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Evaluation of the crossover frequency based on the analysis of room transfer functions through statistical estimators
Applied Acoustics 164 (2020) 107247
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Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
Technical note
Evaluation of the crossover frequency based on the analysis of room transfer functions through statistical estimators Marcio Avelar a,⇑, Paulo Bonifacio b, Luis Sant’Ana a, Eric Brandao c, Elvis Bertoti a, Rodrigo Catai d a
Academic Department of Mechanics, Federal University of Technology, Curitiba 81280-340, Brazil Federal Institute of Technology, Itajai, Brazil c Federal University of Santa Maria, Santa Maria, Brazil d Department of Civil Engineering, Federal University of Technology, Curitiba, Brazil b
a r t i c l e
i n f o
Article history: Received 19 September 2019 Received in revised form 31 January 2020 Accepted 4 February 2020
2010 MSC: 00-01 99-00 Keywords: Schroeder frequency Crossover frequency Modal overlap
a b s t r a c t This paper presents two ways of processing a room’s transfer function with the goal to determine the crossover frequency above which statistical approaches are valid for predicting or evaluating the sound field. One of them uses Kurtosis for analyzing the real and imaginary components of a transfer function. The other one is based on a proper measure developed for the analysis of the phase signal. Both measures are related to the statistical distribution of values within a frequency window. Transfer functions measured in a large reverberation room and in a scale model were analyzed, and provide results for the crossover frequency which are in the expected range. Analytical responses from three shoebox rooms with the same volume and reverberation time on the one hand, and different geometric aspect ratios on the other hand, were also analyzed. Both measures have shown similar and consistent results in these cases, and are sensitive to the aspect ratio from the hypothetical rooms. The effect of increasing sound absorption on experimental and analytical cases were observed, and are also in accordance to the expectation of increasing the modal overlap. Ó 2020 Elsevier Ltd. All rights reserved.
1. Introduction In several practical situations one faces the problem of choosing the most suitable approach to analyze the sound field in a room. For acoustically small rooms, a modal approach is of higher importance, as opposed to acoustically large rooms, when statistical considerations provide more useful models (formulae and methods). A very useful estimation for the decision on whether a room is acoustically small or large is given by an equation to compute the Schroeder frequency. Such an equation is given in terms of the reverberation time and the room’s volume, being related to the frequency above which there is at least an overlap of three modes within the half power band width of one resonance, on average [1]. Although useful, the Schroeder frequency equation is not as precise as some situations may require. When simulating rooms through computer programs based on geometrical acoustics, if one dimension is very different from the others, as a corridor, for example, such a frequency is often underestimated. In another situation, when designing or qualifying a reverberation room, more ⇑ Corresponding author. E-mail address: [email protected] (M. Avelar). https://doi.org/10.1016/j.apacoust.2020.107247 0003-682X/Ó 2020 Elsevier Ltd. All rights reserved.
detailed information about modal overlap and the spacing between natural frequencies (or their effects) would be desirable, as indicated for instance in ISO 10140-5 [2]. A possible practical approach is to verify the standard deviation of the sound pressure level at each frequency band, given a reasonable, statistically significative, number of measurement locations. There are, however, other possibilities to be explored, as exposed in the following. At the condition of high modal overlap, according to Schroeder’s seminal work [3], one may expect the values of the real and imaginary part of a transfer function, considered as random variables, to assume a normal distribution. The present paper proposes the search of the high modal overlap condition though statistical indicators of value distribution from transfer function’s real and imaginary part, as well as of phase values. Two indicators are investigated: 1. Kurtosis, for analyzing the real and imaginary parts of a room’s transfer function; and 2. the maximal deviation from a reference value (uniform distribution) in histograms obtained from phase signals. Instead of using the term ‘‘Schroeder frequency”, which is related to the equation proposed by Schroeder, our search is expected to lead to a ‘‘crossover frequency”. For evaluating the mentioned statistical indicators, results obtained from measurements in a reverberation chamber (with
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geometry close to a shoebox), and in a scaled reverberation room (with irregular geometry) are presented. Analytical responses from three idealized shoebox rooms with same volume, and different degree of symmetry, are also analyzed. 2. Methods According to Jacobsen and Juhl [4], a statistical approach in the study of room acoustics is justified if there is sufficient modal overlap, and experience shows that an overlap above approximately three would be enough, according to Schroeder and Kuttruff [5]. As Kuttruff [6] mentions, in this situation, the room’s transfer function may be described by the composition of a large number of terms which are statistically independent from each other, and tend to produce values that tend to follow the central limit theorem. The transfer function may be expressed, in cartesian and polar coordinates, respectively, as
H ¼ X þ iY ¼ Rei/ :
ð1Þ
Suppose that, above a certain frequency, X and Y represent random processes which are statistically independent, normally distributed, with zero mean, and same variance. Under these conditions, which are very close to what Schroeder [7], Kuttruff and Thiele [8], Polack [9], among others, have observed for an universe of rooms, one may show [10] that the probability distribution function P of the magnitude R is
PðRÞ ¼
R
r2
2
R2 2r ;
ð2Þ
e
which denotes a Rayleigh distribution (in this equation, r denotes the standard deviation of magnitude values). On the other hand, the probability distribution function associated with the phase / is
(
Pð/Þ ¼
1 2p
0
for p < / < p otherwise
;
ð3Þ
which denotes an uniform distribution. Therefore, by means of statistical analysis methodologies, one may test: real and imaginary parts of room transfer functions for a Gaussian distribution; magnitude values for a Rayleigh distribution and/or; phase values for an uniform distribution. The present work does not address tests of magnitude values for Rayleigh distribution, and one expects the uncertainty of the crossover frequency to be larger. In the following, a test procedure for real and imaginary parts of room transfer functions are described at first, followed by a description of a procedure used to analyze the phase signal. 2.1. The use of Kurtosis for testing real and imaginary parts of room transfer functions Kurtosis has many applications in acoustics and vibration (it is used, for example, as an integrity indicator of roller bearings). In one of the most recent works, by Jeong [11], it was shown to be a very useful tool related to the evaluation of diffuseness of a sound field, when analyzing room impulse responses. Kurtosis is a statistical moment of fourth order, related to the ‘‘peakedness” or ‘‘flatness” of a sample distribution. If values of a population follow perfectly a Gaussian distribution, its value is 3. Therefore, one may use a normalized Kurtosis, written as
K¼
Eðp lp Þ4
r4p
3;
ð4Þ
in order to verify if a given collection of values follows a normal distribution. In this equation, and for our purposes, p is the real Xðf Þ or imaginary Yðf Þ part of a room transfer function at a given frequency (as described in Eq. (1)), lp is the mean value within a frequency window, and rp is the standard deviation of values in the same frequency window. When K approaches zero, the distribution of values within a frequency window probably follow a Gaussian curve. In the present case, it is important to note that: the condition of high modal overlap is likely to be met when the Kurtosis of both, real and imaginary parts, are sufficiently close to zero; according to Polack [9], Kurtosis is not a strict test for the probability distribution being normal. However, since the hypothesis that real and imaginary parts assume a normal distribution above a certain frequency has been properly investigated by Schroeder and Kuttruff, and Polack, (for a universe of rooms) it is assumed that if the normalized Kurtosis approaches zero, it is very likely that such a condition is met. The procedure adopted by Jeong [11] was used as reference in regard to the number of samples within a (sliding) frequency window1, although in the present case the study does not address sound field diffusion. This means that, in the cases presented here, care was taken to provide around 1000 samples within a frequency window used for calculating the normalized Kurtosis, at least initially. In order to produce more clear results, in some cases, a smaller population was adopted in the analysis, after checking the result from the 1000 sized population. Kurtosis was obtained from an sliding window on an individual transfer function, which generated a frequency dependent curve. Than an spatial average, obtained from curves after many source-receiver combinations, was taken. Kurtosis values are then plotted against frequency, and analyzed. A reference value for Kurtosis in this analysis is 2, which is used in some applications, according to [12]. That is, in this study, if its values become steadly under 2, the distribution of the transfer’s function real or imaginary parts is very likely to be normal. The transition frequency is the highest after analysis of real and imaginary part, situation which should lead the magnitude values to fit into a Rayleigh distribution. In regard to the value adopted for Kurtosis as a reference, 2 is a first approach. From an overview after analysis of some measurements performed in rooms, and analytical responses of rectangular rooms, by using this value one would obtain crossover frequencies comparable to those estimated through Schroeder’s formula. However, more consistent and directed studies in this regard are still to be done. As an example, a certain shape of a gamma distribution would lead to a Kurtosis value of 2. A more strict approach would be the one reported by Defrance and Polack [13] for estimating the mixing time, i.e., the crossover frequency could, for instance, be defined by the first one when Kurtosis reaches a null value. It should be noted that Kurtosis is an indicator, which has the advantage of being easily computed, on the one hand, while not being properly a normality test, on the other hand. I.e., distributions which are not normal may present a null Kurtosis value. Such distributions are not expected to happen in the studied context. However, in order to be safe in a first approach, KolmogorovSmirnov tests (KS test) were also performed within the same sliding frequency windows for all cases presented here, since Polack 1
In Jeong’s work a time window with a certain number of samples is used.
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[9] indicates it as a reliable method. Results from KS tests are shown only for one of such cases. 2.2. Maximum deviation from an ideal uniform distribution applied to the analysis of phase values Because under a situation of high modal overlap the probability distribution function of the phase, as pointed out in Eq. (3), assume a constant value, it is relatively simple to develop a measure which indicates the approach of such a condition. In the region of first natural frequencies of a room, for a 100 Hz observation frequency window, the phase signal looks like the one illustrated at the top of Fig. 1, after computation from an analytical solution for an exemplary shoebox room. At significant higher frequencies, the phase signal of the same room is depicted at the bottom of the figure. From these signals it is possible to build histograms representing occurrences (in percentage) within a chosen phase angle interval. In the illustrative case presented here, an angle interval of 20 degrees was used, and the resulting histograms are shown in Fig. 2. for both frequency regions. If the phase values in this situation were uniformly distributed, each angle interval would contain 5.56% of the total number of signal samples (100% divided by 18 angle intervals). In both histograms shown in Fig. 2, a horizontal line represents this (ideal) uniform distribution. The maximum deviation from the horizontal, reference line, is used in the present work as an indicator of how close the distribution is from being uniform (for each frequency band under analysis). Variations of this procedure were tested, in which standard deviation, instead of maximum deviation, and maximum deviation from a cumulative distribution function were adopted. Similar results were obtained, but the one described here seemed to show clearer results. The use of different angle intervals were also observed, and intervals of 18 degrees were considered to be adequate. Using small angle intervals lead to ‘‘noisy” frequencydependent curves, while the use of too large intervals lead to imprecise results. Results presented in this paper were obtained from angle intervals of 18 degrees, which means that the histograms contain 20 intervals, and the reference value which indicates the uniform distribution is 5%. This procedure was applied through an sliding window in constant frequency bands, in order not to influence the results by the size of the population. Just like in the procedure which uses Kurtosis, spatial averages for the maximum deviation were taken, and the result graphically displayed. In this initial approach, the criteria for estimating the crossover frequency was considered as being the frequency above which the maximum difference from the uniform distribution happens around a constant value, which may be seen as an analog to a ‘‘background noise”. 3. Materials Measured responses from two rooms with very different volumes were analyzed with both indicators described in Section 2 for an initial approach. Because the volumes are very different, the crossover regions are also expected to be very different. Secondly, analytical transfer functions from three shoebox rooms with the same volume, and different symmetry degrees were compared. In this case, the Schroeder Equation provides the same results for all rooms. 3.1. Measured rooms One of the rooms is composed by a set of two reverberation chambers originally built to be used for sound insulation measurements. In the present case, without the sample of a
3
building element, such as to provide a relatively large room with a shoebox geometry. The geometry is not a perfect cuboid due to the frame which would usually be used to hold samples of building elements, to deviations from the original design, and to the presence of sound source and receiver. The total volume of the room is 108:5 m3 (8:4 m long, 4:8 m wide, and 2:7 m high, with a small volume occupied by the frame). Measurements were performed for one source position and fifteen receiver positions, respecting the critical distance and distance from walls larger than 1 m. A second room was originally constructed with the aim to perform random-incidence scattering coefficient measurements, and does not hold a shoebox geometry. The chamber has a volume of 1:48 m3 , a surface area of 8:12 m2 , and semi-automatic measuring system for moving sound source and microphone. In this case, two source and six microphones positions were adopted for obtaining the impulse responses. Measurements were performed in the empty room, and also after inserting an absorptive foam covering all the floor, except a small area where there is a guide for moving automatically the sound source. The covered area was 1:2 m2 . The measuring systems for obtaining rooms’ characteristic responses were different in both situations. The AD/DA converter and microphone were the same (a Roland Quad Capture interface, a G.R.A.S. measurement microphone model 40AR, a G.R.A.S. preamplifier type 26CA and a G.R.A.S. 12AL power supply), and the sound sources were different. For measuring responses in the large room, a commercial dodecahedron loudspeaker (01 dB Omnidirectional dodecahedron) and power amplifier 01 dB Ampli 12 were used. For measurements in the smaller room, a cubic (with edges measuring 100 mm) self-made loudspeaker driven by commercial power amplifier for monitor reproduction systems was used instead (Alesis RA-100). The sound source used for measurements in the smaller room was built with six speakers with diameter of 2 inches, power handling of 10 W, connected in way to provide a load of 4X. In all cases, the impulse responses were obtained through a deconvolution technique with sine sweep signals used as excitation. The software used for performing the measurements was the same (ITA-Toolbox [14], which is based on Matlab).
3.2. Analytical solutions of three shoebox rooms Responses for three shoebox rooms were obtained by a routine which implements the rectangular room’s Green function, as described by Kuttruff [1]. This routine basically uses modal superposition, and care was taken in order to pre-calculate the modes up to the double of the frequency established as the one of maximal value of interest. Damping was introduced by means of a relation with the stipulated reverberation time. The use of a normal mode model, in this case, is advantageous in relation to the use of a stochastic model, because it provides frequency responses which present a transition from a condition where the statistical approach is not possible to one where such an approach is valid. Such a model is justified by a quote from Jacobsen [15]: ‘‘Strictly speaking, the sound field in a room is governed by causal laws and is therefore in principle predictable in the deterministic sense. However, it is not unreasonable to describe it as a pseudo-random phenomenon when many modes are simultaneously excited”. Consequently, the hypothesis which considers the terms X and Y in Eq. (1) to be random processes turns to be related to a pseudo-random one. The shoebox rooms in this case were imagined to have the same volume and reverberation time, so that the resulting Schroeder frequency is the same for all of them. Their dimensional relations, however, are significantly different. One of these rooms has all dimensions measuring 5 m. The second one is 10 m long, 5 m wide, and 2:5 m high. The third room has a dimensional relation,
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Fig. 1. Transfer functions’ phase within 100 Hz. Low frequency (20 Hz–120 Hz), at the top, and high frequency (1900 Hz–2000 Hz), at the bottom.
according to [16], which leads to natural frequencies more evenly distributed along the frequency domain, and consequently a lower crossover frequency. Its dimensions are: 7:974 m long, 5:819 m wide, and 2:694 m high. Reverberation time in all cases is set to be 10 s, and, because their volumes is also the same (125 m3 ), the Schroeder frequency is approximately 566 Hz. Transfer functions for one source position, and fifteen receivers randomly distributed in the room were obtained. Source and receivers’ positions were set to be at least 0.8 meters from any surface, and from each other. Every receiver’s position was selected to be greater than the critical distance.
4. Results 4.1. From measured impulse responses Impulse responses measured in the larger reverberation chamber were edited in order to cut non-linear effects, as suggested in [17], and lasted 6 s in the analyses reported in this paper. The sample frequency was 48 kHz. Graphics presented in this section were obtained from sliding frequency window containing 800 points. The frequency range of the graphics displayed may differ from case to case, with the intention to balance between details and behavior of the proposed measures, in a broad range. Fig. 3 presents the maximum deviation from the uniform distribution, and results of Kurtosis from real and imaginary parts of the transfer function are shown in Fig. 4, from measurements performed in the larger room. As previously reported, the curves are averaged values from the measures, after processing all transfer
functions. From Fig. 3, it is estimated that the crossover frequency lies around 275 Hz, while Fig. 4 shows that such a frequency is approximately 500 Hz. In fact, checking in the actual graphic, the frequency is 504 Hz for the real part, and 485 Hz for the imaginary part. To have an idea, the Schroeder frequency of this room is estimated to be in the 1=3 octave band with central frequency in 400 Hz. Figs. 5–7 present the maximal deviation from the uniform distribution, and kurtosis from real and imaginary parts, after processing transfer functions measured in the scaled reverberation room, respectively. In this case, the effect of covering the floor with an absorptive foam is also shown, in order to have a first impression. The impulse responses were sampled with a 48 kHz frequency, and lasted 1:4 s, after edition. From the maximal deviation from the uniform distribution (Fig. 5), the estimated crossover frequency is approximately 1250 Hz, when the room is empty, and 590 Hz, when the floor is covered with an absorptive foam. It is noticeable that this measure indicates the increase of modal overlap, when increasing sound absorption. It is also noticeable that the ‘‘background level” of the maximum deviation from the uniform distribution increases from approximately 0:04 to 0:05. However, it is not possible to point out the cause of that without a further study. The effect of increasing sound absorption is also shown by Kurtosis, which indicates a drop from 1520 Hz (the highest estimation from the solid lines in Figs. 6 and 7)) to 720 Hz, in the crossover frequency. From the measured reverberation time, and the room’s volume, the Schroeder frequency is estimated to be in the 1=3 octave band with central frequency of 1600 Hz, when empty, and 1000 Hz, with the absorbent foam.
M. Avelar et al. / Applied Acoustics 164 (2020) 107247
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Fig. 2. Phase histograms within 100 Hz. Low frequency (20 Hz–120 Hz), at the top, and high frequency (1900 Hz–2000 Hz), at the bottom.
Fig. 3. Maximum deviation from uniform distribution obtained from analysis of the signal’s phase. Average results after processing fifteen transfer functions measured in the large reverberation chamber. Estimated crossover frequency: 275 Hz.
4.2. From analytical impulse responses Analytical transfer functions were obtained from analytical impulse responses with a sample frequency of 8000 Hz, which in turn have a time length of 4:1 s.
The maximal deviation from the uniform distribution, obtained from transfer functions for the three shoebox rooms described in Section 3.2, is shown in Fig. 8. As previously mentioned, the value obtained from Schroeder’s formula, to be used as a reference, is 566 Hz. The effect of different symmetry degree is clearly seen,
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Fig. 4. Kurtosis from real and imaginary parts of fifteen transfer functions measured in the large reverberation chamber. Estimated crossover frequency: 500 Hz.
Fig. 5. Maximum deviation from uniform distribution obtained from analysis of the signal’s phase. Average results after processing twelve transfer functions measured in the scaled reverberation chamber, without and with absorptive foam covering its floor. Estimated crossover frequencies: 1250 Hz (empty room), and 590 Hz (damped room).
Fig. 6. Kurtosis from real part. Average results after processing twelve transfer functions measured in the scaled reverberation chamber, without and with absorptive foam covering its floor. Estimated crossover frequencies: 1520 Hz (Empty room), and 720 Hz (damped room).
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Fig. 7. Kurtosis from imaginary part. Average results after processing twelve transfer functions measured in the scaled reverberation chamber, without and with absorptive foam covering its floor. Estimated crossover frequencies: 1120 Hz (Empty room), and 600 Hz (damped room).
Fig. 8. Maximum deviation from uniform distribution obtained from analysis of the signal’s phase. Comparison of analytic responses of rooms with the same volume and reverberation time, and different aspect ratios. Estimated crossover frequencies: 1600 Hz (cubic room), 1200 Hz, and 750 Hz (ESE – Equally Spaced Eigen frequency – Room).
and the crossover frequencies estimated from this measure are: 1600 Hz (cubic room), 1200 Hz, and 750 Hz (room with the eigen modes more uniformly distributed, hereafter designated by ‘‘ESE – Equally Spaced Eigen modes”). In order to observe the effect of damping (reverberation time) on the measures, further transfer functions from the ESE room were analyzed, now with reverberation times set to be 5, and 2 s. In these cases, Schroeder’s formula delivers 400 Hz, and 253 Hz, respectively. The results for the maximum deviation from uniform distribution are presented in Fig. 9. As seen in Fig. 9, the tendency predicted through Schroeder’s equation is clearly show by this measure, although the results are quite different (see Table 1). The effect of different aspect ratios on Kurtosis is shown in Figs. 10 and 11, relative to real and imaginary components, respectively. It is seen that this measure also shows clearly the different crossover frequencies. The estimated values may be seen in Table 1 in bold face. They are similar to the results obtained from the maximum difference from uniform distribution, however slightly smaller. When lowering the reverberation time, Kurtosis shows that the crossover frequency drops, just like the other measure, as observed in Figs. 12 and 13. The estimated values, seen in bold face in Table 1, are also smaller than those obtained through the maximum difference from uniform distribution.
Beyond the analyses presented here, in order to check with a different method the frequency above which the real and imaginary values of the transfer functions tend to follow a normal distribution, a Kolmogorov-Smirnov (KS) test was also performed in a procedure similar to the one used to obtain Kurtosis values. Results are displayed in Figs. 14 and 15, for the analysis of real and imaginary parts, respectively. In this case, the reference value for the Kolmogorov-Smirnov statistics, which varies from 0 to 1, as commonly adopted, is 0.05. Series of values with a result above this reference are considered to follow a normal distribution. Results for the crossover frequency obtained through this test are shown in Table 1, and are larger than those obtained from the measures presented here. For the ‘‘optimal” room with a reverberation time of 10 s, the Schroeder frequency is 566 Hz, the crossover frequency obtained through the maximum difference from uniform distribution is 750 Hz, through Kurtosis is 680 Hz, and through the KS statistics is 870 Hz. For the cubic room, the KS test presented a much higher value, the difference being from approximately 1600 Hz, obtained both through Kurtosis, as well as through the maximum difference from uniform distribution, to 2400 Hz. The results obtained through KS tests, and the KS statistics, assured the tendency observed through the use of Kurtosis and the maximum deviation from uniform distribution. For the cubic room, and the 2.5 vs. 5 vs. 10 m3 , and reverberation time set to
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Fig. 9. Maximum deviation from uniform distribution obtained from analysis of the signal’s phase. Comparison of analytic responses of the ESE (Equally Spaced Eigen modes) room for different reverberation times. Estimated crossover frequencies: 750 Hz, 700 Hz, and 600 Hz.
Table 1 Crossover Frequency (in Hertz). Bold face indicates the crossover frequency obtained through the Maximum Deviation from Uniform Distribution, Kurtosis and KolmogorovSmirnov (KS) test. ESE Room stands for room with ‘‘Equally Spaced Eigen modes”, with dimensions 7:974 vs. 5:819 vs. 2:694 m3 . Room 5 5 5 m3 (RT = 10 s) 2.5 5 10 m3 (RT = 10 s) ESE Room (RT = 10 s) ESE Room (RT = 5 s) ESE Room (RT = 2 s)
Max. Dev. Uniform Distribution
Kurtosis Re
Kurtosis Im
KS Re
KS Im
1600 1200 750 700 600
1220 1070 670 635 450
1580 980 600 400 200
1410 1250 870 750 680
2400 1540 670 478 468
Fig. 10. Kurtosis from real part of transfer function. Comparison of analytic responses of rooms with the same volume and reverberation time, and different aspect ratios. Estimated crossover frequencies: 1220 Hz, 1070 Hz, and 670 Hz.
10 s, the KS tests presented significantly larger crossover frequencies, than those obtained from the Maximum deviation from uniform distribution, and Kurtosis analyses. This leads to the discussion of the adequacy of the criteria used in the present study for both measures. Fig. 16 summarizes the results obtained through the measures introduced in this work, for the data obtained analytically, compared to the Schroeder frequency. In this figure it becomes clear that both measures are sensitive to the room aspect ratio, and to absorption as well, both in similar ways. Kurtosis, with the criteria adopted for this study, presents crossover frequencies systematically lower than those obtained with the maximum deviation from uniform distribution. This difference lies in the range of approxi-
mately 100 Hz. When absorption is increased, the measures present a variation which indicates the increase of modal overlap, as also predicted by Schoeder’s equation, although in a different proportion.
5. Conclusion The measures based on statistical features of rooms’ transfer functions introduced in this paper have shown a potential to be used in practical situations as estimators of the crossover frequency, as observed in the analysis of measured and analytical responses. When comparing the crossover frequencies obtained
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Fig. 11. Kurtosis from imaginary part of transfer function. Comparison of analytic responses of rooms with the same volume and reverberation time, and different aspect ratios. Estimated crossover frequencies: 1580 Hz, 980 Hz, and 600 Hz.
Fig. 12. Kurtosis from real part of transfer function. Comparison of analytic responses of the ESE (Equally Spaced Eigen modes) room for different reverberation times. Estimated crossover frequencies: 670 Hz, 635 Hz, and 450 Hz.
Fig. 13. Kurtosis from imaginary part of transfer function. Comparison of analytic responses of the ESE room for different reverberation times. Estimated crossover frequencies: 600 Hz, 400 Hz, and 200 Hz.
from responses measured in the large reverberation room with those obtained in the (empty) scaled reverberation room, such values are in the expected range of variation, i.e., a lower value for the
larger room, and a higher value for the smaller room. Both are comparable to values estimated through Schroeder’s equation. When introducing an absorptive area in the scale model room, the crossover frequency drops significantly, also as expected because of the increase of modal overlap. This could also be verified in results from analytical transfer functions. In the experimental case, the minimal values of the maximum deviation from uniform distribution increased, and an investigation into this behavior is still to be done. There may be limits for the application of the methods presented here, especially when damping is increased in the room. From the analytical responses of three rooms with same volume, and different aspect ratios, it was possible to clearly observe the sensibility of both measures in relation to such ratios. In this case, results keep a close relation with the distribution of the eigenvalues over the frequency domain. The measures may be used in future work for evaluating how large the variation in results are, as a function of morphology. Kolmogorov-Smirnov tests were also applied to measured responses and, as mentioned by Polack [9], the results are very sensitive to the variation of reverberation time with the frequency. Such tests were successful in the analysis of analytical responses because, in this case, the reverberation time was forced to be
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Fig. 14. Kolmogorov-Smirnov statistics from real part of transfer function. Comparison of analytic responses of rooms with the same volume and reverberation time, and different aspect ratios. Estimated crossover frequencies: 1410 Hz, 1250 Hz, and 870 Hz.
Fig. 15. Kolmogorov-Smirnov statistics from imaginary part of transfer function. Comparison of analytic responses of rooms with the same volume and reverberation time, and different aspect ratios. Estimated crossover frequencies: 2400 Hz, 1540 Hz, and 670 Hz.
Fig. 16. Summarized results from Schroeder’s Equation, Kurtosis applied to the real and imaginary parts of the transfer function, and maximum deviation from uniform distribution (from the transfer function’s phase). ESE Room stands for room with ‘‘Equally Spaced Eigen modes”.
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constant for all frequencies. KS test results confirmed the tendencies observed from the measures presented here, although with significative differences in some situations. Such differences lead also to a discussion related to the criteria adopted for establishing the crossover frequency. When using Kurtosis as indicator, the limit of 2 may be reviewed, for instance, through studies similar to those from Schroeder and Kuttruff [5], and Polack [9]. The effect of variation of reverberation time, as a function of frequency, on the measures presented here is to be further investigated. In a first approach, as observed during the data processing for the results presented in this work, such a sensibility seems to be not very visible. CRediT authorship contribution statement Marcio Avelar: Conceptualization, Methodology, Writing - original draft, Writing - review & editing. Paulo Bonifacio: Conceptualization, Methodology. Luis Sant’Ana: Software. Eric Brandao: Conceptualization, Software. Elvis Bertoti: Conceptualization, Software. Rodrigo Catai: Supervision. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The authors are thankful to the Institute of Technical Acoustics at RWTH-Aachen University, and its contributors for programming the ITA-Toolbox, and leaving it available for use abroad. The
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reviewer is also acknowledged for providing pertinent comments, inquiries, and suggestions. References [1] Kuttruff H. Room acoustics. 5th ed. Spon Press; 2009. [2] I.O. for Standardization. Acoustics – laboratory measurement of sound insulation of building elements – part 5: requirements for test facilities and equipment; 2010.. [3] Schroeder M. Statistical parameters of the frequency response curves of large rooms. J Audio Eng Soc 1987;35(5):299–306. [4] Jacobsen F, Juhl PM. Fundamentals of general linear acoustics. 1st ed.; 2013.. [5] Schroeder M, Kuttruff H. On frequency response curves in rooms. comparison of experimental, theoretical, and monte carlo results for the average frequency spacing between maxima. J Acoust Soc Am 1962;34(1):76–80. [6] Kuttruff H. Acoustics: an introduction. Taylor and Francis; 2007. [7] Schroeder M. Eigenfrequenzstatistik in räumen. Acustica 1954;4(Suppl. 1):456–68. [8] Kuttruff H, Thiele R. Über die frequenzabhängigkeit des schalldruck in räumen. Acustica 1954;4:614–7. [9] Polack J-D. Acoustics, information, and communication. Springer; 2015. Ch. Are impulse responses gaussian noises?. [10] Papoulis A, Pillai SU. Probability, random variables, and stochastic processes. 4th ed. McGraw-Hill; 2002. [11] Jeong C-H. Kurtosis of room impulse responses as a diffuseness measure for reverberation chambers. J. Acoust. Soc. Am. 2016;139(5):2833–41. [12] George D, Mallery P. IBM SPSS statistics 23 step by step. 14th ed. Routledge; 2016. [13] Defrance G, Polack J-D. Acoustics, information, and communication. Springer; 2015. Ch. Estimating the crossover time within room impulse responses.. [14] I. of Technical Acoustics (Aachen University) (february 2019) [cited 6th of february, 2019].http://ita-toolbox.org.. [15] Jacobsen F. The diffuse sound field. statistical considerations concerning the reverberant field in the steady state [Tech. rep.]. The Acoustics Laboratory. Technical University of Denmark; 1979. [16] Cox TJ, D’Antonio P, Avis MR. Room sizing and optimization at low frequencies. J Audio Eng Soc 2004;52(6):640–51. [17] Mueller S, Massarani P. Transfer-function measurements with sweeps. J Audio Eng Soc 2001;49:443–71.
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Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
Technical note
Evaluation of the crossover frequency based on the analysis of room transfer functions through statistical estimators Marcio Avelar a,⇑, Paulo Bonifacio b, Luis Sant’Ana a, Eric Brandao c, Elvis Bertoti a, Rodrigo Catai d a
Academic Department of Mechanics, Federal University of Technology, Curitiba 81280-340, Brazil Federal Institute of Technology, Itajai, Brazil c Federal University of Santa Maria, Santa Maria, Brazil d Department of Civil Engineering, Federal University of Technology, Curitiba, Brazil b
a r t i c l e
i n f o
Article history: Received 19 September 2019 Received in revised form 31 January 2020 Accepted 4 February 2020
2010 MSC: 00-01 99-00 Keywords: Schroeder frequency Crossover frequency Modal overlap
a b s t r a c t This paper presents two ways of processing a room’s transfer function with the goal to determine the crossover frequency above which statistical approaches are valid for predicting or evaluating the sound field. One of them uses Kurtosis for analyzing the real and imaginary components of a transfer function. The other one is based on a proper measure developed for the analysis of the phase signal. Both measures are related to the statistical distribution of values within a frequency window. Transfer functions measured in a large reverberation room and in a scale model were analyzed, and provide results for the crossover frequency which are in the expected range. Analytical responses from three shoebox rooms with the same volume and reverberation time on the one hand, and different geometric aspect ratios on the other hand, were also analyzed. Both measures have shown similar and consistent results in these cases, and are sensitive to the aspect ratio from the hypothetical rooms. The effect of increasing sound absorption on experimental and analytical cases were observed, and are also in accordance to the expectation of increasing the modal overlap. Ó 2020 Elsevier Ltd. All rights reserved.
1. Introduction In several practical situations one faces the problem of choosing the most suitable approach to analyze the sound field in a room. For acoustically small rooms, a modal approach is of higher importance, as opposed to acoustically large rooms, when statistical considerations provide more useful models (formulae and methods). A very useful estimation for the decision on whether a room is acoustically small or large is given by an equation to compute the Schroeder frequency. Such an equation is given in terms of the reverberation time and the room’s volume, being related to the frequency above which there is at least an overlap of three modes within the half power band width of one resonance, on average [1]. Although useful, the Schroeder frequency equation is not as precise as some situations may require. When simulating rooms through computer programs based on geometrical acoustics, if one dimension is very different from the others, as a corridor, for example, such a frequency is often underestimated. In another situation, when designing or qualifying a reverberation room, more ⇑ Corresponding author. E-mail address: [email protected] (M. Avelar). https://doi.org/10.1016/j.apacoust.2020.107247 0003-682X/Ó 2020 Elsevier Ltd. All rights reserved.
detailed information about modal overlap and the spacing between natural frequencies (or their effects) would be desirable, as indicated for instance in ISO 10140-5 [2]. A possible practical approach is to verify the standard deviation of the sound pressure level at each frequency band, given a reasonable, statistically significative, number of measurement locations. There are, however, other possibilities to be explored, as exposed in the following. At the condition of high modal overlap, according to Schroeder’s seminal work [3], one may expect the values of the real and imaginary part of a transfer function, considered as random variables, to assume a normal distribution. The present paper proposes the search of the high modal overlap condition though statistical indicators of value distribution from transfer function’s real and imaginary part, as well as of phase values. Two indicators are investigated: 1. Kurtosis, for analyzing the real and imaginary parts of a room’s transfer function; and 2. the maximal deviation from a reference value (uniform distribution) in histograms obtained from phase signals. Instead of using the term ‘‘Schroeder frequency”, which is related to the equation proposed by Schroeder, our search is expected to lead to a ‘‘crossover frequency”. For evaluating the mentioned statistical indicators, results obtained from measurements in a reverberation chamber (with
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geometry close to a shoebox), and in a scaled reverberation room (with irregular geometry) are presented. Analytical responses from three idealized shoebox rooms with same volume, and different degree of symmetry, are also analyzed. 2. Methods According to Jacobsen and Juhl [4], a statistical approach in the study of room acoustics is justified if there is sufficient modal overlap, and experience shows that an overlap above approximately three would be enough, according to Schroeder and Kuttruff [5]. As Kuttruff [6] mentions, in this situation, the room’s transfer function may be described by the composition of a large number of terms which are statistically independent from each other, and tend to produce values that tend to follow the central limit theorem. The transfer function may be expressed, in cartesian and polar coordinates, respectively, as
H ¼ X þ iY ¼ Rei/ :
ð1Þ
Suppose that, above a certain frequency, X and Y represent random processes which are statistically independent, normally distributed, with zero mean, and same variance. Under these conditions, which are very close to what Schroeder [7], Kuttruff and Thiele [8], Polack [9], among others, have observed for an universe of rooms, one may show [10] that the probability distribution function P of the magnitude R is
PðRÞ ¼
R
r2
2
R2 2r ;
ð2Þ
e
which denotes a Rayleigh distribution (in this equation, r denotes the standard deviation of magnitude values). On the other hand, the probability distribution function associated with the phase / is
(
Pð/Þ ¼
1 2p
0
for p < / < p otherwise
;
ð3Þ
which denotes an uniform distribution. Therefore, by means of statistical analysis methodologies, one may test: real and imaginary parts of room transfer functions for a Gaussian distribution; magnitude values for a Rayleigh distribution and/or; phase values for an uniform distribution. The present work does not address tests of magnitude values for Rayleigh distribution, and one expects the uncertainty of the crossover frequency to be larger. In the following, a test procedure for real and imaginary parts of room transfer functions are described at first, followed by a description of a procedure used to analyze the phase signal. 2.1. The use of Kurtosis for testing real and imaginary parts of room transfer functions Kurtosis has many applications in acoustics and vibration (it is used, for example, as an integrity indicator of roller bearings). In one of the most recent works, by Jeong [11], it was shown to be a very useful tool related to the evaluation of diffuseness of a sound field, when analyzing room impulse responses. Kurtosis is a statistical moment of fourth order, related to the ‘‘peakedness” or ‘‘flatness” of a sample distribution. If values of a population follow perfectly a Gaussian distribution, its value is 3. Therefore, one may use a normalized Kurtosis, written as
K¼
Eðp lp Þ4
r4p
3;
ð4Þ
in order to verify if a given collection of values follows a normal distribution. In this equation, and for our purposes, p is the real Xðf Þ or imaginary Yðf Þ part of a room transfer function at a given frequency (as described in Eq. (1)), lp is the mean value within a frequency window, and rp is the standard deviation of values in the same frequency window. When K approaches zero, the distribution of values within a frequency window probably follow a Gaussian curve. In the present case, it is important to note that: the condition of high modal overlap is likely to be met when the Kurtosis of both, real and imaginary parts, are sufficiently close to zero; according to Polack [9], Kurtosis is not a strict test for the probability distribution being normal. However, since the hypothesis that real and imaginary parts assume a normal distribution above a certain frequency has been properly investigated by Schroeder and Kuttruff, and Polack, (for a universe of rooms) it is assumed that if the normalized Kurtosis approaches zero, it is very likely that such a condition is met. The procedure adopted by Jeong [11] was used as reference in regard to the number of samples within a (sliding) frequency window1, although in the present case the study does not address sound field diffusion. This means that, in the cases presented here, care was taken to provide around 1000 samples within a frequency window used for calculating the normalized Kurtosis, at least initially. In order to produce more clear results, in some cases, a smaller population was adopted in the analysis, after checking the result from the 1000 sized population. Kurtosis was obtained from an sliding window on an individual transfer function, which generated a frequency dependent curve. Than an spatial average, obtained from curves after many source-receiver combinations, was taken. Kurtosis values are then plotted against frequency, and analyzed. A reference value for Kurtosis in this analysis is 2, which is used in some applications, according to [12]. That is, in this study, if its values become steadly under 2, the distribution of the transfer’s function real or imaginary parts is very likely to be normal. The transition frequency is the highest after analysis of real and imaginary part, situation which should lead the magnitude values to fit into a Rayleigh distribution. In regard to the value adopted for Kurtosis as a reference, 2 is a first approach. From an overview after analysis of some measurements performed in rooms, and analytical responses of rectangular rooms, by using this value one would obtain crossover frequencies comparable to those estimated through Schroeder’s formula. However, more consistent and directed studies in this regard are still to be done. As an example, a certain shape of a gamma distribution would lead to a Kurtosis value of 2. A more strict approach would be the one reported by Defrance and Polack [13] for estimating the mixing time, i.e., the crossover frequency could, for instance, be defined by the first one when Kurtosis reaches a null value. It should be noted that Kurtosis is an indicator, which has the advantage of being easily computed, on the one hand, while not being properly a normality test, on the other hand. I.e., distributions which are not normal may present a null Kurtosis value. Such distributions are not expected to happen in the studied context. However, in order to be safe in a first approach, KolmogorovSmirnov tests (KS test) were also performed within the same sliding frequency windows for all cases presented here, since Polack 1
In Jeong’s work a time window with a certain number of samples is used.
M. Avelar et al. / Applied Acoustics 164 (2020) 107247
[9] indicates it as a reliable method. Results from KS tests are shown only for one of such cases. 2.2. Maximum deviation from an ideal uniform distribution applied to the analysis of phase values Because under a situation of high modal overlap the probability distribution function of the phase, as pointed out in Eq. (3), assume a constant value, it is relatively simple to develop a measure which indicates the approach of such a condition. In the region of first natural frequencies of a room, for a 100 Hz observation frequency window, the phase signal looks like the one illustrated at the top of Fig. 1, after computation from an analytical solution for an exemplary shoebox room. At significant higher frequencies, the phase signal of the same room is depicted at the bottom of the figure. From these signals it is possible to build histograms representing occurrences (in percentage) within a chosen phase angle interval. In the illustrative case presented here, an angle interval of 20 degrees was used, and the resulting histograms are shown in Fig. 2. for both frequency regions. If the phase values in this situation were uniformly distributed, each angle interval would contain 5.56% of the total number of signal samples (100% divided by 18 angle intervals). In both histograms shown in Fig. 2, a horizontal line represents this (ideal) uniform distribution. The maximum deviation from the horizontal, reference line, is used in the present work as an indicator of how close the distribution is from being uniform (for each frequency band under analysis). Variations of this procedure were tested, in which standard deviation, instead of maximum deviation, and maximum deviation from a cumulative distribution function were adopted. Similar results were obtained, but the one described here seemed to show clearer results. The use of different angle intervals were also observed, and intervals of 18 degrees were considered to be adequate. Using small angle intervals lead to ‘‘noisy” frequencydependent curves, while the use of too large intervals lead to imprecise results. Results presented in this paper were obtained from angle intervals of 18 degrees, which means that the histograms contain 20 intervals, and the reference value which indicates the uniform distribution is 5%. This procedure was applied through an sliding window in constant frequency bands, in order not to influence the results by the size of the population. Just like in the procedure which uses Kurtosis, spatial averages for the maximum deviation were taken, and the result graphically displayed. In this initial approach, the criteria for estimating the crossover frequency was considered as being the frequency above which the maximum difference from the uniform distribution happens around a constant value, which may be seen as an analog to a ‘‘background noise”. 3. Materials Measured responses from two rooms with very different volumes were analyzed with both indicators described in Section 2 for an initial approach. Because the volumes are very different, the crossover regions are also expected to be very different. Secondly, analytical transfer functions from three shoebox rooms with the same volume, and different symmetry degrees were compared. In this case, the Schroeder Equation provides the same results for all rooms. 3.1. Measured rooms One of the rooms is composed by a set of two reverberation chambers originally built to be used for sound insulation measurements. In the present case, without the sample of a
3
building element, such as to provide a relatively large room with a shoebox geometry. The geometry is not a perfect cuboid due to the frame which would usually be used to hold samples of building elements, to deviations from the original design, and to the presence of sound source and receiver. The total volume of the room is 108:5 m3 (8:4 m long, 4:8 m wide, and 2:7 m high, with a small volume occupied by the frame). Measurements were performed for one source position and fifteen receiver positions, respecting the critical distance and distance from walls larger than 1 m. A second room was originally constructed with the aim to perform random-incidence scattering coefficient measurements, and does not hold a shoebox geometry. The chamber has a volume of 1:48 m3 , a surface area of 8:12 m2 , and semi-automatic measuring system for moving sound source and microphone. In this case, two source and six microphones positions were adopted for obtaining the impulse responses. Measurements were performed in the empty room, and also after inserting an absorptive foam covering all the floor, except a small area where there is a guide for moving automatically the sound source. The covered area was 1:2 m2 . The measuring systems for obtaining rooms’ characteristic responses were different in both situations. The AD/DA converter and microphone were the same (a Roland Quad Capture interface, a G.R.A.S. measurement microphone model 40AR, a G.R.A.S. preamplifier type 26CA and a G.R.A.S. 12AL power supply), and the sound sources were different. For measuring responses in the large room, a commercial dodecahedron loudspeaker (01 dB Omnidirectional dodecahedron) and power amplifier 01 dB Ampli 12 were used. For measurements in the smaller room, a cubic (with edges measuring 100 mm) self-made loudspeaker driven by commercial power amplifier for monitor reproduction systems was used instead (Alesis RA-100). The sound source used for measurements in the smaller room was built with six speakers with diameter of 2 inches, power handling of 10 W, connected in way to provide a load of 4X. In all cases, the impulse responses were obtained through a deconvolution technique with sine sweep signals used as excitation. The software used for performing the measurements was the same (ITA-Toolbox [14], which is based on Matlab).
3.2. Analytical solutions of three shoebox rooms Responses for three shoebox rooms were obtained by a routine which implements the rectangular room’s Green function, as described by Kuttruff [1]. This routine basically uses modal superposition, and care was taken in order to pre-calculate the modes up to the double of the frequency established as the one of maximal value of interest. Damping was introduced by means of a relation with the stipulated reverberation time. The use of a normal mode model, in this case, is advantageous in relation to the use of a stochastic model, because it provides frequency responses which present a transition from a condition where the statistical approach is not possible to one where such an approach is valid. Such a model is justified by a quote from Jacobsen [15]: ‘‘Strictly speaking, the sound field in a room is governed by causal laws and is therefore in principle predictable in the deterministic sense. However, it is not unreasonable to describe it as a pseudo-random phenomenon when many modes are simultaneously excited”. Consequently, the hypothesis which considers the terms X and Y in Eq. (1) to be random processes turns to be related to a pseudo-random one. The shoebox rooms in this case were imagined to have the same volume and reverberation time, so that the resulting Schroeder frequency is the same for all of them. Their dimensional relations, however, are significantly different. One of these rooms has all dimensions measuring 5 m. The second one is 10 m long, 5 m wide, and 2:5 m high. The third room has a dimensional relation,
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Fig. 1. Transfer functions’ phase within 100 Hz. Low frequency (20 Hz–120 Hz), at the top, and high frequency (1900 Hz–2000 Hz), at the bottom.
according to [16], which leads to natural frequencies more evenly distributed along the frequency domain, and consequently a lower crossover frequency. Its dimensions are: 7:974 m long, 5:819 m wide, and 2:694 m high. Reverberation time in all cases is set to be 10 s, and, because their volumes is also the same (125 m3 ), the Schroeder frequency is approximately 566 Hz. Transfer functions for one source position, and fifteen receivers randomly distributed in the room were obtained. Source and receivers’ positions were set to be at least 0.8 meters from any surface, and from each other. Every receiver’s position was selected to be greater than the critical distance.
4. Results 4.1. From measured impulse responses Impulse responses measured in the larger reverberation chamber were edited in order to cut non-linear effects, as suggested in [17], and lasted 6 s in the analyses reported in this paper. The sample frequency was 48 kHz. Graphics presented in this section were obtained from sliding frequency window containing 800 points. The frequency range of the graphics displayed may differ from case to case, with the intention to balance between details and behavior of the proposed measures, in a broad range. Fig. 3 presents the maximum deviation from the uniform distribution, and results of Kurtosis from real and imaginary parts of the transfer function are shown in Fig. 4, from measurements performed in the larger room. As previously reported, the curves are averaged values from the measures, after processing all transfer
functions. From Fig. 3, it is estimated that the crossover frequency lies around 275 Hz, while Fig. 4 shows that such a frequency is approximately 500 Hz. In fact, checking in the actual graphic, the frequency is 504 Hz for the real part, and 485 Hz for the imaginary part. To have an idea, the Schroeder frequency of this room is estimated to be in the 1=3 octave band with central frequency in 400 Hz. Figs. 5–7 present the maximal deviation from the uniform distribution, and kurtosis from real and imaginary parts, after processing transfer functions measured in the scaled reverberation room, respectively. In this case, the effect of covering the floor with an absorptive foam is also shown, in order to have a first impression. The impulse responses were sampled with a 48 kHz frequency, and lasted 1:4 s, after edition. From the maximal deviation from the uniform distribution (Fig. 5), the estimated crossover frequency is approximately 1250 Hz, when the room is empty, and 590 Hz, when the floor is covered with an absorptive foam. It is noticeable that this measure indicates the increase of modal overlap, when increasing sound absorption. It is also noticeable that the ‘‘background level” of the maximum deviation from the uniform distribution increases from approximately 0:04 to 0:05. However, it is not possible to point out the cause of that without a further study. The effect of increasing sound absorption is also shown by Kurtosis, which indicates a drop from 1520 Hz (the highest estimation from the solid lines in Figs. 6 and 7)) to 720 Hz, in the crossover frequency. From the measured reverberation time, and the room’s volume, the Schroeder frequency is estimated to be in the 1=3 octave band with central frequency of 1600 Hz, when empty, and 1000 Hz, with the absorbent foam.
M. Avelar et al. / Applied Acoustics 164 (2020) 107247
5
Fig. 2. Phase histograms within 100 Hz. Low frequency (20 Hz–120 Hz), at the top, and high frequency (1900 Hz–2000 Hz), at the bottom.
Fig. 3. Maximum deviation from uniform distribution obtained from analysis of the signal’s phase. Average results after processing fifteen transfer functions measured in the large reverberation chamber. Estimated crossover frequency: 275 Hz.
4.2. From analytical impulse responses Analytical transfer functions were obtained from analytical impulse responses with a sample frequency of 8000 Hz, which in turn have a time length of 4:1 s.
The maximal deviation from the uniform distribution, obtained from transfer functions for the three shoebox rooms described in Section 3.2, is shown in Fig. 8. As previously mentioned, the value obtained from Schroeder’s formula, to be used as a reference, is 566 Hz. The effect of different symmetry degree is clearly seen,
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Fig. 4. Kurtosis from real and imaginary parts of fifteen transfer functions measured in the large reverberation chamber. Estimated crossover frequency: 500 Hz.
Fig. 5. Maximum deviation from uniform distribution obtained from analysis of the signal’s phase. Average results after processing twelve transfer functions measured in the scaled reverberation chamber, without and with absorptive foam covering its floor. Estimated crossover frequencies: 1250 Hz (empty room), and 590 Hz (damped room).
Fig. 6. Kurtosis from real part. Average results after processing twelve transfer functions measured in the scaled reverberation chamber, without and with absorptive foam covering its floor. Estimated crossover frequencies: 1520 Hz (Empty room), and 720 Hz (damped room).
M. Avelar et al. / Applied Acoustics 164 (2020) 107247
7
Fig. 7. Kurtosis from imaginary part. Average results after processing twelve transfer functions measured in the scaled reverberation chamber, without and with absorptive foam covering its floor. Estimated crossover frequencies: 1120 Hz (Empty room), and 600 Hz (damped room).
Fig. 8. Maximum deviation from uniform distribution obtained from analysis of the signal’s phase. Comparison of analytic responses of rooms with the same volume and reverberation time, and different aspect ratios. Estimated crossover frequencies: 1600 Hz (cubic room), 1200 Hz, and 750 Hz (ESE – Equally Spaced Eigen frequency – Room).
and the crossover frequencies estimated from this measure are: 1600 Hz (cubic room), 1200 Hz, and 750 Hz (room with the eigen modes more uniformly distributed, hereafter designated by ‘‘ESE – Equally Spaced Eigen modes”). In order to observe the effect of damping (reverberation time) on the measures, further transfer functions from the ESE room were analyzed, now with reverberation times set to be 5, and 2 s. In these cases, Schroeder’s formula delivers 400 Hz, and 253 Hz, respectively. The results for the maximum deviation from uniform distribution are presented in Fig. 9. As seen in Fig. 9, the tendency predicted through Schroeder’s equation is clearly show by this measure, although the results are quite different (see Table 1). The effect of different aspect ratios on Kurtosis is shown in Figs. 10 and 11, relative to real and imaginary components, respectively. It is seen that this measure also shows clearly the different crossover frequencies. The estimated values may be seen in Table 1 in bold face. They are similar to the results obtained from the maximum difference from uniform distribution, however slightly smaller. When lowering the reverberation time, Kurtosis shows that the crossover frequency drops, just like the other measure, as observed in Figs. 12 and 13. The estimated values, seen in bold face in Table 1, are also smaller than those obtained through the maximum difference from uniform distribution.
Beyond the analyses presented here, in order to check with a different method the frequency above which the real and imaginary values of the transfer functions tend to follow a normal distribution, a Kolmogorov-Smirnov (KS) test was also performed in a procedure similar to the one used to obtain Kurtosis values. Results are displayed in Figs. 14 and 15, for the analysis of real and imaginary parts, respectively. In this case, the reference value for the Kolmogorov-Smirnov statistics, which varies from 0 to 1, as commonly adopted, is 0.05. Series of values with a result above this reference are considered to follow a normal distribution. Results for the crossover frequency obtained through this test are shown in Table 1, and are larger than those obtained from the measures presented here. For the ‘‘optimal” room with a reverberation time of 10 s, the Schroeder frequency is 566 Hz, the crossover frequency obtained through the maximum difference from uniform distribution is 750 Hz, through Kurtosis is 680 Hz, and through the KS statistics is 870 Hz. For the cubic room, the KS test presented a much higher value, the difference being from approximately 1600 Hz, obtained both through Kurtosis, as well as through the maximum difference from uniform distribution, to 2400 Hz. The results obtained through KS tests, and the KS statistics, assured the tendency observed through the use of Kurtosis and the maximum deviation from uniform distribution. For the cubic room, and the 2.5 vs. 5 vs. 10 m3 , and reverberation time set to
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Fig. 9. Maximum deviation from uniform distribution obtained from analysis of the signal’s phase. Comparison of analytic responses of the ESE (Equally Spaced Eigen modes) room for different reverberation times. Estimated crossover frequencies: 750 Hz, 700 Hz, and 600 Hz.
Table 1 Crossover Frequency (in Hertz). Bold face indicates the crossover frequency obtained through the Maximum Deviation from Uniform Distribution, Kurtosis and KolmogorovSmirnov (KS) test. ESE Room stands for room with ‘‘Equally Spaced Eigen modes”, with dimensions 7:974 vs. 5:819 vs. 2:694 m3 . Room 5 5 5 m3 (RT = 10 s) 2.5 5 10 m3 (RT = 10 s) ESE Room (RT = 10 s) ESE Room (RT = 5 s) ESE Room (RT = 2 s)
Max. Dev. Uniform Distribution
Kurtosis Re
Kurtosis Im
KS Re
KS Im
1600 1200 750 700 600
1220 1070 670 635 450
1580 980 600 400 200
1410 1250 870 750 680
2400 1540 670 478 468
Fig. 10. Kurtosis from real part of transfer function. Comparison of analytic responses of rooms with the same volume and reverberation time, and different aspect ratios. Estimated crossover frequencies: 1220 Hz, 1070 Hz, and 670 Hz.
10 s, the KS tests presented significantly larger crossover frequencies, than those obtained from the Maximum deviation from uniform distribution, and Kurtosis analyses. This leads to the discussion of the adequacy of the criteria used in the present study for both measures. Fig. 16 summarizes the results obtained through the measures introduced in this work, for the data obtained analytically, compared to the Schroeder frequency. In this figure it becomes clear that both measures are sensitive to the room aspect ratio, and to absorption as well, both in similar ways. Kurtosis, with the criteria adopted for this study, presents crossover frequencies systematically lower than those obtained with the maximum deviation from uniform distribution. This difference lies in the range of approxi-
mately 100 Hz. When absorption is increased, the measures present a variation which indicates the increase of modal overlap, as also predicted by Schoeder’s equation, although in a different proportion.
5. Conclusion The measures based on statistical features of rooms’ transfer functions introduced in this paper have shown a potential to be used in practical situations as estimators of the crossover frequency, as observed in the analysis of measured and analytical responses. When comparing the crossover frequencies obtained
M. Avelar et al. / Applied Acoustics 164 (2020) 107247
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Fig. 11. Kurtosis from imaginary part of transfer function. Comparison of analytic responses of rooms with the same volume and reverberation time, and different aspect ratios. Estimated crossover frequencies: 1580 Hz, 980 Hz, and 600 Hz.
Fig. 12. Kurtosis from real part of transfer function. Comparison of analytic responses of the ESE (Equally Spaced Eigen modes) room for different reverberation times. Estimated crossover frequencies: 670 Hz, 635 Hz, and 450 Hz.
Fig. 13. Kurtosis from imaginary part of transfer function. Comparison of analytic responses of the ESE room for different reverberation times. Estimated crossover frequencies: 600 Hz, 400 Hz, and 200 Hz.
from responses measured in the large reverberation room with those obtained in the (empty) scaled reverberation room, such values are in the expected range of variation, i.e., a lower value for the
larger room, and a higher value for the smaller room. Both are comparable to values estimated through Schroeder’s equation. When introducing an absorptive area in the scale model room, the crossover frequency drops significantly, also as expected because of the increase of modal overlap. This could also be verified in results from analytical transfer functions. In the experimental case, the minimal values of the maximum deviation from uniform distribution increased, and an investigation into this behavior is still to be done. There may be limits for the application of the methods presented here, especially when damping is increased in the room. From the analytical responses of three rooms with same volume, and different aspect ratios, it was possible to clearly observe the sensibility of both measures in relation to such ratios. In this case, results keep a close relation with the distribution of the eigenvalues over the frequency domain. The measures may be used in future work for evaluating how large the variation in results are, as a function of morphology. Kolmogorov-Smirnov tests were also applied to measured responses and, as mentioned by Polack [9], the results are very sensitive to the variation of reverberation time with the frequency. Such tests were successful in the analysis of analytical responses because, in this case, the reverberation time was forced to be
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Fig. 14. Kolmogorov-Smirnov statistics from real part of transfer function. Comparison of analytic responses of rooms with the same volume and reverberation time, and different aspect ratios. Estimated crossover frequencies: 1410 Hz, 1250 Hz, and 870 Hz.
Fig. 15. Kolmogorov-Smirnov statistics from imaginary part of transfer function. Comparison of analytic responses of rooms with the same volume and reverberation time, and different aspect ratios. Estimated crossover frequencies: 2400 Hz, 1540 Hz, and 670 Hz.
Fig. 16. Summarized results from Schroeder’s Equation, Kurtosis applied to the real and imaginary parts of the transfer function, and maximum deviation from uniform distribution (from the transfer function’s phase). ESE Room stands for room with ‘‘Equally Spaced Eigen modes”.
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constant for all frequencies. KS test results confirmed the tendencies observed from the measures presented here, although with significative differences in some situations. Such differences lead also to a discussion related to the criteria adopted for establishing the crossover frequency. When using Kurtosis as indicator, the limit of 2 may be reviewed, for instance, through studies similar to those from Schroeder and Kuttruff [5], and Polack [9]. The effect of variation of reverberation time, as a function of frequency, on the measures presented here is to be further investigated. In a first approach, as observed during the data processing for the results presented in this work, such a sensibility seems to be not very visible. CRediT authorship contribution statement Marcio Avelar: Conceptualization, Methodology, Writing - original draft, Writing - review & editing. Paulo Bonifacio: Conceptualization, Methodology. Luis Sant’Ana: Software. Eric Brandao: Conceptualization, Software. Elvis Bertoti: Conceptualization, Software. Rodrigo Catai: Supervision. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The authors are thankful to the Institute of Technical Acoustics at RWTH-Aachen University, and its contributors for programming the ITA-Toolbox, and leaving it available for use abroad. The
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reviewer is also acknowledged for providing pertinent comments, inquiries, and suggestions. References [1] Kuttruff H. Room acoustics. 5th ed. Spon Press; 2009. [2] I.O. for Standardization. Acoustics – laboratory measurement of sound insulation of building elements – part 5: requirements for test facilities and equipment; 2010.. [3] Schroeder M. Statistical parameters of the frequency response curves of large rooms. J Audio Eng Soc 1987;35(5):299–306. [4] Jacobsen F, Juhl PM. Fundamentals of general linear acoustics. 1st ed.; 2013.. [5] Schroeder M, Kuttruff H. On frequency response curves in rooms. comparison of experimental, theoretical, and monte carlo results for the average frequency spacing between maxima. J Acoust Soc Am 1962;34(1):76–80. [6] Kuttruff H. Acoustics: an introduction. Taylor and Francis; 2007. [7] Schroeder M. Eigenfrequenzstatistik in räumen. Acustica 1954;4(Suppl. 1):456–68. [8] Kuttruff H, Thiele R. Über die frequenzabhängigkeit des schalldruck in räumen. Acustica 1954;4:614–7. [9] Polack J-D. Acoustics, information, and communication. Springer; 2015. Ch. Are impulse responses gaussian noises?. [10] Papoulis A, Pillai SU. Probability, random variables, and stochastic processes. 4th ed. McGraw-Hill; 2002. [11] Jeong C-H. Kurtosis of room impulse responses as a diffuseness measure for reverberation chambers. J. Acoust. Soc. Am. 2016;139(5):2833–41. [12] George D, Mallery P. IBM SPSS statistics 23 step by step. 14th ed. Routledge; 2016. [13] Defrance G, Polack J-D. Acoustics, information, and communication. Springer; 2015. Ch. Estimating the crossover time within room impulse responses.. [14] I. of Technical Acoustics (Aachen University) (february 2019) [cited 6th of february, 2019].http://ita-toolbox.org.. [15] Jacobsen F. The diffuse sound field. statistical considerations concerning the reverberant field in the steady state [Tech. rep.]. The Acoustics Laboratory. Technical University of Denmark; 1979. [16] Cox TJ, D’Antonio P, Avis MR. Room sizing and optimization at low frequencies. J Audio Eng Soc 2004;52(6):640–51. [17] Mueller S, Massarani P. Transfer-function measurements with sweeps. J Audio Eng Soc 2001;49:443–71.