Proceedings of Meetings on Acoustics

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1 Proceedings of Meetings on Acoustics Volume 19, ICA 213 Montreal Montreal, Canada 2-7 June 213 Architectural Acoustics Session 1pAAa: Advanced Analysis of Room Acoustics: Looking Beyond ISO 3382 II 1pAAa4. Three-dimensional spatial analysis of concert and recital halls with a spherical microphone array Samuel Clapp*, Anne Guthrie, Jonas Braasch and Ning Xiang *Corresponding author's address: Graduate Program in Architectural Acoustics, Rensselaer Polytechnic Institute, 11 8th Street, Troy, NY 1218, clapps@rpi.edu The most well-known acoustical parameters - including Reverberation Time, Early Decay Time, Clarity, and Lateral Fraction - are measured using data obtained from omnidirectional or figure-of-eight microphones, as specified in ISO Employing a multi-channel receiver in place of these conventional receivers can yield new spatial information about the acoustical qualities of rooms, as well as the potential for new parameters that could have greater predictive power in terms of listeners' subjective preferences. In this research, a spherical microphone array was used to measure the room impulse responses of a number of different concert and recital halls. The data was analyzed using spherical harmonic beamforming techniques, along with other direction of arrival estimation algorithms, to understand how the soundfield evolves spatially over time at different points in the room. The results were compared to geometrical acoustic simulations and used to differentiate between listener positions which exhibited similar values for the standard parameters. In addition, new parameters were examined, including soundfield homogeneity and other spatial ratios. Published by the Acoustical Society of America through the American Institute of Physics 213 Acoustical Society of America [DOI: / ] Received 23 Jan 213; published 2 Jun 213 Proceedings of Meetings on Acoustics, Vol. 19, 152 (213) Page 1

2 INTRODUCTION The ISO 3382 standard defines a method for measuring the acoustics of different spaces (Iso, 29). An omnidirectional source and three types of receivers (omnidirectional microphone, figure-of-eight microphone and binaural dummy head) are used to measure how a room transforms a sound signal as it travels from the source to the receiver, known (in the time domain) as the Room Impulse Response (RIR). A variety of single number metrics may be calculated from the RIR, allowing for a direct numerical comparison of different spaces. Many of these are very well known in the Architectural Acoustics community, including Reverberation Time, Early Decay Time, Clarity, Bass Ratio, Lateral Fraction, and Interaural Cross-Correlation Coefficient. After calculating these parameters, one might ask: can they, by themselves or in combination, predict how listeners might subjectively rate these spaces (especially in the context of rooms used for live, unamplified musical performances)? Leo Beranek undertook a survey of prominent musicians and conductors to come up with a consensus ranking of the world s major concert halls and opera houses (Beranek, 23, 24). Armed with these data, he was able to examine which parameters might be able to predict a hall s ranking. Bass Ratio offered no predictive power whatsoever. Reverberation Time only appeared to divide the halls into two groups, with the better halls exhibiting values around seconds and the lesser halls around seconds. The parameters exhibiting the strongest correlations with subjective preference were Lateral Fraction (LF) and Interaural Cross-Correlation Coefficient (IACC). LF is the ratio of energy arriving laterally (as measured with a figure-of-eight microphone with the lobes pointed towards the side walls) to the total arriving energy. It was found that higher values of LF correlated with more highly regarded halls. IACC is a measure of the similarity between two channels of the RIR recorded with the binaural dummy head. It was found that lower values of IACC (corresponding to a higher degree of dissimilarity between the two ear signals) correlated with more highly regarded halls. Due to the symmetry of the head, it is lateral sound energy that creates dissimilar signals at the two ears. This indicates that spatial aspects of the sound field play an important role in listeners perceptions of concert halls. Yet the standard ways to measure these spatial aspects are either with a very blunt instrument (LF) or a proxy measure (IACC). Thus the need for a method to measure spatial characteristics more precisely and directly. SPHERICAL ARRAY PROCESSING Scattering from a Rigid Sphere It can be shown (Williams, 1999) that the sound field on the surface of a rigid sphere of radius a due a plane wave of wavenumber k = 2πf /c and amplitude P impinging on its surface from the direction (θ i,φ i ) is given in spherical coordinates by: p(θ,φ) = 4πP i n b n (ka) n= n m= n The quantity b n (ka), sometimes referred to as the modal amplitude, is given by: b n (ka) = j n (ka) j n (ka) h (1) n Y m n (θ,φ)y m n (θ i,φ i ). (1) (ka) h(1) n (ka), (2) where j n (ka) are the spherical Bessel functions and h (1) n (ka) are the spherical Hankel functions of the first kind. Yn m (θ,φ) refers to the set of functions known as the spherical harmonics, defined as: Yn m (θ,φ) = 2n + 1 (n m)! 4π (n + m)! P n m (cosθ)eimφ, (3) which form a set of orthonormal basis functions over the sphere, obeying the following orthogonality relation: 2π π Y m n (θ,φ) Yn m (θ,φ) sinθ dθ dφ = δ nn δ mm. (4) Proceedings of Meetings on Acoustics, Vol. 19, 152 (213) Page 2

3 Clapp et al. Degree Phase Order F IGURE 1: The spherical harmonics up to third order, with color scale indicating phase. The -degree spherical harmonics are entirely real-valued. There are 2 n + 1 spherical harmonics per order n, with degree m falling within the range n m n. The spherical harmonics up to third order are shown in Fig. 1. Beamforming The term beamforming" refers to a general set of techniques that can be applied to multi-channel sensors. These techniques involve assigning a weighting factor and/or a time delay to the different channels of a sensor and then summing to achieve spatial filtering. Spherical microphone arrays lend themselves well to directional analysis of sound fields, as their spherical symmetry allows them to look" (or rather, listen") in any direction in 3-D space to determine the characteristics of the sound energy coming from that direction, as shown in Meyer and Elko (24), Rafaely (25), and Li and Duraiswami (27). Returning to the expression for the pressure on the surface of a rigid sphere due to an incident plane wave: n X X p(θ, φ, ka) = 4πP i n b n ( ka) Ynm (θ, φ)ynm (θ i, φ i ). (5) m= n n= Now, we apply a weighting factor to each point on the sphere: Wnm (θ, φ, ka) = Ynm (θ, φ) 4π i n b n ( ka), (6) and then integrate over the entire sphere, using the orthonormality of the spherical harmonics (as shown in Eq. 4) to obtain the following result: Z 2π Z π Wnm (θ, φ, ka) p(θ, φ, ka) sin θ d θ d φ = Ynm (θ i, φ i ). (7) This shows that we can use a rigid sphere to find the spherical harmonic components of the incident plane wave. Beamforming can then be achieved by linear combinations of these spherical harmonic components. Because any function on the surface of a sphere can be expressed as a sum of spherical harmonics, it is possible to create a beampattern of any shape. The ideal beampattern is a delta function in the look direction (θ, φ ): F (θ, φ, θ, φ ) = δ(θ θ )δ(φ φ ). (8) This beampattern is considered ideal" because it only looks in one precise direction; the main lobe is infinitely narrow and there are no side lobes. The ideal beampattern can be expressed as a sum of spherical harmonic components: X n X F (θ, φ, θ, φ ) = 2π Ynm (θ, φ)ynm (θ, φ ). (9) n= m= n Proceedings of Meetings on Acoustics, Vol. 19, 152 (213) Page 3

4 FIGURE 2: Regular beampatterns for orders, from left to right, 1, 2, 3, 6, and 12. Thus, the weights required to generate this beampattern, as a function of position on the sphere (θ, φ), look direction (θ,φ ), and wavenumber ka are given by: w(θ,φ,θ,φ, ka) = n= 1 2i n b n (ka) n m= n Y m n (θ,φ)y m n (θ,φ ). (1) This ideal beampattern requires a continuous spherical transducer, and in practice we are required to sample the surface of a sphere, which limits the order of spherical harmonics that can be used to construct our beampattern. Let us consider an array with Q microphones located at (θ q,φ q ) for q = 1,2,...,Q. We can approximate the orthonormality relation of the spherical harmonics in Eq. 4 with a summation: 4π Q Q q=1 Y m n (θ q,φ q ) Y m n (θ q,φ q )C m n (θ q,φ q ) = δ nn δ mm (11) C m n (θ q,φ q ) represents the quadrature coefficients applied to each channel to ensure that the above relation holds. Now, because we can only determine the spherical harmonic components up to a given order N, we must truncate the expression for the ideal beampattern: N F N (θ,φ,θ,φ ) = 2π n= n m= n Y m n (θ,φ)y m n (θ,φ ), (12) which is known as the regular beampattern of order N. The weights required to achieve this beampattern are given by: w N (θ q,φ q,θ,φ, ka) = N n= 1 2i n b n (ka) n m= n Several examples of regular beampatterns are shown in Fig. 2. Y m n (θ q,φ q )Y m n (θ,φ ) C m n (θ q,φ q ). (13) SPHERICAL ARRAY DESIGN A spherical array was designed and built at Rensselaer Polytechnic Institute for the purpose of carrying out RIR measurements, and is show in Fig. 3. The array consists of 16 Panasonic omnidirectional microphone capsules embedded in the surface of a rigid sphere of radius 2.5 cm. The shell was created with a rapid prototyping 3-D printer. Equal positioning of points on the surface of a sphere is possible for only certain numbers: 4, 6, 8, 12, or 2, which correspond to the vertices of the Platonic solids. For any other number, a different scheme must be used, along with appropriate quadrature coefficients. The channel placement and quadrature coefficients for this array were computed according to Fliege and Maier (1999), who formulated Proceedings of Meetings on Acoustics, Vol. 19, 152 (213) Page 4

5 Clapp et al. Theoretical Beampatterns F IGURE 3: 16-channel microphone array. Frequency 5 Hz 1 khz 2 khz 4 khz 8 khz 16 khz 2nd Order 3rd Order F IGURE 4: 2nd and 3rd order beampatterns produced with the 16-channel array due to simulated single frequency plane waves. the problem as minimizing the potential energy of a number of charged particles on the surface of a conducting sphere. With a 16-channel array, it is theoretically possible to decompose a recorded sound field up to the third order of spherical harmonics. To test the capabilities of this array, plane waves of different frequencies were simulated impinging on the sphere and then beamforming was carried out with weights given by Eq. 13. The results are shown in Fig. 4. The results of these simulations can be best understood by turning to the following figures. The first, Fig. 5, shows modal amplitude curves (given in Eq. 2) for a rigid sphere of radius 2.5 cm. The second, Fig. 6 shows the orthonormality errors for all spherical harmonics up to third order, calculated as deviations from Eq. 11. We can see that at lower frequencies, the higher order spherical harmonic components need to be amplified considerably to achieve the desired beampattern. The orthonormality errors for this array and set of quadrature coefficients are small to non-existent up to second order, but there are significant errors among the third order harmonics, which are amplified by approximately 23 and 18 db at 5 Hz and 1 khz respectively. The second order harmonics exhibit both lower orthonormality errors and require less amplification, resulting in a useable beampattern over a greater frequency range. At high frequencies, the beampattern degrades again, but for different reasons. These wavelengths are similar in size or smaller than the physical spacing between the individual channels on the sphere, resulting in aliasing. In addition, as can be seen in Fig. 5, the higher order spherical harmonic components become more prominent, resulting in truncation errors. Proceedings of Meetings on Acoustics, Vol. 19, 152 (213) Page 5

6 5 1 Amplitude (db) n= n=1 n=2 n=3 n=4 n=5 n= Frequency (Hz) FIGURE 5: Modal amplitude curves for a rigid sphere of radius 2.5 cm..25 n 2 + m + n n 2 + m + n + 1 FIGURE 6: Orthonormality errors for the 16-channel array. Proceedings of Meetings on Acoustics, Vol. 19, 152 (213) Page 6

7 EMPAC Concert Hall 2 Seat 1 Front and Center Seat 3 7th Row, On Aisle Time (ms) -18 Azimuth Azimuth +18 FIGURE 7: Azimuthal plane beamforming for two audience positions in the EMPAC Concert Hall. ROOM IMPULSE RESPONSE MEASUREMENTS Room Impulse Response (RIR) measurements were made using the 16-channel spherical array in a variety of different concert and recital halls throughout Upstate and Western New York. An omnidirectional sound source was used, consisting of a subwoofer and two dodecahedral loudspeakers of different sizes to ensure an omnidirectional signal across all measured frequencies. In each hall, three audience positions were measured: one front and center, one three to four rows back near a side wall, and one seven to eight rows back on an aisle. Displaying data obtained with the spherical array presents a particular challenge because it is essentially four-dimensional data: three spatial dimensions and time. If one wishes to show how the sound field evolves spatially over time in a single plot, one possibility is to examine just the azimuthal plane, which yields a plot similar in appearance to a binaural activity map. This is the approach that I took with the following plots. The examples shown here come from two different venues within the Experimental Media Performing Arts Center on the campus of Rensselaer Polytechnic Institute. The first is a 12-seat shoebox-style concert hall, and the second is a 4-seat drama theater. Fig. 7 shows the azimuthal plane beamforming results for two different audience positions in the concert hall for the first 2 ms of the RIR following the onset of the direct sound. The color scale is in db re: the direct sound. In both seats we see the direct sound followed by two strong lateral reflections from the side walls, a characteristic of the geometry of shoebox-style halls. Rather than simply including these in a quantity we call Lateral Energy," as with the standard parameters, we can see with this method the precise timing and orientation of these lateral reflections, which are believed to be very crucial to the enjoyment of live orchestral performances. In Seat 1, we see four angular positions that exhibit a near absence of sound energy for the duration of the impulse response. It is possible to attribute these to a unique design aspect of the EMPAC Concert Hall, hidden corners," which were designed to lessen room modes in the hall and disturbing reflections from those locations (Mistur and Goebel, 21). These plots also reveal the large differences in the sound field as one moves past the critical distance from the source. In Fig. 8, it is possible to see the different spatial qualities of the sound fields in rooms designed for different purposes. While the concert hall is used primarily for musical performances, which require significant reverberation to add fullness and warmth to the sound, the theater is used mainly for lectures and films, where speech needs to be understood. We can see that the theater features much fewer strong lateral reflections and that the level of reverberant sound energy drops significantly after approximately 1 ms, while the concert hall features a higher number of strong lateral reflections as well as a much higher level of reverberant energy. Proceedings of Meetings on Acoustics, Vol. 19, 152 (213) Page 7

8 2 Comparison of EMPAC Concert Hall and Theater Theater Front and Center Concert Hall 7th Row, On Aisle Time (ms) -18 Azimuth Azimuth +18 FIGURE 8: Azimuthal plane beamforming for audience positions in the EMPAC Concert Hall and Theater. FUTURE WORK One element of future work is to apply Direction of Arrival (DOA) estimation algorithms (c.f. Sun et al. (212)) to the data collected in order to obtain more accurate spatial information. Another element is to use the data to generate new parameters involving spatial information, such as measures of sound field uniformity (c.f. Gover et al. (24)) or other spatial ratios. REFERENCES (29). ISO :29: Acoustics Measurement of room acoustic parameters Part 1: Performance spaces, International Organization for Standardizations, Geneva, Switzerland. Beranek, L. L. (23). Subjective rank-orderings and acoustical measurements for fifty-eight concert halls, Acta Acustica united with Acustica 89, Beranek, L. L. (24). Concert halls and opera houses: music, acoustics, and architecture, 2nd edition (Springer, New York, NY). Fliege, J. and Maier, U. (1999). The distribution of points on the sphere and corresponding cubature formulae, IMA Journal of Numerical Analysis 19, Gover, B. N., Ryan, J. G., and Stinson, M. R. (24). Measurements of directional properties of reverberant sound fields in rooms using a spherical microphone array, Journal of the Acoustical Society of America 116, Li, Z. and Duraiswami, R. (27). Flexible and optimal design of spherical microphone arrays for beamforming, IEEE Transactions on Audio, Speech, and Language Processing 15, Meyer, J. and Elko, G. W. (24). Audio Signal Processing for Next-Generation Multimedia Communication Systems, chapter 3, Spherical Microphone Arrays for 3D Sound Recording", (Kluwer Academic Publishers, Hingham, Massachusetts, USA). Mistur, M. and Goebel, J. (21). The Architecture of EMPAC: The Tangible and the Tantalizing (Rensselaer Polytechnic Institute, Troy, NY). Rafaely, B. (25). Analysis and design of spherical microphone arrays, IEEE Transactions on Speech and Audio Processing 13, Proceedings of Meetings on Acoustics, Vol. 19, 152 (213) Page 8

9 Sun, H., Mabande, E., Kowalczyk, K., and Kellerman, W. (212). Localization of distinct reflections in rooms using spherical microphone array eigenbeam processing, Journal of the Acoustical Society of America 131, Williams, E. G. (1999). Fourier Acoustics: Sound Radiation and Nearfield Acoustical Holography (Academic Press). Proceedings of Meetings on Acoustics, Vol. 19, 152 (213) Page 9