Applying fuzzy set theory to the evaluation of sound quality in halls JAIME RAMIS, MARCELINO FERRI, JAVIER REDONDO, JUAN MARTÍNEZ Departamento de Física Aplicada. VICENTE ESTRUCH, JOSE PASTOR Departamento de Matemática Aplicada. Escuela Politécnica Superior de Gandia. Universidad Politécnica de Valencia Carretera Nazaret-Oliva sn, 46730 Gandia SPAIN
[email protected] [email protected] [email protected] http://www.fisgan.upv.es
Abstract: - The perception of sound quality in halls designed for sound transmission is dependent on several physical and psychological attributes of the message (sound event) and of the channel (the hall itself). A complete study must take into account both kind of attributes, but one can argue that finally the evaluation of a hall is different for each person. However, there must be some criteria about the values of a set of physical parameters [1-5] that drive to an optimum design for a particular purpose (concert hall, reunion hall, lecture room, opera house). In last decades, several authors have defined a considerable amount of physical parameters, each one of them provides an acceptable correlation with a particular sound attribute, such as, reverberation, envelopment, etc. The main problem, which is the evaluation of the global sound quality, is still unsolved and it is a relevant example of a multicriteria decision process; where fuzzy logic seems to be considerably helpful. Key-Words: - Room acoustics. Multicriteria decision. Concert Halls. Psychoacoustics. Fuzzy sets.
1 Introduction Research in auditorium acoustics have reached an important development in the last three decades. There have been proposed different objective measures relating to subjective reactions. Works of Ando [1], Barron [2-3], Beranek [4] and Jordan [5], must be pointed out. On the other hand, relevant contributions of the application of fuzzy set theory for multicriteria purposes in this ambit do not appear until the past decade of 90's [6]. To establish the acoustical attributes of a good auditorium, we have revised the remarks of the previous studies, and also a complementary subjective survey have been undertaken.
2 Main physical parameters relative to sound quality Beranek [4], in his paper "Excerpts for Concert Hall Acoustics", proposes a set of physical "orthogonal design objectives": initial-time-delay gap, reverberation time (RT), loudness level, bass ratio, and a measure of the diffusion of sound field at listener's position known as "inter-aural cross correlation coefficient" (IACC). A brief definition of these parameters is provided. The initial time delay gap is the time elapsed, at listener position, between the arrival of the direct sound and the first significant reflection (significant
reflections are those whose difference respect to the sound field remains smaller than 10 dB). Small times are required, whereas big times produces echo sensation and lost of intelligibility Reverberation time (RT): The impulse response of a closed room, after a small time, (a few ms) presents a tendency close to exponential. The reverberation time is the time elapsed until the moment where the sound power is a fraction 10-6 of the initial sound power. It could also be described as follows RT = τ ⋅ 6 ln(10) where τ is the time constant of the exponential tendency.The perception of reverberation (highly correlated with RT) is one of the main perceptive differences between the sound indoors and outdoors. The loudness level depends directly on the sound intensity at listener position. Some corrections should be made in order to consider non-linearities in frequency and energy response of the hearing system. Middle-high levels are recommended. Bass ratio (BR): The reverberation time can be evaluated for different frequency bands. In acoustics one should consider bass frequencies the octave bands centred on 125Hz and 250Hz, and mid frequencies those centred on 500Hz and 1KHz. The bass ratio is obtained as follows:
BR =
RT(125 Hz ) + RT( 250 Hz ) RT( 500 Hz ) + RT(1kHz )
(1)
IAAC: To evaluate this parameter, measures must be undertaken by means of an artificial dummy head. One shall record the left and right impulse responses (hL(t) , hR(t) ), in a given position of source and receiver, then:
IACC = max
∫t =t hL (t ) ⋅ hR (t + τ )dt 1 (2) ; t2 t2 2 2 ∫ hL (t )dt t =∫t hL (t )dt t =t1 1 τ ∈ (−1ms,1ms) t1=0, t2=1s t2
Other authors [2,5], incorporate parameters such as the "clarity" C80. This parameter belongs to the family of "early-to-late ratios", that compare, for a given impulse response, the energy in the first miliseconds with the resting contribution. Particularly C80=10 log (energy 0-80ms / energy 80ms-∞) ∆p (Pa)
t(s)
Fig.1 Example of a hall's impulse response (1 channel)
More recently, Shuo-Xian Wu [6] applied a fuzzy set based algorithm with four inputs (1) RT, (2) LE, (3) RT'(f), and (4) the maximum variation among the value of 10·log(LE) in the whole room. LE is a estimation of sound envelopment, relatively similar to sound diffusion, that can be estimated by IACC.
2.1 Physical parameters of the proposed model Some parameters that are considered high-priority for several authors, do not seem to correlate with the perception of quality. These result is not contradictory, on the contrary, it shows that the quality of a hall is more than the quality perceived by a single listener. Attributes that must be considered, conjointly with sound quality in a particular position, are the homogeneity in the sound in the whole room, and the reinforcement of the direct sound (mainly in big auditoria). Considering the tendencies of the room acoustics community and the results obtained in the psychoacoustical experiment, the parameters of the
proposed base are as follows.(1)Reverberation time RT. (2) Slope of the tonal curve RT'(f). (3) IACC. (4) Ratio direct vs reverberant sound field, or Reinforcement (F). And finally (5) The gravity centre of the squared impulse response Ts. The first parameter, even when does not present relevant correlation with sound quality in our experiment (Barron [2] defines two families of listeners, with reverberation or intimacy preference) must be included because of its extended use, tradition, and contribution to the average loudness level into the room. The second parameter (proposed by several authors) is the average slope of the tonal curve RT'(f). The tonal curve represents reverberation time versus frequency in octaves, and so RT'(f) (s/oct) evaluates if the response of the hall is "flat" or frequency dependent. This parameter is highly correlated with BR. The third parameter, IACC, has been included because of the notable correlation with the perceived quality found in the survey, and for been an indicator of diffusion. One of the interests of diffusion is its usefulness to evaluate sound field homogeneity, that represents an objective additional to the quality of a single listening position. The ratio between direct and reverberant sound evaluates the contribution of the hall to the perceived loudness, independently of the source's sound power. Usually the direct sound is evaluated in the impulse response in the first 5ms, and in this case we have considered the interval 5-50ms as the useful reinforcement. The parameter is calculated as a "early-to-middle" ratio F=10 log (energy 0-5ms / energy 5-50ms). The last parameter Ts, is found to be highly correlated with "clarity" C80 . However Ts is more stable in front of small variations in the arriving time of significant reflections near 80ms. The temporary scale of a impulse response is found to be the RT, or the time constant τ; thus, Ts has been re-scaled using the following expression
Ts = t ( gr .centre) / t c being t c = 0'05 + τ e
(3) −0 '.05
τ
(s)
(4)
The measure of these physical parameters (by means of an artificial head if necessary) have been undertaken in 16 different situations. A musical event has also been played and recorded (with the dummy head) in these situations, and afterwards a psychoacoustical experiment to evaluate sound quality perceived has been carried out. We will not
show here the complete analysis of the experimental results, but we present a comparison between the chosen parameters. The correlation matrix, on table 1, evidences that some of them should not be considered as orthogonal. In this cases the optimum acoustics is related with the antagonic effect of a given couple: for example, the parameters, IACC and F present a positive correlation, but the optimum situation is found increasing IACC and decreasing the value of F. RT RT'(f) IACC F RT 1 RT'(f) 0.41 1 IACC -0.14 0.33 1 F -0.17 0.20 0.74 1 Ts 0.57 -0.18 -0.77 -0.73
3.2 Rating matrix
Ts
The construction of these matrixes is the main step of the evaluation process, and is here where fuzzy sets are defined. Initially one must built on as many matrix Rk as kind of test (q), and finally the general matrix R shall be calculated . Each element rijk of the
1
matrix (Rk)nm is the membership of a particular value of the i-th parameter to the j-th fuzzy evaluation set (measurement made in k-th test) .
Table 1 Correlation matrix
rijk = µ e j
3 The proposed model The procedure to give the overall evaluation of the system has similar structure to that proposed by Shuo -Xian Wu, adding some considerations and modifying some of the input parameters. First we define a parameters set (P), a test set (T) and a evaluation test (E) P = {p1 ,..., p n }, (5)
T = {t1 ,..., t q },
E = {e1 ,..., em },
(6) (7)
the particular sets, in accordance with our own results and with bibliography recommendations, are as follows (8) P = {RT , RT ' ( f ), IACC , F , Ts} (9) T = { full Hall, empty Hall}, (10) E = {excellent, good , acceptable, bad }, where n=5, q=2 and m=4.
3.1 Weighting vectors In order to assess the effect of different parameters and different tests, two weighting vectors can be assigned as WP = ( wp1 ,..., wp n ) , (11)
WT = ( wt1 ,..., wt q ) being
(12)
∑ wp
(13)
i
= 1 =∑ wt i
(orthogonality), we define tentatively the weighting vector WP = (0'3,0'15,0'2,0'2,0'15) and the test vector WT = (0'65,0'35) . It must be pointed out that, to get an overall evaluation of a hall, one must measure all the parameters, but is not necessary to perform the two test (full, empty room). If only one test is carried out, the vector WT becomes WT=(1,0) or WT=(0,1) respectively.
For the evaluation of concert halls, considering the relevance of each parameter and its independence
(i ,k )
( pi )
(14)
being pi the value of the parameter Pi in a particular situation. We have defined the fuzzy evaluation sets associated to each parameter and kind of test, imposing a normalisation condition as follows m
∑r j =1
k ij
= 1 ( for i = 1,..., n k = 1,..., q) (15)
the membership function that defines each evaluation set is different for each parameter, and can be equivalent or different for each kind of test. In order to simplify the model, the defined membership functions are only dependent on parameter and evaluation set, and independent of the kind of test. The proposed membership functions are presented in the following section. 3.2.1 Membership functions On previous models there have been proposed stepped membership functions dependent on confidence intervals accepted by room acoustics community. For example, for RT its generally accepted than optimum values are found between 1.7s and 2.2 s and the acceptable values are defined by the range 1.4s to 2.8s. Thus, the membership function µe1(RT) proposed in a particular previous model (membership of RT to excellent set) is shown in the Fig. 2 (thick line):
µ excellent(RT), ... , µ bad (RT)
µexcellent(F), ... , µbad(F)
1 0.8
1
0.6
0.8
0.4 0.2 0 1
0.6 1.5
2
2.5
3
3.5 RT(s)
Fig.2 Stepped membership functions proposed for the RT in a particular previous model
We have preferred to choose continuously variable functions, in order to give major flexibility to the model, but respecting the general design recommended. The proposed functions, are shown in the next figures (e1 thick line; e2 dashed-pointed; e3 continuous; e4 dashed).
0.4 0.2 0 -15
-10
-5
0
5
10
15
F(dB)
Fig. 6 Membership functions relative to F µexcellent(TS), ... , µbad(TS) 1 0.8
µexcellent(RT), ... , µbad(RT)
0.6 0.4
1
0.2
0.8
0 0
0.6
0.2
0.4
0.6
0.8
1
1.2
1.4
TS
Fig. 7 Membership functions relative to TS
0.4 0.2 0 1
1.5
2
2.5
3
3.5 RT(s)
Fig. 3 Membership functions relative to RT µexcellent(RT’(f)), ... , µbad(RT’(f)) 1 0.8 0.6 0.4 0.2 0 0
0.05
0.1
0.15
0.2
0.3 RT’(f) (s/oct)
0.25
Fig. 4 Membership functions relative to RT'(f) µexcellent(IACC), ... , µbad(IACC )
Once measured the total set of parameters (n), in all different kinds of test (q), and obtained its membership to the all the evaluating sets (m), is time to build on the different rating matrixes (q)
r11k L r1km Rk = M O M rk L rk nm n1
1 0.8 0.6 0.4
(16)
and the final rating matrix R is a weighted average of all the particular matrixes
0.2 0 0
All the membership function defined has some remarkable particularities. The maximum value of the "bad" sets is 1, whereas the maximum value of the "excellent" sets is 0'7 with only one exception (F). The "good" and "acceptable" sets use to have a value minor than 0'3, and frequently the "acceptable" sets have a shape close to the "negative" of the excellent sets. The "good" sets remains constant where the "excellence" is bigger. All this particularities make this sets close to the semantic sets; for example a very bad auditory is easy to evaluate as bad, but a excellent one sometimes can be judged only as a "good" one, and this is why the membership to the "excellent" sets only arrives to 0'7 whereas the "bad" sets ranges to 1.
q
0.2
0.4
0.6
0.8
1
IACC
Fig. 5 Membership functions relative to IACC
R = ∑ wt k R k k =1
(17)
3.3 Overall evaluation When the aggregate rating matrix has been constructed, we get the overall evaluation vector S, by weighting the rating matrix.
S = WP ⋅ R
(18)
r11 L r1m ( s1 ,..., s m ) = ( wp1 ,..., wp n ) M O M r L r nm n1
(19)
References:
The value of the i-th element of the matrix S denotes the membership of the whole system to the overall i-th evaluation, i.e, if s1=0'4 and s2=0'3 it shows than the room is excellent with a confidence of 0'4 and the room is good with a confidence of 0'3. The single final evaluation must be obtained according to a most approximate principle. If sr=max(S) the final decision will be er if and only if r −1
∑ si ≤ i =1
1 m ∑ si 2 i =1
m
and
∑ si ≤
i = r +1
1 m ∑ si 2 i =1
; (20)
however, if one of this conditions is not verified, the final decision will be: r −1
1 m ∑ si 2 i =1 i =1 m 1 m if ∑ s i > ∑ si 2 i =1 i = r +1
er-1 if er+1
∑ si >
, or
different rooms. Membership functions and weighting vectors are proposed tentatively in accordance with generally accepted recommendations, but only the experimentation and a later optimisation process can drive to a definitively accurate model. The authors offers this communication by way of introduction, so that others may provide their valuable opinions about the acoustical part, or respect to the fuzzy set based approach.
(21) (22)
Finally if comparison between equally qualified rooms, or numerical classification, is required, one can enforce a numerical output NS of the algorithm like this NS = WE ⋅ S ' where WE is the weighting vector of evaluations. We propose the following one: WE=(1,0'5,-0'5,-1), that provides a normalised output between -1 and 1, to quantify the overall evaluation.
4 Conclusion As commented in the introduction, a fuzzy set approach to multicriteria systems can by applied to quantify acoustical quality of concert halls. Taking into account that preferences may depend on the kind of music played, a more detailed evaluation increasing the set of kind of test can be made, but the basic evaluation procedure remains similar to that presented. This main model is being optimised by increasing the measurements and the psychoacoustical survey in
[1] Ando, Concert Hall Acoustics, Springer-Verlag, 1985 [2] Barron M., Subjecive Study of British Symphony Concert Halls, Acustica . Acta Acustica, Vol.66, No.1, 1988, pp. 1-14. [3] Barron M., Spatial Impression and Envelopment in Concert Halls, Proceedings of the Institute of Acoustics, Vol.21, No.6, 1999, pp. 163-170 [4] Beranek L., Excerpts from Concert Hall Acoustics, Applied Acoustics, Vol.31, No.1, 1990, pp. 3-6. [5] Jordan V.L., A Group of Objective Acoustical Criteria for Concert Halls, Applied Acoustics, Vol.66, No.1, 1981, pp. 253-256. [6] Shuo-Xian Wu, Applying Fuzzy Set Theory to the Evaluation of Concert Halls, Journal of the Acoustical Society of America, Vol.89, No.2, 1991, pp. 772-776.
