IN-SITU ACOUSTIC IMPEDANCE ESTIMATION BASED ON SPARSE ARRAY PROCESSING

Transcription

1 INNOVATIONPROJEC T IN-SITU ACOUSTIC IMPEDANCE ESTIMATION BASED ON SPARSE ARRAY PROCESSING DTU, Antoine Richard DTU, Efren Fernandez-Grande, Cheol-Ho Jeong, Jonas Brunskog Brüel and Kjær, Jørgen Hald, Woo-Keun Song, Karim Haddad Bang and Olufsen, Martin Olsen, Morten Lydolf Ecophon, Erling Nilsson Odeon, Claus Lynge Christiensen Abstract: This project investigates the estimation of the acoustic impedance of given material samples from measurements with a spherical microphone array. The pressure measured by the array is used to detect the incident and reflected waves, using Compressive Sensing for obtaining a finer spatial resolution. The impedance of the material is then estimated based on reconstructing the sound pressure and the particle velocity fields on the surface of the material. The estimation is also compared to measurements with a double-layer array and an intensity probe. The methods provide good estimates of the impedance and absorption, consistent with standardized measurements. The results highlight the importance taking into account the sample s finite size and possible reflections from the room, which is particularly critical at low frequencies. There is an interesting potential for using the spatial information to compensate for the effect of edge diffraction and reflections.

2 COLOPHON Publisher Danish Sound Innovation Network Technical University of Denmark Richard Petersens Plads, Building 321, 2800 Kongens Lyngby, Denmark November 2015 About the publication This publication and possible comments and discussions can be downloaded from The content of this publication reflects the authors point of view and not necessarily the view of the Danish Sound Innovation Network as such. Copyright of the publication belongs to the authors. Possible agreements between the authors might regulate the copyright in detail. About the network Danish Sound Innovation Network is an innovation network funded by the Danish Agency for Science, Technology and Innovation. The Network is hosted by the Technical University of Denmark and is headed by Director, Associate Professor, PhD Jan Larsen. Danish Sound is the facilitator of the national ecosystem for SOUND, creates value for all parts in the value chain and contributes to growth and wealth in Denmark. Network membership is free of charge and open for all. Registration at This publication is the result of an innovation project, an instrument to strengthen the cooperation between knowledge institutions and private companies. The primary goal is to promote innovation by combining accessible/existing research and technologies with creative uses in order to facilitate the creation of new products, services or experiences. Innovation projects are mainly short term feasibility studies conducted on a pre-competitive level. 2 Dansk Sound Innovation Network

3 PREFACE This publication is the result of an innovation project entitled In-situ acoustic impedance estimation based on sparse array processing. The project is financed by the Danish Sound Innovation Network through a grant from the Danish Agency for Science, Technology and Innovation. The project is completed in the period September- November 2015 and managed by DTU, project manager Efren Fernandez-Grande. The other participants from DTU are Antoine Richaud (PI), Jonas Brunskog and Cheol-ho Jeong. The other project participants are: Karim Haddad, Jørgen Hald, Woo-Keun Sung (Brüel and Kjær); Martin Olsen, Morten Lydolf(Bang and Olufsen), Erling Nilsson (Ecophon) and Claus Lynge Christensen (Odeon). INTRODUCTION BACKGROUND In room acoustics, simulations and predictions depend on the acoustic properties of the materials that are present in the room. A good understanding of these acoustic properties is therefore needed to ensure correct estimations. An essential property of materials is their surface impedance and absorption coefficient. The existing standardized measurement methods rely on idealized sound field models and do not provide angle dependence information. Spherical array processing can be used for sound field reconstruction [1, 2] and the development of the Spherical Equivalent Source Method (S-ESM) proved to be particularly accurate, especially when using the sparsity of the setup [3, 4, 5]. The S-ESM has shown a promising potential to measure the impedance and the absorption coefficient of a material, via the reconstruction of the sound field on the material s surface [6]. This can in principle be performed in-situ, thus not relying on laboratory conditions. OBJECTIVE The goal of this project is to test the impedance estimation method based on the S-ESM [6] in both laboratory conditions and in a conventional room, which poses a challenging situation. Additionally, the study of small material samples, where diffraction effects from the edges become critical, is examined. The results are compared with other measurement techniques, including the use of a planar array, an intensity probe, an impedance tube, and a room acoustic simulation tool (ODEON). IMPACT/EFFECT The project investigates new applications with microphone array systems, useful for the manufacturers and consultants that would benefit from such measurement tools. Material providers/manufacturers (Ecophon) can benefit from a characterization of the acoustic behavior of their materials in-situ. This can also be beneficial for room acoustic simulations, which could make use of the measured absorption or impedance impedance in-situ. Finally, these methods can potentially be adapted to smaller enclosures, such as car or vehicle interiors, which is of interest for noise control and car audio system design. This DS project has led to a PhD project at DTU (Acoustic Technology, Department of Electrical Engineering), to examine the acoustic characterization of materials from microphone array measurements. It is expected that both surface impedance and scattering will be examined in this project. 3 Dansk Sound Innovation Network

4 SUMMARY OF CONCLUSIONS The method provides sensible estimates of the impedance and absorption coefficients in situ, in particular for the mid-frequency range. It was not possible to yield a good estimate at large angles of incidence (higher than 45 ). The finiteness of the absorber results in oscillations, especially at low frequencies. Short distances between measurement positions and sample favor a better estimate of the properties (thus planar geometries can be more favourable). The algorithm can discriminate secondary sources if these ones are not too strong. This also holds for spurious reflections in a room, which can influence the sound field if they are strong enough. Good results can be obtained in ordinary rooms, although a low frequency limit linked to the room's dimensions appears. The measurements are repeatable. The calculation of the absorption coefficient is more sensitive to instabilities than the calculation of impedance. Overall, a good agreement is found between the double layer array and the spherical array measurements. Some discrepancies occur due to the different processing, but the overall behavior of the materials is consistent. The main limitations of the proposed spherical array measurements are: o the need to estimate the noise floor manually (which increases the processing time and reduces the frequency resolution), o the rather long calculation time, o the greater distances to the sample, or difficulty for conformal measurements. The ODEON Genetic Material Optimizer (GMO) tool was used to determine the absorption of the material in-situ and compare with the other methodologies. There is good agreement at high frequencies (above 500 Hz). However, at low frequencies, the strong presence of modes is not suited for the ODEON calculation, resulting in deviations in the absorption estimates. METHOD AND RESULTS THEORY The method used in this project relies on the analytical description of a spherical sound field scattered by a rigid sphere and a reflection model on the materials to be studied. The fundamental methodology has been published previously in [1-5]. An overview of the array processing theory can also be found in the Appendix. The fundamental idea relies on expressing the sound pressure on the rigid sphere array as the superposition of the waves generated by LL point sources, located at (rr 0,ll, Ω 0,ll ) [3] LL nn pp tt (aa, Ω kk ) = jjjjjjqq ll aa 2 h (2) nn (kkrr 0 ) h (2) nn (kkkk) YY nn mm (Ω kk )YY mm nn Ω 0,ll, (1) ll=1 nn=0 mm= nn where aa is the radius of the sphere, an the angular position is Ω kk. The vector of measured pressures pp tt (size KK) and the vector of source strengths qq (size LL) can be expressed in matrix form as 4 Dansk Sound Innovation Network

5 pp tt = GG NN qq, (2) where GG NN is a transfer matrix of size KK LL relating the source amplitudes (i.e. volume velocities) and the measured pressures. The source coefficient vector qq is determined using Compressive Sensing (CS), i.e. minimizing the l_1 norm of the vector, to enhance spatial resolution. For this, the CVX MATLAB package was used [8]. This estimated qq coefficients are used to extrapolate the sound field to a different surface than measured (in this case the surface of the absorber). IMPEDANCE AND REFLECTION MODEL This array reconstruction method (described in Refs. [1-5] and Appendix) makes it possible to estimate the sound field on the material s surface, namely the pressure pp and the normal component of the particle velocity uu nn. Consequently the normalized surface impedance of the material can be calculated as ZZ = 1 pp. (3) ρρρρ uu nn The absorption coefficient is calculated from the impedance and the angle of incidence ψψ as [9] 2 ZZ cos(ψψ) 1 αα = 1 ZZ cos(ψψ) + 1. (4) This estimation assumes a locally reacting and infinite material, and plane wave incidence onto it. METHOD The method is illustrated in Figure 1. The array is positioned close to the material s surface, and the sound field is generated by an omnidirectional loudspeaker. The measured pressure on the array is represented as a combination of elementary spherical waves (monopoles, referred to as equivalent sources). In this project, 256 equivalent sources uniformly distributed on a sphere are used, centered on the sample s surface, and chosen to include the source and its image. The reconstruction surface is a square grid of 10 x 10 cm2, consisting of 21 x 21 equally distributed points. The pressure and the normal component of the particle velocity are reconstructed on each point of the grid and the surface impedance is calculated at each point with (3). The estimated impedance is a spatial average over the reconstruction area. The sound field is generally dominated by the direct field and a specular reflection on the material, although other components may appear, such as diffraction from the edges of the material, or reflections if the measurements are performed in a room. These effects of diffraction and unwanted reflections are compensated for, by means of discarding the corresponding equivalent sources from the reconstruction. 5 Dansk Sound Innovation Network

6 Figure 1 - Method principle The coefficient vector of the equivalent sources qq is strongly influenced by the estimate of the noise-floor εε. Generally the SNR estimate is lower than the single transducer SNR (about 25 db lower). It includes effects of transducer mismatch, positioning errors, etc. In this project, the noise-floor estimate was chosen manually, related to an estimated SNR as εε = pp tt 2 10 SSSSRR eeeeeeeeeeeeeeee 20, (5) and SSSSRR eeeeeeeeeeeeeeee varying between 5 and 40 db. The manual selection required a longer processing time, consequently the calculations were limited to the center frequencies of third octave bands from 100 Hz to 4000 Hz. The effect of this is later analysed in the results. EXPERIMENTS The proposed method was tested under in an anechoic room and in a lightly damped room, using three different medium-small sample sizes (4.3 m 2, 1.4 m 2 and 0.1 m 2 ), and both a porous material as well as a porous material covered by a light membrane. The setup is shown in Figure 2. A detailed description of the set-up is included in this section. Figure 2 - Experimental setup 6 Dansk Sound Innovation Network

7 SPHERICAL ARRAY The spherical array used in the measurement is manufactured by Brüel & Kjær. It consists of a rigid sphere of radius 9.75 cm containing 64 microphones, and can sample up to 7 th order spherical harmonics. The valid frequency range of the array is approximately [6] of 56 < f <3919 Hz. The studies conducted in this project were therefore limited to a range of khz. In the measurements, the array s center was placed at cm from the sample s surface. Figure 3 - Spherical array with 64 channels SOURCE The source used was an omnidirectional source manufactured by B&K (Omni-source), which is placed 1.5 m from the rigid backing and at different angles (0, 15, 30, 45, 60, 75 ), radiating pink noise. MATERIAL SAMPLES POROUS ABSORBERS The tested samples are made from the ceiling absorber INDUSTRY MODUS manufactured by ECOPHON. It consists of mineral wool, and its behavior modeled with Miki s model [10] based on the material s flow resistivity of N.s.m -4. The samples were framed in wooden boards. Three different sizes were tested: Large absorber (180 cm x 240 cm x 10 cm), Medium absorber (60 cm x 240 cm x 10 cm), Small absorber (30 cm x 30 cm x 10 cm). Figure 4 - Photos of the different tested absorbers. ABSORBERS WITH FOIL The medium and the small samples presented previously were covered with a thin plastic foil in order to modify their high frequency behavior. 7 Dansk Sound Innovation Network

8 Figure 5 - Photos of absorber samples with foil. CAR SEAT SAMPLE A sample provided by B&O was also investigated. It consists of a rigid plate, foam and a leather lining. Its dimensions are 50 cm x 50 cm x 8 cm. Figure 6 - Car seat sample provided by B&O. ROOMS ANECHOIC ROOM Initially, the measurements were done in DTU s anechoic room. A rigid backing (large wooden plate) of about 3.5 m x 3.5 m was used, on which the samples were placed, at the center. The spherical array was hung above the sample. Figure 7 - Setup in the anechoic room 8 Dansk Sound Innovation Network

9 ORDINARY ROOM The measurements were also conducted in a conventional lightly damped room to examine the robustness of the method. The samples were placed vertically on an empty rigid wall. The source and array were installed on tripods. Figure 8 - Setup in the laboratory room COMPARISON WITH OTHER METHODS The results were compared to results from a double layer array, an intensity probe and an impedance tube, in order to obtain a benchmark of the properties of the material. PLANAR DOUBLE LAYER ARRAY A double-layer planar array containing 64 microphones was also used in the measurements. The processing relies on SONAH (Statistically Optimized Near-field Acoustical Holography), where the pressure in a given region is expressed as a sum of elementary plane waves (propagating and evanescent), used for reconstructing the sound field. This characterization enables one to estimate the impedance in the vicinity of the array [11]. INTENSITY PROBE The absorption coefficient of the materials was also estimated with an intensity probe. The field above the material is scanned to yield the surface impedance. It should be noted that this method assumes free-field conditions, and that hypothesis is not fulfilled in an ordinary room. IMPEDANCE TUBE MEASUREMENT This standardized method makes it possible to calculate the impedance and the absorption coefficient of a small sample placed in a tube at normal incidence [12]. ADDITIONAL MEASUREMENTS SOUND FIELD WITH TWO SOURCES In the anechoic room, a second source was placed around the absorber in order to imitate a reflection on a rigid wall. The goal is to determine how to handle such strong reflections in calculating the material properties, and to anticipate a more complex room situation. 9 Dansk Sound Innovation Network

10 Figure 9 - Setup with two sources REPEATABILITY The same measurement was done several times to study the repeatability of the method. CORNER Measurements were also obtained with the array and the source standing close to a corner. The goal of this setup is to assess the influence of stronger reflections in the room. Figure 10 Setup close to a corner 10 Dansk Sound Innovation Network

11 MEASUREMENT PLAN The table below summarizes the measurements carried out. Note that several of the measurements were not analyzed and are not presented in this report. These data will serve as a measurement data base for the participants, and for reference in their ongoing collaboration. The processed results are indicated in bold in the last column of Table 1. Anechoic room measurements Measurement Sample Angle of incidence Impedance characterization Large sample 0, 15, 30, 45, 60, 75 Medium sample 0, 15, 30, 45, 60, 75 Small sample 0, 15, 30, 45, 60, 75 Medium sample with foil 0, 15, 30, 45, 60, 75 Small sample with foil 0, 15, 30, 45, 60, 75 Test with two sources Large sample S1<S2, S1 S2, S1>S2 Ordinary room measurements Measurement Sample Angle of incidence Impedance characterization Large sample 0, 15, 30, 45, 60 Medium sample 0, 15, 30, 45, 60 Small sample 0, 15, 30, 45, 60 Medium sample with foil 0, 15, 30, 45, 60 Small sample with foil 0, 15, 30, 45, 60 Car seat sample 0, 15, 30, 45, 60 Repeatability Medium sample 0 Corner influence Large sample 0, 45 RESULTS Table 1 - List of measurements ANECHOIC MEASUREMENTS INFLUENCE OF NOISE ESTIMATION This section investigates the influence of the choice of the noise floor estimate εε on the result, in the case of the medium sample without foil measured in the anechoic room. The impedance and the absorption coefficient are calculated for the following cases: A low SNR estimate: the resulting solution only consists of the direct source and its image (if it exists), A high SNR estimate: additional sources (mostly diffraction) appear in the solution, 11 Dansk Sound Innovation Network

12 Discarding spurious sources: The same high SNR estimate, but the sources not corresponding to the source and image source are discarded. The results are shown in Figure 11. Figure 11 - SNR estimates used in the calculation Figure 12 - Calculated impedance for different noise estimations. 12 Dansk Sound Innovation Network

13 The following observations could be made: Figure 13 - Calculated absorption coefficient for different noise estimations. The absorption coefficient is higher for the low SNR estimate as some of the power is discarded as noise. For instance, the image source is easily confused with noise at high frequencies. The high SNR estimate results in a fair improvement, although the error is greater at low frequencies. Discarding spurious sources results in a better low frequency estimation. These additional sources, usually found at the equator of the array, are a result of diffraction from the sample s edges. ANGLE DEPENDENCE The measurement and the impedance estimation were performed on the large sample for 6 different source positions (angles of incidence 0, 15, 30, 45, 60 and 75 ). 13 Dansk Sound Innovation Network

14 Figure 14 - Evolution of the estimated impedance with the incidence angle Figure 15 - Evolution of the estimated absorption coefficient with the incidence angle The estimation is less accurate at larger angles of incidence. The curves follow the theory quite well up to 45. At large angles of incidence, both sources are located closer to the equator, thus being easily confused with diffraction effects. This results in more errors both in localization and power estimation, as shown in Figure 16, which compares the angles 0 and Dansk Sound Innovation Network

15 Figure 16 - Equivalent sources at 500 Hz for the angles of incidence 0 and 75 INFLUENCE OF ABSORBER SIZE Due to the observations above, the focus for examining smaller absorbers is limited to angles of incidence of 0 and 30. The different sample sizes are plotted together. Additional graphs can be found in the appendix (see Table 2). Material Incidence angle Plotted results Location Porous absorber without foil Porous absorber without foil 0 Small sample Medium sample Large sample Impedance tube Miki s model 30 Small sample Medium sample Large sample Miki s model Analyzed in the report Appendix Porous absorber with foil 0 Small sample Medium sample Impedance tube Porous absorber with foil 30 Small sample Medium sample Appendix Analyzed in the report Table 2 - Available results for the study of the absorber size in anechoic conditions. 15 Dansk Sound Innovation Network

16 POROUS ABSORBER WITHOUT FOIL Figure 17 - Estimated impedance for the porous absorber samples without foil with an incidence angle of 0. Figure 18 - Estimated absorption coefficient for the porous absorber samples without foil with an incidence angle of 0. There is a low frequency limit, more critical for the small sample. This behavior consists of oscillations around the expected absorption value, and is a consequence of diffraction by the sample edges. The first peak at 125 Hz appears consistently for the three cases; it is attributed to the finiteness of the rigid backing plate. Indeed, the corresponding wavelength agrees with its dimensions. At high frequencies, similar results are obtained for the three samples. The impedance tube measurements are consistent with the array method at high frequencies, and it seems that Miki s model underestimates the absorption coefficient. 16 Dansk Sound Innovation Network

17 Figure 19 - Setup with the small absorber. It should be noted that in the case of the small absorber, the array and the sample are almost the same size (Figure 19). These results indicate that very small absorbers pose a challenge, as discarding diffraction sources is not possible or only brings a partial improvement. Measuring closer to the sample would improve the results. POROUS ABSORBER WITH FOIL - 30 Figure 20 - Estimated impedance for the porous absorber samples with foil with an incidence angle of 30. Figure 21 - Estimated absorption coefficient for the porous absorber samples with foil with an incidence angle of 30. The samples finiteness results in low frequency oscillations, and the small sample is too small to obtain a sensible estimate. The 125 Hz peak attributed to the backing plate is also visible. 17 Dansk Sound Innovation Network

18 The high frequency behavior is modified by the presence of the foil. The added mass results in an increase of the imaginary part of the impedance and a decrease in the absorption coefficient. This high frequency behavior was not observable at normal incidence, as shown in the appendix. SOUND FIELD CREATED BY TWO SOURCES A measurement using two loudspeaker was carried out. The motivation is to test the ability of the method to discriminate more than two sources: ideally one would recover two direct sources and two image sources. Different relative powers were tested (same level, Loudspeaker S1 louder, and Loudspeaker S2 louder) and the impedance was calculated assuming that the actual source was S1 (the closest one). Therefore, the objective is to evaluate how well the contribution of S2 can be eliminated from the calculation. Figure 22 shows the resulting equivalent sources for the three measured configurations at 500 Hz. Figure 22 - Sound field with two sources, resulting equivalent sources at 500 Hz. In configuration 1, all sources are recovered and the source S2 appears weaker as it is located further away. In configuration 2, S2 is totally discarded due to its weaker power and its larger distance from the array. This case is then very similar to the previous anechoic measurements. In configuration 3, S2 and its image are well characterized but additional noise sources appear in the solution. The contribution of S1 is not identified. In general, the dominant components of the sound field are correctly estimated, which is promising for measurements in rooms. If the objective is to recover the reaction of the sample to the first source S1, then the third configuration can easily be discarded. Moreover, the first configuration constitutes a more critical setup, for which the four expected sources appear clearly. For this case, only the equivalent sources corresponding to S1 and its image were kept. Figure 23 and Figure 24 show the resulting impedance and absorption coefficient of these two first configurations, compared with the results obtained for one source at 30 as reference, which is the closest angle of incidence. 18 Dansk Sound Innovation Network

19 Figure 23 - Estimated impedance for a sound field created by two sources. Figure 24 - Estimated absorption coefficient for a sound field created by two sources. The estimation is correct for all cases above 300 Hz. The presence of the second source disrupts the results at lower frequencies. Looking at the equivalent sources, for example at 125 Hz in Figure 25, the source localization is less efficient at low frequencies, probably due to a higher coherence in the sensing matrix GG NN. The equivalent sources accounting for S1 and its image tend to be shifted towards S2, especially if the two sources have an equivalent power. 19 Dansk Sound Innovation Network

20 PP 11 PP 22 PP 11 > PP 22 Figure 25 - Equivalent sources at 125 Hz for a sound field created by two sources. Configuration 1: similar power. Configuration 2: S1 stronger than S2. These results show that one can estimate the tested material s properties in the presence of additional sources and/or reflections, provided that these are not too strong. Below 200 Hz the estimation is notably worsened by the presence of additional sources. CONCLUDING REMARKS This first step of measurements showed that it is possible to measure the surface impedance, and hence the absorption coefficient, of a given material sample under the following conditions: The angle of incidence is smaller than 45, The studied frequency is between 200 Hz and 4000 Hz (the higher frequency limit is due to the array s properties while the lower limit is related to the setup limitations), The sample is not too small Additional sources may be present provided that they are not too strong. Several aspects need to be investigated further: A better understanding of the edge diffraction, A better frequency resolution at low frequencies, Imposing additional conditions on the solution. ORDINARY ROOM MEASUREMENTS The results in an ordinary room are likely to be altered by a higher background noise and additional reflections. The noise floor estimate was again chosen manually, by allowing additional sources to be part of the solution, although they are discarded in the reconstruction phase. The study focused on several aspects: The influence of the sample size, as it was done in the anechoic measurements, The repeatability of the measurements for one given case, The effect of strong reflections, when the setup is placed close to a corner. 20 Dansk Sound Innovation Network

21 INFLUENCE OF ABSORBER SIZE As similar conclusions could be drawn for all the studied cases, only the results for the porous absorber without foil at normal incidence are presented in this section. The other measurements can be found in the Appendix. Figure 26 - Estimated impedance in a room for the porous absorber without foil at normal incidence. Figure 27 - Estimated absorption coefficient in a room for the porous absorber without foil at normal incidence. Below 200 Hz, the results are not reliable. In this range, the wavelength is comparable to the room dimensions, and the influence too strong to separate the reflected waves. Figure 28 shows the estimated equivalent sources at low frequencies. Above 200 Hz, the estimation is very close to the results obtained in the anechoic room. 21 Dansk Sound Innovation Network

22 Figure 28 - Equivalent sources for the large sample at normal incidence in an ordinary room at 160 Hz. REPEATABILITY The repeatability was studied for the medium sample without foil at normal incidence, where the measurement was performed 5 times. Figure 29 - Repeatability of the measurement, estimated impedance. 22 Dansk Sound Innovation Network

23 Figure 30 - Repeatability of the measurements, estimated absorption coefficient. The impedance graph shows that the estimation is fairly stable. The variations are amplified at low frequencies for the absorption coefficient, because the ratio in the formula (4) is larger at low frequencies. This underlines a stability issue when this ratio becomes too large, when ZZZZZZZZ(ψψ) + 1 is small: strong variations can be expected for smaller values of αα. CAR SEAT SAMPLE As a supplementary test, a sample imitating a car seat was also studied in the room. Although no theoretical basis or reference was available for this absorber, the calculation was done at several angles of incidence. Nevertheless, the obtained results being largely unstable, partly because of the size of the sample and because of the relatively large distance between the surface and the array, the results and the analysis are presented in the appendix. ADDITIONAL TEST CORNER INFLUENCE In this section, the measured impedance and absorption coefficient obtained when the array is placed close to a corner are compared with the previously measured data at normal incidence. Figure 31 - Measured impedance when the source and the array are placed close to a corner. 23 Dansk Sound Innovation Network

24 Figure 32 - Measured absorption coefficient when the source and the array are placed close to a corner. The stronger reflections result in a worse estimation at low frequencies. Additional smaller oscillations can be seen in both Figures up to 1600 Hz. As illustration, the equivalent sources at 200 Hz are compared in Figure 33. The more challenging scenario becomes apparent. Figure 33 - Comparison of the equivalent sources at 200 Hz. In the corner measurement, the location of the direct source is misestimated. It is represented by two equivalent sources, one of which is discarded, reducing considerably the incident power in the calculation, and hence the absorption coefficient. 24 Dansk Sound Innovation Network

25 COMPARISON WITH OTHER METHODS The comparison was made for each case (different rooms, different samples, and different angles of incidence). The results presented here focus only on a few relevant cases: Large sample without foil at 0, comparison with the two rooms; Medium sample without foil at 0, comparison with the two rooms; Medium sample with foil in the anechoic room, at angles 0 and 30. In addition, the large sample was studied in a rectangular room, using Odeon. The other comparison results can be found in the appendix. LARGE SAMPLE WITHOUT FOIL AT 0 Figure 34 - Comparison of different methods for the large sample at normal incidence in the anechoic room. In anechoic conditions, the two array measurements agree well with the impedance tube measurement above 300 Hz. The planar array is only plotted from 200 Hz. It would be expected that greater deviations than at higher frequencies would occur. The measurements indicate that the absorption coefficient is higher than predicted by Miki s model. 25 Dansk Sound Innovation Network

26 Figure 35 - Comparison of different methods for the large sample at normal incidence in the ordinary room. In the ordinary room, both array methods show more oscillations than in the anechoic measurements, although the overall estimate is satisfactory from 200 Hz upwards. The absorption coefficient tends to be higher than in anechoic conditions. The intensity probe measurement is faulty as the free-field hypothesis does not hold in a room. MEDIUM SAMPLE WITHOUT FOIL AT 0 We examine the medium sample for all the methods. The results in both rooms are displayed. Figure 36 - Comparison of different methods for the medium sample at normal incidence in the anechoic room. There is a good agreement between all the methods at high frequencies. Below 500 Hz, where the size of the absorber gains a more dominant role, small deviations can be observed. 26 Dansk Sound Innovation Network

27 Figure 37 - Comparison of different methods for the medium sample at normal incidence in the ordinary room. The intensity probe method is not suited to measurements in rooms. The high frequency result is consistent for the array methods and impedance tube measurement. Below 500 Hz, larger discrepancies appear, compared to the anechoic measurement. The spherical array measurement favors the direct source relative to its image, resulting in an overestimation of the absorption coefficient. The estimation from the planar array is better for this sample, presumably because of being placed closer to the material s surface, which therefore minimizes edge diffraction. MEDIUM SAMPLE WITH FOIL IN THE ANECHOIC ROOM Figure 38 - Comparison of different methods for the medium sample with foil at normal incidence. 27 Dansk Sound Innovation Network

28 Figure 39 - Comparison of different methods for the medium sample with foil at an angle of incidence of 30. The normal incidence results show a clear drop for the planar array and the intensity probe, but this is not visible for the spherical array. The drop is visible at 30 and is quite consistent with the planar array curve. USING ODEON TO CALCULATE ABSORPTION COEFFICIENTS A study was carried out to estimate the absorption coefficient of the studied sample using the Odeon roomacoustics software. The large sample was placed in a small room with rigid walls and impulse responses were measured at different source and receiver positions. The absorption coefficient was estimated from the measured reverberation times, and from a Genetic Material Optimizer (GMO), which adapts the absorption data in a room simulation in order to fit with measured room acoustic parameters. Due to the relatively small size of the room, the best results were obtained with the GMO, based on the parameters T30, ts, in combination with the parameters C50 and C80 which are dependent on position. As an energy-based method, the study is made in octave bands and the domain of validity is above the Schroeder frequency 500 Hz in the room in use. Due to the small size of the room, the experimental set-up is ill-suited to ODEON, as the strong presence of modes poses a challenge. Above 500 Hz, the GMO results agree well with the other methods. It is expected in future collaboration to study materials with absorption below unity, and placed in a larger room. A detailed report of this study can be found in the Appendix. CONCLUDING REMARKS The following conclusions could be drawn from the comparisons: The two array methods proved to be more robust in a room environment than the intensity probe method. Generally a good agreement between the two array methods and the impedance tube measurement could be found at mid-high frequencies. At lower frequencies, the results tended to differ more in the ordinary room setup. 28 Dansk Sound Innovation Network

29 The planar array is advantageous for small samples, since it can be placed closer to the materials surface, and therefore reduces edge diffraction effects. Good agreement was found between the results from Odeon and the other methods above 500 Hz. These comparisons underlined also the need for a more precise reference in terms of impedance and absorption coefficient, especially for oblique incidence. The presented method still requires some improvement, particularly: The estimation of the noise floor is critical in the resulting equivalent source representation and thus the accuracy of the estimation. Narrowband processing: in this report only selected frequency lines were analyzed due to the lack of automatic regularization criteria, which makes the processing more cumbersome than automatic methods. CONCLUSION The results presented in the report showed that it is possible to draw a good estimate of a material s impedance from in situ measurements with a spherical array. Promising results were obtained in lightly damped rooms with finite absorbers, although a clear low frequency limit was identified. However, due to the time frame of the project, many theoretical aspects that could greatly improve the quality of the results had to be left aside. The main areas of improvement are the following: The choice of the noise floor in the optimization algorithm needs to be automatized. This would allow a faster processing and a better frequency resolution. The characterization of edge diffraction with point sources needs to be formalized and validated, so that the sample finiteness can be compensated. The grazing incidence case should be studied further. The elimination of additional reflections works when they are weak enough. The case of strong reflections should be investigated, which would enable one to obtain better results in rooms. That is why these measurements led to the creation of a PhD project, where these issues as well as other aspects concerning the acoustical properties of surfaces will be investigated. REFERENCES [1] E. Williams and K. Takashima, "Vector intensity reconstructions in a volume surrounding a rigid spherical microphone array," J. Acoust. Soc. Am., vol. 127, pp , [2] F. Jacobsen, G. Moreno-Pescador, E. Fernandez-Grande and J. Hald, "Near field acoustic holography with microphones on a rigid sphere," J. Acoust. Soc. Am., vol. 129, pp , [3] E. Fernandez Grande and T. Walton, "Reconstruction of sound fields with a spherical microphone array," in Internoise, [4] E. Fernandez-Grande and A. Xenaki, "Sparse acoustic imaging with a spherical array," in Euronoise, 29 Dansk Sound Innovation Network

30 Maastricht, [5] E. Fernandez-Grande and A. Xenaki, "Compressive sensing with a spherical microphone array," J. Acoust. Soc. Am. (EL), accepted, to appear [6] A. Richard, "In-situ measurement of the acoustic properties of materials using a spherical array," [7] E. G. Williams, "Spherical Waves," in Fourier Acoustics - Sound Radiation and Nearfield Acoustical Holography, [8] M. Grant and S. Boyd, "CVX User's Guide," [9] S. I. Thomasson, "On the absorption coefficient," Acustica, vol. 44, no. 4, pp , [10] Y. Miki, "Acoustical properties of porous materials - Modifications of Denaly-Bazley models," Journal of the Acoustical Society of Japan, no. 11, pp , [11] J. Hald, "Patch holography in cabin environments using a two-layer handheld array with an extended SONAH algorithm," in Euronoise, [12] ISO Acoustics - Determination of sound absorption coefficient and impedance in impedance tubes - Part 2: Transfer-function method. [13] W. M. Candes E., "An Introduction to Compressive Sampling," IEEE Signal Processing Magazine, pp , Dansk Sound Innovation Network

31 APPENDIX USING ODEON TO CALCULATE ABSORPTION COEFFICIENTS 31 Dansk Sound Innovation Network

32 Using Odeon to calculate absorption coefficients This report is a part of the Danish Sound Innovation project titled In-situ acoustic impedance estimation based on sparse array processing. In this report, it is investigated if sound absorption coefficients estimated using the above array method leads to room acoustic simulation results, which are in better agreement with measured values than would be the case if absorption coefficients were obtained using the absorption measurement method given in ISO 354. Using genetic algorithms from within Odeon [1], it is estimated which absorption coefficients of the sample in question gives the best agreement between room acoustics parameters measured in a number of receiver positions with different source positions. Taking into account room acoustics parameters such as C 50 and C 80 which varies with position, the non-diffuse behavior of the room is taken into account. The test sample A test sample of a porous material from Ecophon with the dimensions (l,d,h)= (2.45 m, 0.10 m, 1,85 m) were used. The absorption coefficients of this material have been measured in a number of laboratories according to the ISO 354 method as part of a Round robin on the ISO 354 method (results not published yet). The results of the measured absorption coefficients can be seen in figure 1. It is seen that absorption coefficients at several octave bands are above 1.0 which is commonly seen for porous materials measured in a reverberation chamber according to ISO 354. The difference between measurements of absorption coefficients measured in different laboratories are typically within 10% above 250 Hz, typically ranging between 1.0 and below 250 Hz the disagreements are larger. Although it is physically impossible to absorb more than 100 % of the incident sound energy, some of the absorption coefficients are above 1.0. This is presumably because extra absorption occurs at the edges of the 10 m 2 absorber sample used in the measurements according to ISO 354 where the reflected sound wave from the absorber and the hard surface of the reverberation chamber is out of phase. 1

33 Figure 1. Absorption coefficients of test material as measured according to ISO 354 in 6 different laboratories. Absorption coefficients of the same material were also measured using an impedance tube, a planar array and a spherical array the spherical array being the topic investigated in this Danish Sound innovation project. These results along with estimates according to the Miki model are shown in figure 2a and figure 2b. It is noted that all results are less or equal to 1, that there is not a large difference between results with 0 and 30 degrees incidence and that results are close to 1 from the 500 Hz band an up. There are no data for absorption coefficients at gracing incidence i.e. angles greater than 80 degrees which may be significantly smaller. The 4 different methods are largely in agreement for frequencies above 125 Hz. 2

34 Figure 2a Absorption coefficients measured using planar and spherical arrays and compared to impedance tube and the Miki model at 0 degrees incidence. Figure 2b Absorption coefficients measured using planar and spherical arrays and compared to impedance tube and the Miki model at 30 degrees incidence. 3

35 The main difference between results according to ISO 354 and the array methods are that the latter always gives results below 1 and considerable lower absorption coefficients below 500 Hz. Absorption coefficients range in the Odeon software The Odeon room acoustics model is (simplified a bit) based on the Image source model for early reflections and a ray radiosity model for late reflections. The calculation model in Odeon is energy based thus makes use of absorption coefficients in octave bands. Upon reflection at a surface, a fraction of the sound energy, proportional to the absorption coefficient of the surface, is absorbed. This means all absorption coefficients must be within the range 0 to 1.0. Therefore any values greater than 1.0 must be truncated to values within 0 to 1.0 even if results given by ISO 354 exceeds this range. Whether the absorption coefficients should be truncated to 1.0 or e.g is an open question - measured according to ISO 354, even coefficients below 1 may be overestimated. Material sample and test room limitations The test material which was available for the measurements has the dimensions (l,d,h)=(2.45 m, 0.10 m, 1.85 m) so this limits the experiments to a fairly small room where the sample has a significant impact on acoustics. For this reason and because the room was available, the flanking transmission room at Acoustic Technology at the Technical university of Denmark was used. The room has the dimensions (l,w,h)=(4.38 m, 3.30 m, 2.95 m). The room is not ideal for comparison with simulations made in an energy based model because the Schroeder frequency below which energy based models such as those in Odeon are unreliable because of the low modal density. A prediction model like Odeon also performs better in geometries with higher complexity. P3 P P2 P1 Odeon Licensed to: Odeon Restricted version - research and teaching only! 4

36 Seconds[s] Figure 3. Odeon room model of the flanking transmission room at Acoustic Technology at the Technical university of Denmark with absorption sample, receiver positions and source positions used in experiments. Measurements setup Impulse responses were measured in the room with and without the absorption sample present. A total of 12 source-receiver pairs was used, three microphone positions and four loudspeaker positions. The Odeon room acoustics software was used in order to measure impulse responses and to derive room acoustics parameters in octave bands. An Omni directional loudspeaker, a dodecahedral made and owned by Acoustic Technology DTU and an omni directional microphone was used with the software. Measured reverberation time T 20 and T 30 are displayed in figure 4. Values for T 20 and T 30 are in good agreement, the T 20 values being marginal shorter than T 30,, which is expected when absorption is not completely evenly distributed. In particular for the empty room state, reverberation is lower for T 20 than for T 30 this is probably because the ceiling is made of plasterboard which gives higher absorption at low frequencies than the materials of the rest of the surfaces (concrete). The measured reverberation time in empty conditions is around 2.5 seconds at 500 Hz, so we will expect that the Schoeder frequency becomes F Schroeder= 2000*Sqrt(T/V)= 2000*Sqrt(2.5/(4.38*3.30*2.95))=484 Hz. The 500 Hz octave band starts at 355Hz so we should not rely too much on results below the 500 Hz octave which will be heavily influenced by modes. This is unfortunate as the largest differences between absorption data estimated using the ISO 354 and the spherical array method is found below 500 Hz. 3,00 2,50 2,00 1,50 1,00 0,50 T20 and T30 with and without sample 0, Frequency[Hz] T20, without sample T20, with sample T30, without sample T30, with sample Figure 4 Reverberation T20 and T30 in room with and with absorption sample. Values are averages of 12 source receiver pairs. Deriving absorption coefficients from measurements of reverberation time If the room had been large and diffuse enough, there was only one surface type in the empty state and the test sample had been 10m 2, we would be able to derive absorption coefficients according to ISO 354. The room though is too small, there is too little diffusion in the room (empty room with rigid walls) and the test sample was only 4,5m 2. Even so, for comparison we estimate the absorption coefficients from the measured reverberation times without absorption sample (Table 1) and with the absorption sample (Table 2). The method we use is similar to the ISO 354 method but we 5

37 do subtract the area of the wall behind the absorptive material, when calculating the absorption, because of the small size of the room. Measurements are also done in full octaves not in 1/3 octaves as given in ISO 354. This report describes the measurements and simulations in a small room with and without a test material, using Odeon to estimate the absorption coefficients. The flanking transmission room at DTU appear to have concrete walls, another type of concrete floor and plasterboard ceiling and the room and the test material has dimensions as on page 4. We use 12 measuring pairs of three microphone positions and four loudspeaker positions. The Omni directional loudspeaker is the dodecahedral made and owned by DTU. The microphone was an omni directional microphone plugged to a computer using the Odeon software for measuring the impulse response using the sweep method. The measured reverberation time in empty conditions is up to 2.5s, so we will expect a Schroeder frequency around 484Hz. The 500 octave band starts at 355Hz, and we mainly look at the octave bands Hz, being aware that results in the 500Hz octave band is also influenced much by modes. If assuming that all surfaces in the room without the absorber present was of the same material, then the absorption properties of that material would be as listed in table 1. However, it does appear that materials are different on walls, floor and ceiling indeed the room is not diffuse field. Hz Average 0, ,084 Table 1. Absorption coefficients of walls in empty room, estimated from measurements only Hz Average ,944 Table 2 Absorption coefficients of absorption sample, estimated from measurements only It is noted that the absorption coefficients taken from the measurements in the flanking transmission room in table 2 are lower than values obtained according to ISO 354 and the array methods for 500 Hz and up most noticeable are the bands 500 and 1000 Hz where the coefficients are as low as 0.63 and Estimating absorption coefficients taking into account non diffuse behavior of room In order to take into account the none-diffuse behavior of the room, two computer models of the room was made. One in empty state and one with the sample present. In both cases Odeon was used to estimate absorption coefficients of surfaces in the room, using the measured data to match with the simulated. The absorption coefficients from the empty room model were used for input in the model with the test material. In order to match the simulated room best against the measured one, the Genetic Material Optimizer (GMO) which is a tool built into Odeon was applied [1]. This method tests different material combinations in order to find the ones giving the best agreement between selected measured and simulated room acoustic parameters at each octave band, while preserving variation with position over the 12 source receiver pairs. Parameters for estimations At first the chosen parameters were T 20 and T 30. This however later ended in Odeon giving the same absorption coefficients for the test material as the derived earlier, with a drop around Hz (see table 2). The parameters EDT, T 15, T 20 and T 30 had one problem in common, the differences between the different receivers were very small, making it hard for Odeon to judge which surfaces contributes with how much absorption. The chosen parameters were therefore changed to T 30, t s, C 50 and C 80. These parameters varies significantly with position, taking into account the none diffuse acoustic behavior of the room, making it easier for the GMO-tool to give meaningful results. 6

38 Simulation and estimation in empty room In the empty state there were assumed to be different hard materials on floor, ceiling and walls. After optimization using the GMO-tool, these absorption values ended as seen in Table 3 (for the parameters T 20 and T 30) and Table 4 (for the parameters T 30, t s, C 50 and C 80). Hz Walls 0,28 0,01 0,007 0,011 0,031 0,043 0,047 0,033 Floor 0,01 0,008 0,013 0,037 0,024 0,045 0,058 0,04 Ceiling 0,077 0,128 0,127 0,101 0,051 0,037 0,019 0,02 Table 3. Estimated absorption coefficients for three different materials in empty state. Used parameters: T 20 and T 30 Hz Walls 0,024 0,007 0,006 0,01 0,018 0,029 0,022 0,004 Floor 0,054 0,011 0,007 0,01 0,03 0,021 0,029 0,008 Ceiling 0,173 0,191 0,187 0,171 0,097 0,112 0,147 0,159 Table 4. Estimated absorption coefficients for three different materials in empty state. Used parameters: T 30, t s, C 50 and C 80. Simulation and estimation in room with test material In the model with the test material the GMO-tool was used in order to estimate the effective absorption coefficients of the test sample, taking into account the none diffuse behavior of the room. The GMO-tool was allowed to change the absorption coefficients found in the empty room conditions with 50% and the test material 100% [1]. The GMOtool was used in two different cases with different parameters, using values according to the empty conditions model. The Genetic algorithm may find different results if running the search process multiple times as the random materials used in the search process are not identical. We can t know for sure that we have found the global minimum for the fitness function. For this sake the table 5 and table 6 below shows two different estimates for each case, and an average. In Table 5 the results of using T 20 and T 30 as parameters in the GMO-tool are shown, and in Table 6 the results using T 30, t s, C 50 and C 80 are shown. Test material 63 Hz 125 Hz 250 Hz 500 Hz 1000 Hz 2000 Hz 4000 Hz 8000 Hz Estimate Estimate Average 0,321 0,9835 0,9925 0,5955 0,581 0,878 0,9995 0,989 Table 5. Estimated absorption coefficient of test material. Used parameters: T 20 and T 30. Test material 63 Hz 125 Hz 250 Hz 500 Hz 1000 Hz 2000 Hz 4000 Hz 8000 Hz Estimate 1 0,149 0,91 0,985 0,986 0,977 0,925 0,918 0,989 Estimate 2 0,053 0,98 0,995 0,997 0,984 0,936 0,933 0,975 Average 0,101 0,945 0,99 0,9915 0,9805 0,9305 0,9255 0,982 Table 6. Estimated absorption coefficients of test material. Used parameters T 30, t s, C 50 and C 80. The resulting average absorption coefficients on the walls, ceiling and floor after the GMO-tool was used in the room with test material and parameters T 30, t s, C 50 and C 80 can be seen in Table 7. These are not noted for the other case. 7

39 Absorption coefficients 63 Hz 125 Hz 250 Hz 500 Hz 1000 Hz 2000 Hz 4000 Hz 8000 Hz Walls 0,023 0,011 0,024 0,027 0,02 0,039 0,039 0,01 Floor 0,072 0,013 0,007 0,015 0,022 0,021 0,02 0,007 Ceiling 0,216 0,211 0,206 0,083 0,081 0,078 0,098 0,142 Table 7. Estimated absorption coefficients on the walls, ceiling and floor in the empty room state. Used parameters T 30, t s, C 50 and C 80. These absorption values are compared to the ones found in the empty room model to test the model. In figure 5 we see that the materials are very comparable. 0,3 Comparison of absorption coefficients of same materials 0,25 0,2 0,15 0,1 0, Hz Frequency[Hz] Walls1 Floor1 Ceiling1 Walls2 Floor2 Ceiling2 Figure 5. Estimated absorption coefficients of wall materials in the empty room state. Used parameters T 30, t s, C 50 and C 80. Comparison of results The two different cases of using the GMO tool gave very different results. These are compared to the ISO 354 inspired method used on the same room and test sample and the Spherical array method used on the same material but in an anechoic chamber. It is assumed that Spherical array is a good guess of the correct absorption coefficients, where the ISO 354 is assumed to have difficulties in the small room influenced by modes. The graphs show results 8

40 Absorption for frequencies much lower than 500 Hz, but the results at low frequency below 500Hz are unreliable because of the Schroeder frequency. 1,4 1,2 1 Comparison 1 of absorption coefficients using different methods 0,8 0,6 0,4 0, Frequency[Hz] Spherical array Odeon GMO using T30, Ts, C50 and C80 ISO 354 inspired method on small test room Odeon GMO using T20 and T30 Figure 6. In this graph there is a strong correlation between the ISO 354 on small test room and Odeon GMO (T20 and T30), and also a strong correlation between Odeon GMO(T30, Ts, C50 and C80) and the average of multiple different tests of the material in other rooms. This indicates that using Odeon in a room that is not a perfectly diffuse reverberation room can give more accurate than the ISO 354 results if used with the right parameters. In the second comparison we only use the Odeon GMO method with the room acoustic parameters T30, Ts, C50 and C80, seeming to be the best method to take into account the non-diffuse behavior of the room. This is compared to other methods on the same test sample but in other rooms under suiting conditions. These methods the spherical array again, but also the planar array, Miki s model and using an impedance tube. 9

41 Absorption Comparison 2 of absorption coefficients using different methods 1,2 1 0,8 0,6 0,4 0, Frequency[Hz] Spherical array Planar array Odeon GMO using T30, Ts, C50 and C80 Impedance tube Miki's model Figure 7. In this graph there is not much difference from the absorption using the different methods in suitable situations and using Odeon in the very small room. The result lower than 500 Hz is less reliable because of the Schroeder frequency for the room. Conclusions The purpose of this rapport is to test and compare different methods to determine the absorption coefficients of a test material. This rapport describes in more detail the method of using the Genetic Material Optimizer (GMO) in the Odeon software. This method can use different room acoustic parameters, and two cases of parameters are tested. One using T 20 and T 30, and another case using T 30, t s, C 50 and C 80. Besides these two cases a calculation using a ISO 354 inspired method is made on the same test room. All these results are compared to measurements and calculations made by other methods. The room is poorly suited for this task because it is too small and box-shaped, and therefore very influenced by modes. Odeon is an energy based modeling tool, so these modes cannot be part of the calculations. Therefore only frequencies above 500 Hz is reliable. This is unfortunate and is clearly visible in the results. It is hard to say if the array or ISO 354 is preferable. It is worth noticing that estimates from Odeon GMO and ISO 354 gives higher values at low frequencies than the array methods. Presumable this is due to the absorption caused by the edge effects. Even if the array method gives mere correct results it does not solve the problem that edge effects does give extra absorption in real rooms where hard and soft materials joins. Algorithms taking this into account in simulations needs to be developed. When comparing the different estimations using different methods it is seen that the Odeon GMO method using T 20 and T 30 gives results comparable to using the ISO 354 inspired method in the room. These estimations however, have a dip around Hz. On the other hand the Odeon GMO method using T 30, t s, C 50 and C 80 have comparable results to the other methods done earlier on this test material, which gives you the impression that Odeon GMO can be a more accurate method in cases where the room is not completely diffuse. However, using reverberation time alone in the Odeon GMO tool is not recommended for this case, because the reverberation time for the different positions is very similar. 10

42 Further work Because of the small size of the room, it is still not found out how Odeon GMO works with lower frequencies. This could be useful because it is here the different methods very the most. This however, requires a room where even low frequency modes have little influence. Another interesting case would be to test a material with a lower absorption, for example around 0.5 for all frequency bands, and see if this would result in bigger differences between different methods for measuring absorption coefficients. Or an implementation in Odeon making it possible to have higher absorption coefficients than 1, could make better comparison available. References [1] Claus Lynge Christensen, George Koutsouris and Jens Holger Rindel. Estimating absorption of materials to match room model against existing room using a genetic algorithm. Forum Acusticum 2014, Krakow, Poland [2] 3 ISO Acoustics - Measurement of room acoustic parameters - Part 1: Performance spaces, Geneva: International Organization for Standardization, [3] ISO 354. Acoustics Measuring of sound absorption in a reverberation room. [4] Odeon Room Acoustics Software User s Manual - Appendix Example of measured decay curve: Example of Odeon Genetic material optimizer window: 11

43 12

44 THEORY OF THE S-ESM METHOD [3], [4], [5] SPHERICAL SOUND FIELD SCATTERED BY A RIGID SPHERE When handling scattering by a given rigid sphere, the use of a spherical coordinate system where the sphere is centered at the origin proves to be convenient. In that case, a point in space is determined by its radius rr, its polar angle θθ and its azimuth angle φφ. The radial dependence and angle dependence of the pressure field are studied separately with a different set of functions [7]: The angle dependence are characterized with spherical harmonics, given by the formula: YY nn mm (θθ, φφ) = 2nn+1 (nn mm)! PP 4ππ (nn+mm)! nn mm (cos θθ)ee jjjjjj mm, wherepp nn is the associated Legendre function of the first kind. The radial dependence can either be described with the spherical Bessel functions (jj nn (kkkk) and yy nn (kkkk)) or the spherical Hankel functions (h nn (1) (kkkk) and hnn (2) (kkkk)), depending on the problem. Using these functions, it is possible to express the complex pressure on a rigid sphere of radius aa at an angular position (θθ kk, φφ kk ) noted Ω kk when the sound field is generated by LL point sources, located at (rr 0,ll, Ω 0,ll ) [3]: LL nn pp tt (aa, Ω kk ) = jjjjjjqq ll aa 2 h (2) nn (kkrr 0 ) h (2) nn (kkkk) YY nn mm (Ω kk )YY mm nn Ω 0,ll. (6) ll=1 nn=0 mm= nn In order to simplify the numerical calculation, the sum is usually truncated to NN. If KK different positions on the sphere are studied, each point can be determined with the same method, which underlines a linear relation between the vector of measured pressures pp tt (size KK) and the vector of source strengths qq 00 (size LL): pp tt = GG NN qq. (7) GG NN is a transfer matrix of size KK LL and takes into account the setup parameters, such as the source and receiver positions, the scattering by the array or the studied frequency. EQUIVALENT SOURCE METHOD The equivalent source method consists in representing the sound field with a set of LL arbitrarily chosen sources and trying to determine the adequate source strengths qq in equation (7) to account for a set of measured pressures pp tt. Once the equivalent source strengths have been determined, they can be used to reconstruct the sound field in a given region. It must be noted that the scattering by the sphere has been accounted for in the calculation of GG N, which ensures that the reconstructed field should correspond to the one without the presence of the sphere. The main issue with such a problem is that it is usually underdetermined (KK LL), which results in the multiplicity of mathematical solutions to (7). That is why additional knowledge on the physical solution. One useful important hypothesis is that if the arbitrary sources are properly chosen, only few of them are needed to describe the sound field. Therefore a sparse solution should be favored. This can be done by trying to minimizing the norm L-0 which counts the number of non-zero components of a vector of the solution: argmin qq 0 subject to GG NN qq pp tt 2 εε. (8) qq Such a formulation poses however a problem as it is not a convex expression, making it computationally heavy. However, if pp tt is sufficiently sparse, and if the columns of GG NN are incoherent, changing the norm L0 into a norm L1 in equation (8) results in the same solution. The modified problem being convex, it is much easier to solve [13]:

45 argmin qq 1 subject to GG NN qq pp tt 2 εε. (9) qq The advantage of this setup, known as Compressive Sensing, is that one recovers a sparse solution, which is suitable for sound fields generated by a few sources only. Therefore, the solution is expectedly more accurate than the result of traditional energy-based methods such as the least-squares method, as they usually result in smeared peaks around the actual source positions. The problem however depends on the choice of a noise floor εε. If εε is too low, then noise is included in the solution, which disrupts the representation of the sound field and may violate the sparsity assumption. On the other hand, if εε too high, some relevant components of the sound field may be discarded as noise, which would also affect the quality of the reconstruction. ADDITIONAL GRAPHS FOR THE STUDY OF SAMPLE SIZE IN ANECHOIC MEASUREMENTS POROUS ABSORBER WITHOUT FOIL Dansk Sound Innovation Network

46 POROUS ABSORBER WITH FOIL Dansk Sound Innovation Network

47 ADDITIONAL GRAPHS FOR THE STUDY OF SAMPLE SIZE IN AN ORDINARY ROOM POROUS ABSORBER WITHOUT FOIL Dansk Sound Innovation Network

48 POROUS ABSORBER WITH FOIL 0 POROUS ABSORBER WITH FOIL Dansk Sound Innovation Network

49 RESULTS AND ANALYSIS FOR THE CAR SEAT SAMPLE Figure 40 - Estimated impedance for the car seat sample. Figure 41 - Estimated absorption coefficient for the car seat sample. 49 Dansk Sound Innovation Network

50 The resulting plots show that the estimated impedance is not stable; it is especially hard to perceive a trend above 1000 Hz. The same conclusions are logically applicable to the absorption coefficient results. It might be possible though that the range Hz is characterized by a high absorption, whereas absorption decreases outside of this range. Figure 42 - Equivalent sources for the car seat sample at normal incidence for three frequencies (200 Hz, 500 Hz, 1000 Hz) The study of the equivalent sources (Figure 42) shows that the source localization is actually effective; the attributed source strengths are more questionable. As it was seen for the small sample earlier, it is again possible that scattering from the edge influences the estimated power of the image source, thus reducing the apparent absorption coefficient. Bringing the source closer to the array would limit this effect. In addition, as it was seen in the repeatability results, the absorption calculation is quite unstable for lower values of αα, which would be the case above 1 khz. 50 Dansk Sound Innovation Network

51 COMPARISON BETWEEN THE DIFFERENT METHODS LARGE SAMPLE 0 INCIDENCE ANECHOIC ROOM LAB ROOM 51 Dansk Sound Innovation Network

52 30 INCIDENCE ANECHOIC ROOM LAB ROOM 52 Dansk Sound Innovation Network

53 MEDIUM SAMPLE WITHOUT FOIL 0 INCIDENCE ANECHOIC ROOM LAB ROOM 53 Dansk Sound Innovation Network

54 30 INCIDENCE ANECHOIC ROOM LAB ROOM 54 Dansk Sound Innovation Network

55 MEDIUM SAMPLE WITH FOIL 0 INCIDENCE ANECHOIC ROOM LAB ROOM 55 Dansk Sound Innovation Network

56 30 INCIDENCE ANECHOIC ROOM LAB ROOM 56 Dansk Sound Innovation Network

57 SMALL SAMPLE WITHOUT FOIL 0 INCIDENCE ANECHOIC ROOM LAB ROOM 57 Dansk Sound Innovation Network

58 30 INCIDENCE ANECHOIC ROOM LAB ROOM 58 Dansk Sound Innovation Network

59 SMALL SAMPLE WITH FOIL 0 INCIDENCE ANECHOIC ROOM LAB ROOM 59 Dansk Sound Innovation Network

60 30 INCIDENCE ANECHOIC ROOM LAB ROOM 60 Dansk Sound Innovation Network

61 CAR SEAT SAMPLE IN THE LAB ROOM 0 INCIDENCE 30 INCIDENCE 61 Dansk Sound Innovation Network