Rendering of Directional Sources through Loudspeaker Arrays based on Plane Wave Decomposition

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1 Rendering of Directional Sources through Loudspeaker Arrays based on Plane Wave Decomposition L. Bianchi, F. Antonacci, A. Sarti, S. Tubaro Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano Piazza Leonardo da Vinci,, 1, Milano, Italy lbianchi/antonacc/sarti/tubaro@elet.polimi.it Abstract In this paper we present a technique for the rendering of directional sources by means of loudspeaker arrays. The proposed methodology is based on a decomposition of the sound field in terms of plane waves. Within this framework the directivity of the source is naturally included in the rendering problem, therefore accommodating the directivity into the picture becomes much simpler. For this purpose, the loudspeaker array is subdivided into overlapping sub-arrays, each generating a plane wave component. The individual plane waves are then weighed by the desired directivity pattern. Simulations and experimental results show that the proposed technique is able to reproduce the sound field of directional sources with an improved accuracy with respect to existing techniques. I. INTRODUCTION In this paper we propose a method for the rendering of directive sound sources by means of a loudspeaker array. The problem of sound field rendering is of growing interest in multimedia for potential applications in immersive gaming, telepresence and all the scenarios in which the delivery of a realistic sound impression is an issue, and rendering virtual sources along with their directivity is of crucial importance for improving the immersivity. Different solutions exist in the literature for the purpose of sound field rendering. The most popular ones are Wave Field Synthesis (WFS) [1], []; Ambisonics and Higher Order Ambisonics (HOA) [] ; and least squares solution of the rendering equation (LS) [], [5]. One main issue in rendering systems is that of rendering a radiation pattern of the virtual sound sources different from the omnidirectional one. There are evidences in the literature [6] that the disparities in the sound field at different listening positions created by the directional properties of sources greatly contribute in making realistic and immersive the sound scene. The importance of directive sources comes at light also when we are interested in rendering a desired reverberant effect. The accuracy of this kind of rendering, indeed, depends simultaneously on the shape and properties of the boundaries MMSP 1, Sept. - Oct., 1, Pula (Sardinia), Italy /1/$1. c 1 IEEE. and on the directivity characteristics of the source [6]. Some works in the literature focus on this topic, in both nearfield [7] (for Ambisonics) and farfield [8] (for WFS). In [6] the author proposes a methodology for synthesizing WFS filters through a limited set of spherical harmonics. Similarly, in [9] the authors integrate directional sound sources in the WFS framework by means of circular harmonic expansion. With respect to previous works in the literature, authors in [9] focus on the efficiency of the implementation. In this paper we propose an alternate methodology, partially inspired by the novel idea of plenacoustic rendering. The plenacoustic function, first presented in [1], refers to a description of the acoustic pressure as a function of position and time. In particular, in [1] a method is proposed to infer the sound pressure at arbitrary time instants and spatial positions, based on a preliminary sampling. In [11] authors partially redefine the concept of plenacoustic function including directional information and devise a solution for measuring the plenacoustic function by means of a microphone array. Authors in [11] adopt a parameterization of the plenacoustic function that describes the sound field in terms of acoustic rays. The microphone array operates as an Observation Window, as it samples rays passing through it. More specifically, a uniform linear array of microphones is used and it is subdivided into sub-arrays. Each sub-array senses acoustic rays passing through its center (the central microphone of the sub-array). The acoustic rays passing through the whole extension of the array are then found by considering adjacent sub-arrays. In this paper we aim at using the plenacoustic framework for sound field rendering purposes. The basic idea is that of reversing the plenacoustic sensing paradigm adopted in [11]. A loudspeaker array, therefore, acts as an Emission Window (EW) and it is subdivided into sub-arrays, each in charge of rendering the planar portion of the acoustic wavefront that, departing from the virtual acoustic source, passes through the considered sub-array. The overall acoustic wavefront is obtained by the superposition effect after a phase alignment stage, which guarantees that the planar wavefronts produced by all the sub-arrays are aligned in phase. In order to render planar acoustic wavefronts, the beamforming filter of each sub-

2 array is obtained by means of the Superdirective Beamforming (SD), described in [1]. Coming back to the main purpose of this work, the directivity is implemented in the plenacoustic rendering framework by weighing the planar wavefronts of each sub-array through the directivity pattern of the virtual source to be rendered. The main advantage of the proposed technique lies in the fact that the directivity of the sound source is easily included in the rendering framework, whereas other techniques, such as WFS, need to adopt a preliminary analysis based on circular harmonics. Simulations and experimental results confirm that a more accurate rendering is possible with respect to other techniques available in the literature. The rest of the paper is organized as follows: section II formulates the problem and introduces the notation. Section III illustrates the theory of plenacoustic rendering. Moreover, in Section III, Superdirective Beamforming is summarized for what concerns the rendering from individual sub-arrays. Section IV discusses the problem of integrating the directivity into the framework and extends the application to wideband signals. Finally, section V shows simulations and experimental results to prove the accuracy of the methodology presented in this paper. II. PROBLEM STATEMENT Let us consider an array of M loudspeakers. Transducers locations in Cartesian coordinates are x m = [ ] T x m y m, m =,..., M 1. We adopt the linear array configuration with uniform spacing q along the x axis and we fix the reference frame so that y m = y for all the loudspeakers. The line on which the array lies determines two half-spaces. We are interested in rendering the sound field in the half-space y > y. We assume the sound source to be located in the origin of the reference frame, i.e. in x s = [ ] T. Any other location of the source can be considered as a roto-translation of the coordinate system and therefore is included in the current framework. The virtual source emits the signal s(t) and it is also characterized by the polar pattern, where α is the angle with respect to the x axis and ω is the radial frequency. Our goal is to render the sound field generated by the virtual source, including the radiation pattern. As depicted in Fig. 1, the array is decomposed into maximally overlapped sub-arrays, each consisting of W adjacent loudspeakers. The number of sub-arrays is M W + 1. Our goal is to operate a plane-wave decomposition, and each sub-array is in charge of reproducing a component of the overall soundfield. A generic point in space is denoted with the symbol x = [ x y ] T, and the spacetime dependent sound pressure measured in x at time t is p(x, t). Let us consider for a while that the signal s(t) is narrowband, and the wavenumber vector is k and k = k. Later on, we will extend our discussion to wideband signals by means of Fourier analysis. In a -dimensional sound field the dispersion equation can be written as k = (ω/ν) = kx +ky, where k x and k y are the components of the wavenumber vector k and ν is the speed of sound. We can therefore write 1 M W 1 M W y 1 W 1 W M W 1 M W M M 1 q x r α x y=y Figure 1: Reference frame and decomposition of the loudspeaker array into sub-arrays. that k y = k kx. Considering a plane wave travelling in direction α, the Cartesian components of its wavenumber vector are k x = (ω/ν) cos α and k y = (ω/ν) sin α. In this context we can reformulate the Plane Wave Decomposition (PWD), described in [1] for -dimensional fields, as p(x, t) = D(k x )e j(kt x ωt) dk x, (1) R The -dimensional position vector x is related to polar coordinates by x = r cos α and y = r sin α. The spacefrequency pressure sound field is p(x, ω) and its relationship with is p(x, ω) = π e j ω ν (ˆk T x) dα, () where ˆk = k/k = [ cos α sin α ] T is the wavenumber versor. Eq. () states that p(x, ω) can be thought as the sum of infinite plane waves, weighted by the polar pattern. As a matter of fact, a number of decompositions can be conceived for the pressure sound field. A popular one is the Rayleigh I integral, which claims that the sound field produced by a source in a volume V can be reproduced by an infinite distribution of secondary omnidirectional sound sources located on the boundary S of V. The Rayleigh I integral is used for sound field rendering purposes by Wave Field Synthesis (WFS) methodology [1], []. In this paper we aim at approximating the PWD in () by a finite sum of plane-wave components, rendered by the loudspeaker array. More specifically, each sub-array is in charge of reproducing a plane-wave component with travel direction α i, where α i is the angle formed by the x axis and the position vector of the central loudspeaker of the ith subarray, and it is given by ( ) ( ) yi+ W/ y α i = arctan = arctan ; () x i+ W/ x i+ W/ the operator denotes the floor truncation. Notice that we could also demand each sub-array to produce a set of plane waves rather than only a single plane-wave component. Some preliminary results showed that a tradeoff issue between the computational cost (the burden of the rendering engine is proportional to the number of components rendered by each sub-array) and the accuracy of the rendering exists. Therefore

3 we decided to synthesize only one plane wave for each subarray. A choice of this sort is not novel in the literature. As an example, in WFS a windowing function is introduced, which selects the active loudspeakers based on considerations on the geometry of the rendering problem []. Note also that the angle α i coincides with the angle between the virtual source and the loudspeaker. If a different location of the source would be rendered, () modifies accordingly. The distance between the origin and the central loudspeaker of the ith sub-array is r i = x i+ W/ + y i+ W/. () The space-frequency pressure sound field p(x, ω) is then approximated by p(x, ω) M W i= d i (ω)e j ω ν (ˆk T i x) = p P (x, ω), (5) where ˆk i = [ ] T cos α i sin α i. In section IV we derive the expression of the weight d i (ω) in order to reproduce the polar pattern. Section III, on the other hand, concerns the synthesis of planar wavefronts by sub-arrays. III. RENDERING PLANE WAVES In this section we investigate on the synthesis of plane waves through beamforming. The idea of applying beamforming tecniques to loudspeaker arrays is based on the duality between microphones and loudspeakers and it has been previously investigated in the literature [1], [15]. In our framework many beamforming techniques can be employed to synthesize plane waves from each sub-array. We have adopted, among the possible choices, the data-independent reformulation of the MVDR beamformer, at our knowledge introduced in [1]. In order to include in the framework wideband signals, we operate a subdivision of the frequency axis into L sub-bands. Let ω l be the central frequency of the lth sub-band and ω s,i = ω l ν q cos α i (6) is the spatial frequency of a plane wave with time radial frequency ω l and travel direction α i, i =,..., M W. We can express the beam-steering vector for the ith sub-array as ã i (l) = [ e j W/ ωs,i e j W/ ωs,i ] T. (7) Denoting the noise covariance matrix for the lth sub-band with R(l), the MVDR spatial filter f i (l) is obtained as [16] fi (l) = arg min b(l) H R(l) b(l) b(l) subject to b(l)hã i (l) = 1, where ( ) H denotes the Hermitian transpose. The solution of (8) is R 1 (l)ã i (l) fi (l) = ã H i (l) R 1 (l)ã i (l), (9) Under the assumption of spherical isotropic or diffuse noise field [1], the noise covariance matrix R can be replaced by (8) F \ W. F M W \ W d (l) e jω l τ d M W (l) e jω l τ M W Merge \ M h(l) Figure : Block diagram of the proposed rendering framework. the coherence matrix Γ. The element at row r and column c of the coherence matrix is [1] ( ) Γ(l) c π xr x c F s l r = sinc, (1) c L where x r and x c are the coordinates of the r and cth elements of the loudspeaker array, respectively; and F s is the sampling frequency. The beamforming filters of the ith sub-array are collected in the matrix F i as F i = [ fi ()... fi (L 1) ]. (11) IV. RENDERING THE SOURCE DIRECTIVITY The polar pattern is the energy of the sound field in far field conditions all around the source location. For a pointlike source, the polar pattern is related to the sound field on a circle of radius r through [1] ( ) e j(ω/ν)r 1 = p(α, r, ω), (1) r where the sound field has been expressed, for convenience, in polar coordinates. In order to accommodate the synthesis of the directivity in the proposed framework, we set the weights d i (ω) in (5) to be equal to the directivity function evaluated at angles α i, i.e. d i (l) = D(α i, ω l ). (1) Before approaching the problem of finding a rendering filter of the mth loudspeaker from different sub-arrays, we must align in phase the beamforming filter f i. In particular, we need to ensure that the the phase in sub-band l is coherent with the position of the virtual source. At this purpose we introduce the vector τ i as the travel time from the virtual source to all the loudspeakers in the ith sub-array, i.e. τ i = [ xi+ ν... x i+w 1 ν ] T. (1) In order to obtain the rendering filter h(l) for the whole array, as depicted in the block diagram in Fig., we define a specific merging operation to sum all the filter weights relative to the same loudspeaker. At this purpose the beamforming coefficient f i,m i (l) of the (m i)th loudspeaker in the ith subarray is weighed to satisfy the Constant Overlapp and Add constraint.

4 More specifically, the rendering coefficient relative to the mth loudspeaker can be expressed as h m (l) = i 1 i=i fi,m i (l) e jωlτ i,m i d i (l) z(m i), (15).5.5 where i = max(, m W ), i 1 = min(m W, m + W ). The index i = i,..., i 1 spans all the sub-arrays that include the mth loudspeaker and τ i,m 1 is the (m i)th element of τ i. The windowing function z is a squared Hanning window [17] of length W, i.e. z(w) = (.5 ( 1 cos ( ))) πw, w =,..., W 1. W 1 (16) The rendering coefficients h m (l) are collected into the frequency responses h m as h m = [ h m ()... h m (L 1) ]. (17) Finally, the matrix H collects the frequency responses h m as H = h.. (18) h M (a) R{ p D (x u, ω} (b) R{ p P (x u, ω} The M time-domain rendering filters h are obtained with the L-point Inverse Fourier transform over the rows of the matrix H. In particular, } h m (n) = IDFT l { hm = 1 L 1 h m (l)e jπnl/l (19) L is the nth element of the filter to be applied to the mth loudspeaker and L is the number of taps of the filter. l= V. RESULTS In this section we present some simulations and experiments to validate the presented technique. We set up an array composed by M = loudspeakers, with a uniform spacing of q = 6. cm. The spatial Nyquist frequency is f = ν/(q).7 khz, and ν = m/s is the speed of sound at. The amplitude of all the depicted sound fields has been normalized, so that the amplitude range is between 1 and 1. A. Simulations Following results are presented for frequencies in the set 5 Hz, 5 Hz, 1 khz, khz. We compare the results obtained with the proposed rendering framework (PR) and methods available in the literature, namely Geometric Rendering (GR) []; and an implementation of Wave Field Synthesis (WFS) that accommodates the rendering of directive sources [9]. In order to simulate the sound fields rendered by the above techniques, we define a grid of control points x u inside the listening area. The listening area has dimensions m m and it is centered at ( m, m). For the proposed methodology, the sound field p P (x u, ω) is simulated at the control points by (c) R{ p G (x u, ω} (d) R{ p W (x u, ω} Figure : Real part of the sound field of a dipole at frequency f = 1 khz (Fig. a) compared with sound fields rendered by PR (Fig. b), GR (Fig. c) and WFS (Fig. d). Amplitudes have been normalized between 1 and 1. applying the rendering filter h(ω) to the propagation function, i.e. p P (x u, ω) = M 1 m= G(x u x m, ω) h m (ω), () where G(x u x m, ω) is the Green s function at x u, given the loudspeaker position x m. Similarly, we compute the sound fields p G (x u, ω) and p W (x u, ω) of GR and WFS, respectively, where the rendering filters are obtained through a duly implementation of the methdologies proposed in [] and [9]. Simulated sound fields allow us to qualitatively compare the accuracy of rendering for different techniques. In order to get a more quantitative result, we simulate the sound field on T = 18 points on a circle of radius r mic = m. Fig. a shows the real part of the desired sound field R{ p D (x u, ω)}, generated by a dipole placed in x s = [ ] T and emitting a narrowband signal at frequency f = 1 khz. Figs. b, c and d show the real part of the sound fields rendered by PR, GR and WFS, respectively. In these simulations we use W = loudspeakers in each sub-array. In order to make easier the comparison between the different rendering methodologies, Fig. shows for PR ( D P ), GR ( )

5 (a) f = 5 Hz (b) f = 5 Hz (a) α C = 9 (b) α C = (c) f = 1 khz (d) f = khz Figure : Desired directivity patterns of a dipole (, red), compared to the directivity pattern rendered by PR ( D P, green), GR (, blue) and WFS ( D W, black) at frequencies 5 Hz (Fig. a), 5 Hz (Fig. b), 1 khz (Fig. c), khz (Fig. d). and WFS ( D W ), compared to the desired pattern ( ) at frequencies 5 Hz, 5 Hz, 1 khz, khz. The directivity patterns are shown for α α α M W, as outside this range all the rendering techniques fail, as also shown in [18]. At low frequencies all the techniques, except GR, produce similar results. On the other hand, at high frequencies the proposed rendering methodology avoids the ripples introduced by GR and follows the desired pattern more closely than WFS. It is also worth noticing that the proposed rendering technique has a computational burden that is comparable to WFS and lower than GR, as the latter requires the inversion of the propagation matrix. In order to show the capability of the proposed methodology to render different directivity patterns, Fig. 5 shows the desired cardioid pattern ( ) and the directivity patterns synthesized by the proposed methodology ( D P ) and GR ( ). The directivity pattern rendered by WFS is not shown because its synthesis would require a preliminary analysis of the desired sound field. For these simulations we use W = 1 loudspeakers in each sub-array. In detail, Figs. 5a, 5b, 5c and 5d show the comparison of directivity patterns synthesized to reproduce a source with a cardioid pattern oriented at directions α C = 9, 1, 15 and 18, respectively. The (c) α C = 15 (d) α C = 18 Figure 5: Desired directivity pattern (, red) oriented at direction α C, compared to the directivity patterns synthesized by PR ( D P, green) and GR (, blue). directivity patterns synthesized by the proposed methodology better approximate the desired pattern with respect to GR, avoiding both the artifacts at angles close to α and α M W and the ripples in the central region of the plots. B. Experiments We proceed to the evaluation on real world data of the proposed methodology, by showing the sound fields measured using the methodology described in [19], [] and already adopted in [18], [1], for the sake of evaluating the accuracy of sound field rendering techniques. We sample the sound field with a cardioid microphone at N mic = 18 positions on a circle of radius ρ = 95 cm centered in x = [ m m ] T. In order to avoid artifacts due to the interference of microphone bodies with the propagating wavefronts, a sequential measurement technique is used. More specifically, the microphone is moved by a precision stepper motor. The sound field is then reconstructed inside the measurement area using an interpolation and extrapolation procedure based on the decomposition of the sound field into circular harmonics. Fig. 6 shows desired and PR sound fields (using W = ) at f = 1 khz. For reasons of accuracy of the visualization we show the measured sound field only in a circle of radius ρ. Despite the artifacts due to the reconstruction methodology, the rendering result is quite accurate. In particular, the proposed methodology enables the

6 .5.5 (a) Measured sound field..5.5 (b) Desired sound field. Figure 6: Sound field of a dipole generated by a real-world implementation of the proposed rendering framework, measured using [19], []. Amplitudes have been normalized between 1 and 1. correct reproduction of the low-energy area in front of the loudspeaker array (α = 9 ). Notice that the desired sound field exhibits a complex structure, as the two lobes of the dipole pattern have opposite phases, and therefore the listening area is splitted in two parts with wavefronts with an offset of λ/, λ = πν/ω being the wavelength. Nonetheless, the rendering engine is able to accurately reproduce this scenario. VI. CONCLUSION In this paper we have presented a methodology to render directional virtual sources for sound field rendering applications. Our methodology is inspired by the idea of plenacoustic function. The loudspeaker array acts as an Emission Window and it is subdivided into sub-arrays. Each sub-array is in charge of rendering a portion of the overall acoustic wavefront. The decomposition of the complex sound field into basis functions is accomplished by means of Plane Wave Decomposition. In this scenario, the rendering of directive sources turns out to be a fairly simple procedure, as the directivity pattern becomes a weighting of the rendering filter. Both simulations and experimental results showed that the proposed methodology is more accurate and less computationally intensive with respect to other techniques available in the literature. From a computational point of view, the fact that the directivity information is inherently accommodated in the framework enables to avoid the preliminary analysis of the sound field, required in WFS. Moreover, the proposed methodology avoids the numerical issues due to the inversion of the propagation matrix, required in LS methods, resulting in a more efficient implementation. As in WFS, the proposed technique does not require to define a listening area. In [19] the author shows that the directivity pattern can be measured through Plane Wave Decomposition, which can be interpreted as multiple beamforming. The duality between directivity pattern and multiple beamforming paves the way to accommodate data-driven rendering in the proposed framework, as already done in WFS. REFERENCES [1] A. Berkhout, D. D. Vries, and P. Vogel, Acoustic control by wave field synthesis, J. Acoust. Soc. Am., vol. 9, pp , May 199. [] J. Ahrens, R. Rabenstein, and S. Spors, The theory of wave field synthesis revisited, in Proc. AES Conv. 1, Amsterdam, NL, May 17, 8. [] M. A. Gerzon, Periphony: With-height sound reproduction, J. Audio Eng. Soc., vol. 1, no. 1, pp. 1, 197. [] F. Antonacci, A. Calatroni, A. Canclini, A. Galbiati, A. Sarti, and S. Tubaro, Soundfield rendering with loudspeaker arrays through multiple beam shaping, in Proc. 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