Proceedings of Meetings on Acoustics

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1 Proceedings of Meetings on Acoustics Volume 19, ICA 2013 Montreal Montreal, Canada 2-7 June 2013 Architectural Acoustics Session 1pAAa: Advanced Analysis of Room Acoustics: Looking Beyond ISO 3382 II 1pAAa1. Theoretic considerations on how the directivity of a sound source influences the measured impulse response Ingo Witew*, Tobias Knüttel and Michael Vorländer *Corresponding author's address: Institute of Technical Acoustics, RWTH Aachen University, Aachen, 52072, NRW, Germany, Ingo.Witew@akustik.rwth-aachen.de In previous investigations it has been shown that the directivity of a measurement sound source has a significant influence on the measured room impulse response (RIR). Using a specialized method of analysis the sources influence can be identified even in the very late part of the RIR even in very reverberant environments. These results seem to be surprising at first and contradict intuitive expectations. In this contribution the findings are briefly discussed and the congruence with general room acoustic theory is revised and discussed. Published by the Acoustical Society of America through the American Institute of Physics 2013 Acoustical Society of America [DOI: / ] Received 15 Jan 2013; published 2 Jun 2013 Proceedings of Meetings on Acoustics, Vol. 19, (2013) Page 1

2 INTRODUCTION Interest in the directivity of measurement sound sources rose with the availability of modern measurement techniques such as correlation measurements using electroacoustic sound sources. Initial studies focused on the question of which sound sources are appropriate to be used in acoustic measurements. Major studies in this context are the works of Pelorson et al.[1] and Bradley et al.[2]. Both groups compared the measurement results of different kinds of loudspeakers to each other and determined that the standard deviation of results from some particular sources had such extend these devices were identified as unsuitable. Although the origin of these deviations was associated to the radiation pattern of the sources a detailed analysis of the respective sources directivities was not pursued. Initial approaches to clearly show how the directivity of a sound source effects the measurement in auditoria have been made in Aachen, Germany[3,4]. In these studies different dodecahedron loudspeakers were placed on an electronically controlled turntable. Between repeated measurements the source was turned by a fixed angle. From these measured room impulse responses (RIRs) different acoustical quantities were calculated and plotted over the angular position of the turntable. This data shows that the variance of the different metrics increases over frequency with the likewise dependent directivity of the sound source. This question was picked up by San Martin, who assisted during these initial measurements. He conducted additional measurements in different auditoria in Spain[5] and used geometric room acoustic simulation software to generate a statistically sufficient basis for an in-depth discussion[6]. These results confirm the initial findings and allow a closer look how these effects occur in different auditoria. It can be seen that the early part of the impulse response is most significantly affected by the directivity of the sound source. A distinct influence of the source s directivity on the late part of the RIR was neither in the simulation results nor in the measured results evident. PROPERTIES OF SOUND SOURCES Although the requirements on measurement sound sources expressed in ISO 3382[7] are abstract, in practice sources using dodecahedron cabinets with surface mounted oscillatory membranes are wide spread. Other regular polyhedron configurations as discussed by Leishman[8] find, despite having acceptable radiation patterns, little distribution. Special concepts such as volume displacement sources with small aperture openings (e.g. B&K Omnisource) are also used seldomly. FIGURE 1. Measured directivity of a dodecahedron loudspeaker at 2 khz: the directivity shows a three fold (120 degrees) rotational symmetry around its vertical axis. The directivity of a given dodecahedron loudspeaker at the 2 khz-third-octave band is shown in Figure 1. All regular polyhedron designs have in common, that they have some kind of rotational symmetry. In this example a three-fold (120 degrees) symmetry can easily be observed. This pattern clearly relates to the geometry of the cabinet. Proceedings of Meetings on Acoustics, Vol. 19, (2013) Page 2

3 CONCEPT TO IDENTIFY THE SOUND SOURCE PROPERTIES IN MEASUREMENTS Starting point is the discussion of the measurement setup as it was used in previous studies[3-5] (i.e. dodecahedron sound sources placed on turntables). Due to the source s symmetry the distribution of energy radiated into the room will be very similar every 120 degree turn yielding likewise similar RIRs every 120 degrees. The search for periodic patterns in the properties of measured RIRs can be helpful to identify the influence of the loudspeaker s directivity on the RIR. In the following impulse responses with their different reflections are considered. The amplitude and the arrival time of the direct sound in the RIR obviously depend on the distance between the source and the receiver and the consequential propagation loss of the spherical wave. In a second factor the amplitude depends also on the directivity of the sound source and the damping at the direction of the direct sound path. For reflections the relation is almost the same but additional amplitude damping and time delay needs to be considered due to the longer path the sound wave has to travel to interact with any number of walls through its propagation. The influence of the directivity on reflections is likewise considering the emergent direction of the respective sound path. In a next step different RIRs are compared to each other that were derived in identical acoustic environments with only the orientation of the sound source differing from one individual measurement to the other. Considering the aspects discussed earlier it is obvious that the time of arrival of the direct sound and the reflections is constant as sound propagation, reflection and scattering is a deterministic process. Differences should be limited to the energy that is introduced into each sound path due to the sources directivity and, hence, should only be detectable in the amplitude of the direct sound and the reflections. Finding any such periodic patterns of the directivity and likewise in the energies arriving at the microphone can be used to determine the influence of the loudspeaker and its directivity. MEASUREMENTS IN DIFFERENT ROOMS Acoustic measurements very similar to those described in earlier investigations[3-5] were carried out in 4 rooms with different acoustic characters (see Table I). In each of the rooms, the same dodecahedron loudspeaker was placed on an electronic turntable. It was turned in 5 degree steps conducting a measurement for each angle. A number of microphones were placed in the room at arbitrary positions. The measurement series was repeated with the source placed at different locations in the room. Exponential sweeps were used for the excitation. Measurement sets were done fully automated and during the sequence nobody was inside the room. The room was left for 30 minutes to equilibrate. Repeated measurements during this period monitored the relaxation process and ensured that the room was acoustically stable before the actual set of rotation measurements started. TABLE I Acoustical properties of investigated rooms: the critical time t 0 denotes the time after the impulse start when the measured signal of the room impulse response falls below the 10 db level above the noise floor. Room name Volume [m³] Surface [m²] Reverberation time T 40 at 2 khz [s] Critical time t 0 at 2 khz [s] Seminar Room, Aachen Aula I, Aachen Philharmonie, ca Berlin Reverberation Chamber, Aachen The measurement series described in the previous paragraph yield sets of 72 RIRs each corresponding to one rotational angle of the loudspeaker. Data analysis was done in 5 steps to extract the influence of the loudspeaker. All nonlinearities at the end of the RIRs were cut off[9]. The signal length was chosen to have an FFTdegree of at least twice the length of the RIR-signal before it falls below the measurement noise. This ensured the second half of the measured signal only included the background noise and the nonlinearities and can hence be cut off automatically. Proceedings of Meetings on Acoustics, Vol. 19, (2013) Page 3

4 The stet of RIRs was time shifted as a whole so that the earliest direct sound in the set was positioned at t=0. This shifting is important to make the sets measured in different auditoria or at different sourcereceiver combinations comparable to each other as they will have the same time code. This shifting has no influence on the temporal relation of the RIRs within one set to each other as all RIRs of a set are shifted by the same amount. In addition the time shifting has not influence on the delay of any reflections to the direct sound as this remains untouched. All impulse responses were band filtered into third-octave bands. The filtered signals were time windowed with overlapping hamming-windows. The windows had a length of 10 ms with a respective 5ms rise and fall time. The overlap was chosen as 50% of the window time. Besides the need to keep the amount of data at a manageable size windowing is important for a second reason. A sample-accurate discussion of the impulse response is not very practical since the transient response of the loudspeakers membrane is much longer (frequency dependent, but in the magnitude of about 5 ms). The length of the time window was chosen for practical reasons to 10 ms to ensure it is small enough to preserve a high temporal resolution. A too small a window will have a negative effect that comes from the property that in many impulse responses there is a considerable time lag between the direct sound and the arrival of the first reflection (i.e. ITDG[10]). Too many windows falling into this part of the impulse response will yield a result (at these windows) that is dominated by noise effects. Larger windows quickly approach the lengths of integration ranges of room acoustic single number quantities (e.g. C 50 etc.). This analysis would provide few new findings compared to previous studies. Furthermore the pursued discussion is aimed on how the impulse response is affected. The energy average was calculated for each window. The overlap of the window ensured that the total energy of the RIR is preserved. FIGURE 2. Angular signal of the first 10 ms of a set of RIRs at 2 khz: the energy arriving at the microphone during a certain time interval clearly shows the periodic character of the directivity when plotted over the different rotational orientations of the measurement speaker. In the continuing line of argument a focus is placed on a given time window and it is analyzed how the energy arriving within that certain time interval changes for every orientation of the source. This discussion considers an energy-over-angle signal which is shown in Figure 2. Such a signal will be referred to as an angular signal. The signal, shown in the figure above clearly shows the three fold (120 degree) periodicity originating from the sound sources rotational symmetry. Also some higher order periodicities and a constant amount of energy can be observed. In order to better display a further processing step is conducted, namely, a discrete Fourier transform and normalization is carried out according to Eq. 1: N 1 n i2 N n e, and 1 n0 (1) Proceedings of Meetings on Acoustics, Vol. 19, (2013) Page 4

5 In this notation the index identifies the running variable: n denotes the angular signal of energies over the rotational angle n, N the total number of samples / angles of the angular signal and the rotational spectrum of periodicities. The normalization of the rotational spectrum allows for displaying the fraction of total energy carried by each of the periodicities and the constant part. FIGURE 3. Rotational spectrum of the angular signal shown in Figure 3: a Fourier spectrum of the angular signal helps to better display the distribution of energy among the different periodicities. The rotational spectrum of the angular signal shown in Figure 2 is displayed in Figure 3. The 120 degree periodicity is the most dominant one, followed by the 60, 40 and 30 degrees periodicities. Since they correspond to the rotational symmetries that can be found in the directivity pattern of the loudspeaker they can be recognized as characteristic periodicities (CPs). Other periodicities that do not correspond to those geometric symmetries are referred to no-characteristic periodicities (ncps). In a last step this analysis is applied to the entire RIR. In order to demonstrate how the prominence of different source periodicities depends on the frequency, Figure 4 shows the different directivities of the dodecahedron loudspeaker along with the rotational spectrum (ranging from 20 to 360 degrees) as a function of time. (a) 500 Hz (b) 1.6 khz (c) 4 khz FIGURE 4. Measured directivities of speaker and corresponding periodicities in RIRs at different frequencies: the change of the directional pattern in the speaker s directivity is reflected in the measured periodicities of the RIRs (Aula I). Proceedings of Meetings on Acoustics, Vol. 19, (2013) Page 5

6 More details about the temporal course of the 120 degree CP can be seen in Figure 5. Here the average over all source receiver combinations measured in Aula I is shown. It can be observed that in the very beginning (until about 50 ms after the direct sound) there is a strong drop-off of the 120 degree periodicity. After that the signal remains almost at a constant level only slightly decaying until it starts dropping to noise level at the end (here at about 1.1 s). Compared to the also shown 90 degree ncp a significant difference in energy can be observed. FIGURE 5. Averaged 120 and 90 degrees periodicities over all source- and receiver positions measured in Aula I at 2 khz: the mean of the 120 degree periodicity can be used as a representative for the CPs and the 90 degree periodicity for the ncps when focusing on the temporal developing of the periodicities. To conclude the discussion of measured data the influence of the source s directivity on RIRs as measured in different auditoria can be seen in Figure 6. The averaged 120 degree periodicity is prominent in all measured auditoria. The final drop-off to noise level coincides well with the critical times t 0 in the respective measurements (see Table 1). Although the special properties of the reverberation chamber yield results that differ from the results measured in the other auditoria but in a central finding it can be said that the influence of the loudspeaker can be detected until the very end of the RIR through the significant level of the 120 degree periodicity. FIGURE 6. Averaged 120 degrees periodicities of different rooms at 2 khz: comparing the temporal trend, it can be observed that the CPs remain prominent until the very end of the signal in all of the investigated rooms. Proceedings of Meetings on Acoustics, Vol. 19, (2013) Page 6

7 The presented findings seem to contradict intuitive expectations in which some kind of mixing in the diffuse sound field might be anticipated to cancel out the detailed properties of the sound source. In the next section the consistency with general room acoustic diffuse field theory is discussed. THEORETIC CONSIDERATIONS Starting point of this discussion will be the diffuse sound field and its properties. One of the properties of such an acoustic environment is that there is no net energy flow at any given point. This is synonym with the observation that the amount of energy arriving at a point from a certain solid angle integrated over the whole duration of the sound decay is independent from the orientation of that solid angle. This implies that in a diffuse sound field the sound paths in a room are evenly distributed and well mixed. Next a source and receiver with omnidirectional directivities is considered. With the reciprocity principle valid an exchange of the source with the receiver and vice versa will only exchange the signs of the sound propagation vectors and hence on both the receiver and the source side the originating and arriving trajectories of the sound paths are evenly distributed. The paths along which sound propagates in a room are solely depending on the geometry of the room. The line or argument is therefore viable for any room geometry in which a diffuse sound field can exist. Introducing a directivity to the source will not change the propagation paths but only the energy weighting of the different paths. As a result each sound path carries the energy scaled by the directivity of the source at that given source trajectory point. With sound propagation being a deterministic process the rotation of the sound source as it is discussed in the presented measurements will not change the paths along which sound propagates in each step of the turn. Only the distribution of energy within the different paths will be modulated by the directivity of the source at the respective starting point of the ray. This concept is visualized in Figure 7. FIGURE 7. Sound propagation in rooms: when rotating a loudspeaker during a measurement, the energy carried by different sound paths is scaled by the directivity of the speaker the sound paths themselves remain the same. Only the amplitude, but not the position of the individual reflections in time changes in the echogram. From a technical point of view it has to be realized that the modulation of the sound paths arriving at the receiver are related to each other through the sound source and its directivity. Never the less the uniform distribution of originating trajectories from the source allows the conclusion that the different rotational signals and their rotational phases associated to the sound paths are uncorrelated to each other. Next the time structure of the RIR is considered. In a fixed length time interval the number of reflections arriving at the receiver increases with the square of time. This means that the number of uncorrelated angular signals increases likewise. Picking out a single periodicity (This detached discussion is allowed since the periodicities are linearly independent from each other.) reduces the problem to the mathematical problem of a sum of uncorrelated periodic signals with random phase relations and amplitudes but the exact same frequency (here: periodicity). Proceedings of Meetings on Acoustics, Vol. 19, (2013) Page 7

8 The sum of these signals will also contain only the frequency innate to the summands. The only solution that could lead to a sum that is identical to zero can be excluded from a probabilistic point of view. This is independent from the number of summands. SUMMARY AND CONCLUSION In this contribution a method of analysis is presented to identify the influence of a sound source s directivity in a set of measured impulse responses through a specialized measurement sequence. It is shown that the influence of the source is detectable through the entire duration of the impulse response. Although this might not intuitive it is in line with theoretic considerations that are discussed in this paper. ACKNOWLEDGMENTS It is noted that the results and the line of argument discussed in this paper are a condensed version of a manuscript that has been submitted for consideration for publication to the Journal of the Acoustical Society. The authors gratefully acknowledge the funding by Deutsche Forschungsgemeinschaft DFG. REFERENCES 1. X. Pelorson, J.-P. Vian, J. D. Polack, On the variability of room acoustical parameters: reproducibility and statistical validity, Applied Acoustics, 37, (1992). 2. J. S. Bradley, R. E. Halliwell, Sources of error in auditorium acoustics measurements, Proc. ICA Beijing, 14, Paper F3-1 (1992). 3. M. Vorländer, I. B. Witew, Uncertainties in measurement of spatial parameters in room acoustics, J. Acoust. Soc. Am., 116, 2483 (2004). 4. I. B. Witew, G. K. Behler, Uncertainties in measurement of single number parameters in room acoustics, Proc. Forum Acusticum, Budapest, Hungary (2005). 5. R. San Martin, I. B. Witew, M. Vorländer, M. Arana, Influence of the source orientation on the measurement of acoustic parameters, Acta Acustica united with Acustica, 93, (2007). 6. R. San Martin, M. Arana, Uncertainties caused by source directivity in room-acoustic investigation, J. Acoust. Soc. Am., 123, EL (2008) 7. ISO :2009, Acoustics Measurement of room acoustic parameters, International Standardization Organisation, Geneva T. W. Leishman, S. Rollins, H. M. Smith, An experimental evaluation of regular polyhedron loudspeakers as omnidirectional sound sources, J. Acoust. Soc. Am., 120, (2006). 9. S. Müller, P. Masserani, Transfer-Function Measurements with Sweeps Director s cut including previously unreleased material and some corrections, J. Audio Eng. Soc, 49 (6), (2001) 10. L. Beranek, Concert Halls and Opera Houses Music, Acoustic, and Architecture, 2 nd. Ed., Springer Verlag, New York, p. 27, 2004 Proceedings of Meetings on Acoustics, Vol. 19, (2013) Page 8