the simulated measurements is a 9.75 cm radius rigid sphere array with 64 microphones centered at the origin of coordinates. The simulated sound field is generated by a set of monopoles with equal volume velocities Q= m 3.s -1. These monopoles are distributed according to various configurations meant to vary the isotropic conditions and the phase distribution of the resulting sound field. The total sound pressure (incident plus scattered fields) measured on the rigid sphere at (a, Ω m ) is generated by L point sources as [10] L p t (a, Ω m ) = jρcq l a 2 h n(kr s,l ) h n (ka) Y n m (Ω m )Y m n (Ω s,l ), l=1 n n=0 m= n (8) where a is the radius of the sphere, and Q l and (r s,l, Ω s,l ) are the volume velocity and the position of the point source l, respectively. Subsequently, a plane-wave basis of 1000 plane waves, which directions of propagation are distributed over a sphere centered at the origin of coordinates, is used to estimate the wavenumber spectrum. The complex coefficient vector c corresponding to the angular spectrum is estimated using equation (1). This is an ill-conditioned and underdetermined problem that must be inverted using regularization. Tikhonov regularization [12] is applied, along with the L- curve criterion as a parameter-choice method [12]. Normally distributed noise of 30 db signal-tonoise ratio is added to the simulated measurements. Figure 2 illustrates three distributions of monopoles that conform three different sound fields, along with the spherical harmonic expansion of the magnitude of the wavenumber spectrum resulting from these distributions (displayed in terms of the moments energy n m= n A mn 2 ). The positions of the monopoles are chosen in order to vary the isotropic conditions of the resulting wave field. In all three cases, all the sources are placed at a distance of 3 meters from the surface of the array (black point on the figure), ensuring equal phases in the resulting angular spectrum. The first case (left) displays a uniform distribution of point sources over a sphere centered at the origin of coordinates (their positions being derived from the solution to the socalled Thomson problem). In the second and third cases (center and right, respectively), the monopoles are distributed over half a sphere and a quarter of a sphere, respectively. In the fully isotropic case (figure 2, left), it is apparent that the magnitude of the wavenumber spectrum can be represented using the zero-th order spherical harmonic only, yielding an isotropy indicator close to unity. As the isotropy of the sound field decreases (figure 2, center and right), higher- 7
8 order moments are needed to describe the magnitude of the angular spectrum (dipole (i.e. n=1) when the contributing waves cover half of the solid angle; dipole (i.e. n=1) and quadrupole (i.e. n=2) when they cover a quarter of the solid angle). To examine the influence of the phase of the incoming waves, figure 3 illustrates two distributions of monopoles radiating in phase, and uniformly distributed over the spherical coordinates Ω = (θ, φ). The magnitude of the corresponding wavenumber spectrum can be represented using the zero-th order spherical harmonic only. The upper case corresponds to all the monopoles being placed at a distance of 3 meters from the surface of the array, ensuring equal phase in the resulting wave field. In the second case (bottom), the distance from the surface of the array is random, with values between 2 and 6 meters. This results in a random phase distribution of the corresponding wave field. In the case where the resulting waves have equal phase (figure 3, top), it is apparent that the phase of the wavenumber spectrum can be represented using the zero-th order spherical harmonic only. Conversely, in the case where the incoming waves have random phases (figure 3, bottom), higher-order moments are needed to describe the phase of the angular spectrum, which is in turn an indication of the random phase distribution (or rather the non-constant phase) of the corresponding waves. This observation has further implications (for instance regarding the intensity field properties) that are not addressed here. In addition, it is interesting to note the residual energy in the higher-order moments of the magnitude expansion (figure 3, bottom center) due to the spherical spreading when the sources are positioned at a random distance from the surface of the array. 4 Discussion The theory presented in this paper is based on direct observations of the pressure field, which provides a complete characterization of the wave field in terms of both isotropic conditions and phase distribution. In this regard, the underlying theories of [6] and the present paper are different. The former is based on a multipole expansion of the spatial correlation function of the sound pressure, and consequently ignores the phase distribution of the waves (as an illustration, we can here consider the sound field in a spherical room with a source in its centre: using the metric suggested in [6] it would seem that the sound field is diffuse, whereas it is clear that this is not the case. The metric presented in the present paper would deem it as isotropic, but non-diffuse, due to the deterministic distribution of the phase of the waves, as illustrated in figure 3). However, the validity of the suggested method at low frequency is still to be examined, as the random wave model is no longer sustainable in the region of low modal overlap, where the sound field is likely to be dominated by a single mode [13]. 5 Conclusion This study examines a wavenumber approach to characterize diffusion, based on the spherical harmonic expansion of the angular spectrum of an arbitrary sound field. Numerical tests using arbitrary distributions of point sources indicate the validity of the suggested method and confirm the well-established idea that an isotropic sound field is not necessarily diffuse (although any 8
9 Figure 2: Spherical harmonic expansion of the magnitude of the wavenumber spectrum resulting from a set of monopoles placed at a distance of 3 meters from the surface of the array. The point sources are uniformly distributed over a sphere (left); distributed over half a sphere (center); distributed over a quarter of a sphere (right). Frequency: 1000 Hz. Figure 3: Spherical harmonic expansion of the magnitude and phase of the wavenumber spectrum resulting from a set of monopoles uniformly distributed over the full solid angle. The point sources are placed at a distance of 3 meters from the surface of the array (top); at random distances from the surface of the array (bottom). Frequency: 1000 Hz. 9
10 diffuse sound field is isotropic). A simple metric is suggested that has potential to serve as an isotropy indicator in complex spaces. Besides isotropy, the results highlight the importance of a phase indicator that considers the random phase (or not) of the waves in the sound field. Acknowledgments This work is founded by the Oticon Foundation. References [1] T. J. Schultz, Diffusion in reverberation rooms, J. Sound and Vib. 16, (1971). [2] F. Jacobsen, The Acoustics Laboratory, Technical University of Denmark, Report No. 27. The diffuse sound field (1979). [3] R. V. Waterhouse, Statistical properties of reverberant sound fields, J. Acoust. Soc. Am. 43, (1968). [4] R.J. Cook et al., Measurement of correlation coefficients in reverberant sound fields, J. Acoust. Soc. Am. 27, 1072 (1955). [5] F. Jacobsen and T. Roisin, The coherence of reverberant sound fields, J. Acoust. Soc. Of Am., 108, (2000). [6] K.J. Ebeling, Statistical properties of random wave fields, in Physical Acoustics, Principles and Methods, edited by W.P. Mason and R.N. Thurston (Academic Press, New York, 1984), Vol. XVII, Chap. 4, pp [7] B. Rafaely, Fundamentals of Spherical Array Processing, Springer Topics in Signal Processing 8, Springer-Verlag Berlin Heidelberg, [8] F. Jacobsen, G. Moreno-Pescador, and E. Fernandez-Grande, Near-field acoustic holography with microphones on a rigid sphere (L), J. Acoust. Soc. Of Am., 129(6), (2011). [9] E. Fernandez-Grande, Reconstruction of arbitrary sound fields with a rigid-sphere microphone array, 166 th Acoust. Soc. Am. Meeting, San Francisco, 2-6 th December, 134, 3998 (2013). [10] E. Fernandez-Grande, Sound field reconstruction using a spherical microphone array, J. Acoust. Soc. Am. 139 (3), [11] M. Nolan, E. Fernandez-Grande and C.-H. Jeong, Characterization of diffusivity based on spherical array processing, Proceedings of Internoise 2015, San Francisco, California. [12] P.C. Hansen, Discrete Inverse Problems: Insight and Algorithms, vol. 7 of Fundamentals of Algorithms, SIAM, Philadelphia, PA, [13] F. Jacobsen and A.R. Molares, Sound power emitted by a pure-tone source in a reverberation room, J. Acoust. Soc. Am. 126 (2), (2009). 10
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