Experimental investigation on varied degrees of sound field diffuseness in enclosed spaces

Transcription

1 22 nd International Congress on Acoustics ` Isotropy and Diffuseness in Room Acoustics: Paper ICA Experimental investigation on varied degrees of sound field diffuseness in enclosed spaces Bidondo, A (a), Xiang, N. (b), Herder, J. (b) (a) Dipl.Sound Engineering Program. Universidad Nacional de Tres de Febrero, Argentina, abidondo@untref.edu.ar (b) Graduate Program in Architectural Acoustics, Rensselaer Polytechnic Institute, Troy New York. xiangn@rpi.edu, john.g.herder@gmail.com Abstract Sound field diffusion in enclosures should be experimentally quantified based on measured room impulse responses. A parameter, the sound field diffusion coefficient (SFDC) is under development. This parameter includes relative global crossover time and its standard deviation of their value over frequency bands. The SFDC expresses the reflection's amplitude control and temporal distribution uniformity, using both broadband and third octave-band energy-decay compensated impulse responses and taking reference with those parameters from a set of impulse responses synthesized with Gaussian white noise. In an attempt to demonstrate the quantification capability of the SFDC, a systematic investigation is conducted whereby varied room configurations using carefully designed scattered interior surfaces are examined with the hypothesis that varied degrees of surface scattering will ultimately lead to varied degrees of sound field diffusion in the enclosure. To this end, a scale-model room is established with interior surface configurations ranging from totally plane surfaces to diffusely reflecting surfaces that cover large portions of the enclosure's interior area. This paper discusses the experimental design and evaluates the results of data collected using systematic modifications of varied degrees of surface scattering, each with combinations of different source orientations and microphone positions. Keywords: Sound field diffusion, SFDC, crossover time, systematic investigation.

2 Introduction For decades, diffusion of a sound field has been a phenomena without exact and precise definition, measurement methods and / or parameters to quantify it. Some attempts were made by: a) counting peaks of impulse responses registered at different places inside a room, b) analyzing the curvature of decays, c) the variation of reverberation time with position, and d) analyzing the uniformity of sound energy captured by a highly directional rotating microphone. Later, a numerical method to describe the probability of existence of high amplitude local peaks came out. To continue the efforts listed above, the main objective of this research is to systematically investigate the relationship between different degrees of sound field diffusion and a SFDC coefficient, under development. The developed estimator, SFDC, quantifies two sound field attributes in one physical measure: the degree of reflection amplitude control and the degree of gaussianity of reflections, from a decay cancelled room impulse response (RIR). Both results are compared with the average outcomes using Gaussian White Noise. The main purpose of this research is to investigate the relationship between the developed SFDC estimator and the diffuseness of the sound field. To achieve this, a systematic investigation varying the diffusing surfaces inside a test enclosure are carried out using measurements of corresponding RIRs and estimations of the SFDC. 2 Sound Field Diffusion Coefficient (SFDC) 2.1 General description The Sound Field Diffusion Coefficient, SFDC [1], is the degree of amplitude control and gaussianity of a set of temporal reflections (equation 1) in a monophonic room impulse response (RIR) after subtracting the energy decay and normalizing it with respect to its reverberation time. This analysis is done in three time intervals of the RIR: an early part, a late part, and the sum of both. The temporal separation is called tsplit. In addition, by means of reference values of amplitude control and the degree of gaussianity extracted from synthetic gaussian white noise (GNW) RIRs, the SFDC can be expressed in GWN units (GWNu) [2]. Amplitude _ control Gaussianity _ of _ reflection s _ distribution SFDC (1) The SFDC's gaussianity of reflection s temporal distribution is termed distr_coef and is inversely proportional to the normalized kurtosis, k 0, of the existing reflections in the selected time interval as 1 distr _ coef k0. (2) 0,02 On the other hand, the k value takes the ratio of Eq. 3, by one-third octave bands, to a group of sensitivity values (by one-third octave bands) to overcome what is called the wave problem. R( k) z( k) (3) R TOTAL 2

3 where t 2 2 R TOTAL h ( t) dt (4) t1 t 2 2 k h R k ( t) dt (5) t1 The larger that k is found to be in a RIR, the less amplitude control is shown due the existence of discrete high-amplitude reflections; thus a low k value represents high reflection energy control or uniformity of amplitudes in the analyzed RIR. Then, the general and conceptual equation of the SFDC is 1 SFDC dist _ coef normalized k. (6) 2.2 Numerical results of SFDC The SFDC calculation outputs estimators in [GWNu] over third octave frequency bands using the early and late portions, as well as the total experimentally measured RIRs, with the tsplit considered as the time limit between early and late sound fields. A set of 3 single overall results are produced for each sound field configuration: overall early, late, and total SFDC are calculated as averaged SFDCs of selected frequency bands. Furthermore, the relative crossover time, Ct-r [ms], is calculated. Analyzing a room in a statistical manner describes the time limit where its response changes from a reflections with clearly identifiable provenience (i.e. the early sound field, with memory) regime to an indistinctly identifiable provenience and memoryless regime (the late sound field [3]), simply from a RIR (not knowing the volume of the room). A strictly stationary system, where probabilistic distribution is constant in time (Gaussian in this case) and every given portion of samples has zero mean and finite power, is defined as a second order process with constant mean. In these types of processes, named Wide Sense Stationary Gaussian processes, the experimental observation of the autocorrelation function (ACF) provides all the joint distributions, which describe the whole collection of random variables. Thus, autocorrelation fully describes the WSSG process under analysis. Then, the autocorrelation of RIRs may show the moment when a system turns stochastic, losing all memory and self - similarity. The boundary between both room behaviors, splitting the early sound field from the late one, (defined as the crossover time), is associated to the correlation duration, m, of the ACF. To calculate this from the absolute value of the ACF over the squared RIR [4], the following equation m 1 2 d 1 2 acf acf 0 d (7) is applied with () being the normalized ACF function. 3

4 The above integral defines the correlation duration as the half base length of a rectangle of height 1, the area of which is equal to the area of the normalized ACF curve () over the axis. Then ct, relative to the RIR, will be the duration of the normalized height (amplitude) integral of the absolute value of the squared RIR ACF, expressed as C 2 relative acf normalized IR t) 0 ( dt t, (8) where Ct-r is calculated for the full bandwidth IR and for each third octave band, jointly with its normalized standard deviation. 3 Systematic investigation In order to vary just one physical property while keeping the others unchanged to the greatest extent possible, the diffusely reflected surfaces of a test room are systematically changed, and the results of the SFDC processing over a set of experimentally measured RIRs are obtained. A systematic investigation is developed using a scale-modeled room by adjusting portions of flat and diffusive surfaces. 3.1 Room Conditions Three different cases of diffusely reflected surfaces are studied inside an 8 th scale-modeled room: Condition 1, with all flat surfaces (no diffusers) inside, Condition 2, with 43% of its interior surfaces covered by mixed diffuser types (random and QRD 7), and Condition 3, with 86% of its surface area coated with mixed diffuser types (random and QRD 7). Photographs in Figure 1 show the three room conditions. The top cover of the model is featured with a non-diffusing surface. Figure 1. From left to right: Condition 1, without diffusers, Condition 2, with 43% diffusing surfaces, and Condition 3, with 86% diffusing surfaces. A grid of 50 receiver positions is defined inside the test room, as depicted in Figure 2. 4

5 Figure 2. Scale model with the receiver s positions grid. On top of the scale-modeled room, a step-motor-driven scanning mechanism outside the room moved a foot containing two strong magnets, while the microphone holder inside also contain two strong magnets. In this way, the microphone movement was solely controlled via magnetic coupling through the top cover of the room, which is made of 8mm thick Plexiglas. The entire scanning mechanism is defined over two perpendicular axes (x-y) with a microphone repositioning uncertainty on the order of 1.0 mm [5]. An ultrasonic sound source in dodecahedron form covering a frequency range from 1 khz to 45 khz was positioned inside the room far from the receiving grid points measuring a RIR at every receiver position. To remove any possible irregularity of source radiations, the sound source was rotated over 36 fixed orientation angles. For each source orientation and each receiver position, the room impulse responses were measured experimentally. In this study, 12 rotation angles are involved to obtain a mean value of the SFDC, which is comparable with the results from averaging all 36 angle rotations. A correlation technique is employed for measuring all of the RIR s using a logarithmic sine sweep. 3.2 Signal to noise ratio All of the measurements are automated and robotically assisted, using the same microphone and under the same conditions. In general, the peak-to-noise ratio (PNR) of all the experimentally measured RIRs across all frequency bands under investigation were higher than 60 db. 3.3 Temporal split in room impulse responses The temporal split, tsplit, of every RIR for the early and late separation, is set as the calculated relative crossover time, Ct-r [ms], for every calculation. As Ct-r changes when the diffusion increases, the analyzed time intervals (early and late) for calculation of the SFDC also changes, allowing the comparison of the early and late physical sound field results among unique RIRs. 5

6 4 Results Calculation of all SFDC s are carried out for 14 receiver positions inside the scale-modeled room, chosen from well spread receiver positions over a grid 5 by 10 and 12 rotation angles (30 degrees each) at each receiver position, giving a total of 504 calculations. The chosen physical positions can be seen in Figure 3. Table 1 lists the averaged results. Table 1. Global results and Conditions physical values. CONDITION Global values SFDC Early 0,922 1,651 1,913 SFDC Late 1,122 1,475 1,581 SFDC Total 0,740 0,959 1,028 Ct-r [ms] 65,935 30,646 27,500 Rt30 2,04 0,89 0,74 Total inner surface [m2] 210,42 216,04 231,37 Room Volume [m3] 202,65 211,17 234,72 Diffuser's percentage Diffuser's surface [m2] 0 92, ,9782 Figure 3. Source and receiver positions in the model room. Red dots represent 14 studied receiver positions; the blue circle represents the rotating sound source position. Box Wisker plots from Figure 5 show the median, the 25% and 75% quartile boxes, and maximum and minimum values for all the processed IRs. Plots a), b) and c) from Figure 5 show the distributions of the SFDC results around the median values for Conditions 1, 2 and Diffusion variation and SFDC Figures 4 a), b) and c) illustrate the SFDC results estimated from the experimentally measured RIRs over third octave frequency bands for the early, late and total sound fields, for the three different room conditions. Figure 5 shows the overall values in Box Whisker plots. 6

7 Figure 4. a) Left: Early SFDC results as a function of frequency, b) Center: Late SFDC results as a function of frequency, c) Right: Total SFDC results as a function of frequency. All values were averaged over the 14 receiver positions, for Condition 1 (black line), 2 (red line) and 3 (blue dashed line). Displayed values are not surface volume corrected. Figure 5. Box Whisker plots. a) Left plot: Overall Early SFDC for the three acoustical conditions. b) Center plot: Overall Late SFDC for the three acoustical conditions. c) Right plot: Overall Total SFDC for the three acoustical conditions. Displayed values are not surface volume corrected. 4.2 Relative crossover time (Ct-r) Figure 6. Box - Whisker plots on Ct-r for Conditions 1, 2 and 3, showing median values, upper and lower limits. 7

8 Figure 6 and Table 1 convey that the Ct-r is lower for the condition with diffusers than without diffusers. 5 Discussion From scale-down room analysis, the condition with the largest reverberation time yields a Schroeder frequency of 247 Hz. For this reason, the lowest SFDC analysis frequency is set as 315 Hz. As the maximum frequency radiated by the sound source was 45 khz in the 8 th scale- model, the maximum analysis frequency for the original-sized room is set to 3.15 khz to assure valid results of the SFDC. Although Condition 3 doubles the surface covered by Condition 2 (i.e. 86% vs 43% of the coated surfaces being diffuse), the QRD diffusers were not modulated, so repetition of the sequence did not maintain the scattering capabilities of just one sequence. This effect can be seen in figure 7, which shows the scattered energy as a function of the output angle of QRD 7 diffusers. Figure 7. Scattering from N = 7 QRDs at 3,000 Hz for a different number of periods. Left: 1 period; middle: 6 periods; right: 50 periods. Locations of lobes and directions of similar level are marked by radial lines at ±76, ±40, ±19 and 0 (from [6]). Interior volume and surface area had little change in Condition 2 and 3 with respect to Condition 1. These can be corrected assuming sound field diffuseness is directly proportional to the diffuser s coating surface and inversely proportional to the volume of the room. Equations 9, 10 and 11 show the SFDC difference,, from Conditions 2 and 3, taking Condition 1 as a reference. sf Volume sf SFDC Surface final, (9) SFDC reference where Volume Condition sfvolume, is the volume scale factor, (10) Volume reference Surface Condition sfsurface, is the surface scale factor. (11) Surface reference The corrected differential, values from the (reference) Condition 1, are listed in Table 2. 8

9 Table 2. Corrected sound field diffusion coefficient (SFDC) for overall, late and total results, for interior volume and surface variations. No Volume & Surface Correction Volume & Surface corrected CONDITION Diffusers coating percentage [%] Overall SFDC Late 1,122 1,475 1,581 Overall SFDC Total 0,740 0,959 1,028 Overall SFDC Late (from reference -initial state) 0,353 0,460 Overall Total (from reference -initial state) 0,220 0,288 REFERENCE Overall SFDC Late (from reference - initial state) 0,375 0,544 Overall SFDC Total (from reference - initial state) 0,234 0,343 Although Condition 1 has no diffusers inside, it shows a degree of SFDC different from zero. As the SFDC parameter is not based on quotients of different intervals of RIR s energy nor energy decay (because SFDC calculations come after the RIR s decay is cancelled), the results of this experiment seem to reflect the sound field diffusion behavior while making changes in coatings and their covered surfaces. 6 Conclusions The sound field diffusion coefficient (SFDC) is shown to reflect a change of properties of the early, late and total sound field with regard to diffusely reflecting surface variation in a test room. A diffusion quantity, SFDC, towards 1 (one) would indicate that the reflections in the analyzed time interval has reached the mean properties of Gaussian White Noise (GWN). No physical diffusers in a room does not indicate no diffusion in its sound field (SFDC = 0). Every room will reach a certain degree of diffusion after a certain amount of time. Considering the SFDC as a sound field diffusion estimator: Without volume and surface corrections (taking Condition 1 as a reference room without diffusers), this experiment showed that Condition 2, with 43% scattering surfaces, presented 77% more overall early diffusion and 33% more overall late diffusion, meaning a 31% increase in overall Total diffusion and a 53% less Ct-r than Condition 1. In a similar fashion, Condition 3, with 86% scattering surfaces, presented 102% more overall early diffusion and 42% more overall late diffusion, meaning almost 40% more overall total diffusion and almost 59% less Ct-r than Condition 1. With volume and surface corrections, Condition 2 presented 33.4% more overall late diffusion, meaning 31.6% more overall total diffusion than Condition 1. Similarly, Condition 3 presented 48.4% more overall late diffusion, representing a 46.3% overall total diffusion increase from Condition 1. 9

10 This experimental investigation shows that diffusers reduce the Ct-r and the mean Ct-r standard deviation between frequency bands for each receiver point. A similar rate change on SFDC and Ctr was also found between conditions, considering that both quantities are obtained in a very different mathematical way. This reflects coherency between estimators and the analyzed acoustical conditions. All rooms, except anechoic chambers or a few special cases (including spherical enclosures, parallel reflecting walls), have a certain amount of diffusion and a certain value of Ct-r. Estimated diffusion in terms of the SFDC represents the degree of similarity between the properties of a group of reflections inside a time period of a room impulse response (RIR) to the average of the same properties of GWN. The above discussion may imply that adding diffusers in a room should not be intended just for the increase of the diffusion (quantity), but to force (with diffusers) the reduction of the Ct-r. The less the Ct-r, the earlier the room will add random reflections. To reduce the Ct-r means that the system will become memoryless earlier than without diffusers. Adding diffusers into a room forces its functioning to acquire more randomness, earlier. The above discussion of results show a relationship between the SFDC, sound field diffusion and diffuser coated surfaces. Room volume, room interior surface, scattering coefficients, diffuser location and SFDC relationships need further explorations. Acknowledgments The authors are especially grateful to Craig Schaeffer, who built the scale-model rooms with automatic high resolution scanning system in the scope of his MS degree thesis. Thanks are also due to Dr. Philip Robinson, who measured diffusion coefficients of the used diffusors. Also to Javier and Sergio Vazquez who permanently assisted on the SFDC development. References [1] Bidondo, A. Measuring Sound Field Diffusion: SFDC. 139 th Audio Engineering Convention. N.Y. USA [2] Bidondo, A. Gaussian white noise impulse responses as absolute diffusion reference values. ICA Buenos Aires. Argentina [3] Xiang, N., Sessler, G. Acoustics, Information, and Communication. Memorial Volume in Honor of Manfred R. Schroeder. Chapter 4: Estimating the crossover time within room impulse responses. Springer [4] Hidaka, T.; Yamada, Y.; Nakagawa, T. A new definition of boundary point between early reflections and late reverberation in room impulse response. Proceedings of Forum Acusticum [5] Xiang, N.; Escolano, J.; Navarro, J. M.; Jing, Y. Investigation on the effect of aperture sizes and receiver positions in coupled rooms. Journal of The Acoustical Society of America, volume 133 (6), June 2013, [6] Cox, T.; D antonio, P. Acoustic absorbers and diffusers. Theory, Design and Application. Chapter 9: Schroeder diffusers. 2 nd ed. Taylor & Francis