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A calculation of Lx and Leq noise evaluation indices by use of statistical information on the noise level fluctuation, and its microcomputer-aided on-line measurement
Applied Acoustics 25 (1988) 33-47
A Calculation of L~ and Lea Noise Evaluation Indices by Use of Statistical Inforvhation on the Noise Level Fluctuation, and its Microcomputer-Aided On-Line Measurement Y. Mitani & M. Ohtaf Faculty of Engineering, Hiroshima University, Shitami, Saijo-cho, Higashi-Hiroshima City, 724, Japan (Received 29 May 1987; accepted 28 January 1988) ABSTRACT In this paper, a precise calculation method of Lx and Leq noise evaluation indices is theoretically proposed using the statistical information on the noise level fluctuation. More specifically, this method is given in a generalizedform universally applicable to the arbitrary non-Gaussian distribution, including a well known simplified expression derived under the assumption of a standard Gaussian distribution as the first approximation. Furthermore, based on this estimation method, a specific on-line measurement system is constructed with the aid of a microcomputer by introducing an iterative process for extracting the statistical information on the noise level fluctuation. Finally, the effectiveness of the proposed method has been confirmed experimentally by applying it to actual road traffic noise.
INTRODUCTION As is well known, two noise evaluation indices, Lx and Leq , play an important role in the field o f noise evaluation and regulation problems. In order to evaluate these two indices, the usual measurement methods are given respectively according to each original definition. That is, an Lx noise evaluation index is defined as the noise level exceeded by x percent throughout a total measurement time interval. In contrast, an Leq noise evaluation index is defined as a constant noise level whose noise energy value is equal to an averaged energy o f the noise level fluctuation over a total f To whom correspondence should be addressed. 33 Applied Acoustics 0003-682X/88/$03"50 © 1988 Elsevier Science Publishers Ltd, England. Printed in Great Britain
34
Y. Mitani, M. Ohta
measurement time interval. Nowadays, in an actual measurement, the noise level fluctuation is very often measured in a quantized amplitude form at every discrete time period using a digital-type instrument. In the case where these data are used to evaluate Lx and Leq, the following fundamental problems still remain as seen from the viewpoint of signal processing: (1)
In order to evaluate Leq, it is necessary to obtain many level data with a fairly fine sampling period, since the noise energy fluctuation after the anti-logarithmic transformation of the sampled level datum fluctuates with large excursions, as compared with the original decibel-scaled level fluctuation. (2) In principle, the mean value of the noise energy fluctuation should be given by the sample mean operation based on the noise energy data especially with an equally quantized energy amplitude. If one uses the noise energy fluctuation after transforming the measured noise level fluctuation with an equally quantized level amplitude into the energy scale through the anti-logarithmic transformation, a calculation error of the energy mean will occur because of the above level quantization. (3) In order to evaluate Lx, one must select the sampling period to ar,~ appropriate small value. The errors in L x due to this sampling period have already been investigated by many researchers (e.g. see Rel~ t and 2). Furthermore, the value of Lx when x is small (e.g. L~ and L 5) is statistically unstable in comparison with that of the median, since the number of data exceeding x percent is small.
In relation to the above problems, according to the current measurement techniques, the sampling period may greatly influence the accuracy of the measurement result. Especially in the case of the accuracy of L~q, a fine sampling period related to the time constant of the integration giving the noise level will generally give a good approximation to the results obtained with a true integration. Therefore, if one constructs the measurement system with the use of a microcomputer according to current practice, a huge memory capacity is needed for a long-term measurement with such a fine sampling period. From the above practical point of view, it is very convenient to utilize explicitly the statistical information on the decibelscaled level fluctuation itself with the aid of the theory for evaluating Lx and Leq (e.g. see Ref. 3). The above information is fairly stable and reliable based on the averaging operation supported by a large number of data. Thus, in order to extract the statistical information, the sampling period of the noise level fluctuation can be grosset than that of current practice mentioned above. At the same time, each order moment statistics can be successively obtained by introducing an iterative calculation process. Therefore, the
Noise level fluctuation and noise indices
35
problem with a huge memory capacity can be solved by using this procedure. In this report, a unified method for measuring two noise evaluation indices, Lx and L~, is theoretically proposed by using the statistical information directly on the decibel-scaled level fluctuation under consideration. It must be noticed that the proposed method is generally applicable to any kind of random phenomena of a non-Gaussian property. Furthermore, the proposed method is a generalized one containing a well known simplified estimation method derived under the assumption of a standard Gaussian distribution. Based on this estimation method, a specific on-line measurement system for the sequential measurement is constructed using a digital sound level meter and a microcomputer. Finally, the effectiveness of the proposed method has been confirmed experimentally by applying it to actual road traffic noise.
G E N E R A L A L G O R I T H M FOR ESTIMATING L~ Let us consider the noise level fluctuation L (L a= 10 loglo E/E 0 = M l n E/Eo, M ~ 10/ln 10, where E is the noise energy fluctuation, and E 0 is the reference noise energy usually taken as 10-'2 W/m 2) of an arbitrary non-Gaussian distribution type. According to the previous study, 4 a generalized expansion-type expression for estimating Le~ is given by using the cumulant statistics of L, as follows (see Appendix A):
Leq
lOloglo-
(E) -o =
x2
+
~¢a
+
x4
+
~cs + 120M
=/a + 0"l15a 2 + 8"84 x 10-3xa + 5"09 x 10-4x4 +2"34 x 10-sx 5 + ' "
(1)
where xn (n = 1, 2 .... ), la (= ~cl)and a 2 (= x2) denote the nth order cumulant, the mean value and the variance of L. Here, ( * ) is an averaging operation with respect to the random variable *. From eqn (1), it is possible to estimate Lea generally by reflecting not only lower order cumulants but also higher order cumulants in a hierarchical form. It should be noted that the above estimation formula agrees completely with a well known simplified estimation formula, s derived under the assumption of a standard Gaussian distribution as the first approximation: L~q =/a + 0.115o-2
(2)
since higher order cumulants xn (n = 3, 4,...) become zero for this special case. Thus, this evaluation method shows a generalized form including the well known simplified estimation method as a special case.
Y. Mitani, M. Ohta
36
To establish an evaluation algorithm with the aid of a microcomputer one must obtain the cumulant K, (n = 1,2 .... ) by spending a little a m o u n t of its memory. To achieve this purpose, one can first calculate the ruth order m o m e n t o f L, especially by means of the following iterative process:. (L')u-
N-l N
1 ( L " ) N ~ +---L"N u
~3}
F r o m eqn (3), one can obtain the mth order m o m e n t ( L " ) N at the Nth measurement time based on the memorized past value of the ruth order moment, ( L " ) N 1, at the ( N - 1 ) t h measurement time and the present datum, L N, at the Nth measurement time, in an iterative form. After obtaining the mth order m o m e n t within the specific measurement time interval using the above procedure, the resultant m o m e n t (L"'.:. (n = 1, 2,...,m), can be transformed into the cumulant he, (n = 1,2 ..... m~, as follows: 6 h-, : (LS,
h'2 = ( L 25 -- ( L ) h 1
h."3 = ( L 3) -- 2 ( L ) K 2 -- (L2)h'l 1,:4 = ( L 45 - 3 ( L ) K 3 -- 3(L2)h-2
--(L3)h-i
Ks = ( L 5) - 4 ( L ) K a
-- 4(L3)K2
-
6(L2)h'3
--
(L4)h-i. . . .
(4)
Thus one can estimate the objective Leq by substituting the calculated value of cumulant statistics into eqn (1).
G E N E R A L A L G O R I T H M F O R E S T I M A T I N G L,. In order to estimate the arbitrary L x noise evaluation indices (e.g. L~, L~;, Lso, L9o,...), let us introduce the statistical Hermite series expansion-type expression 7 of the cumulative probability distribution universally applicable to the noise level fluctuation, L, of an arbitrary non-Gaussian distribution type, as follows (see Appendix B):
Q(L)=
N ( ~ ; / 1 , O.2} d ~
-LK~32N(L;p,O.2)H2 DO"
2 f L -- fl~ h4 N ( L ; lt, O" )H3 { ........ J 24O"3 "\ a / • ~,~s 4 N ( L ; I~, ae}H4 1 ZUO.
....
i51
Noise level fluctuation and noise indices
37
where N(~; #, 0"2) denotes the Gaussian distribution function defined as: , o.2) =zx
x / ~1t r exp ( - (~ - g)2/2tr2)
(6)
Moreover, H.(.) denotes the nth order Hermite polynomial, and its value can be calculated by the following recursive formula suitable for computer processing:8 H.(0 = ~H._ 1(~) - (n - 1)H._ 2(~)
(7)
In addition, the calculation of eqn (6) can be easily performed by using the well known error function, as follows: fL
N(~;/t, tr2)d~ = 1
1 E r f c {L -/z'~
(8)
where Erfc (.) denotes the error function, defined as: Erfc (~) ~
exp -'2 dt
(9)
At this time, the value of Erfc(0 can be accurately calculated by the following approximation: 9 2
- - E r ~ ( ~ ) + 1/[1 +0.0705230784~ +0.0422820123¢ 2 + 0.0092705272~ 3 + 0-0001520143~ 4 + 0.0002765672~ s + 0.0000430638~6] 16
(10)
Since the relationship between Q(L) and Lx is given by: x
1 100- Q(Lx)
(11)
and Q(L) is a monotone-increasing function, the objective Lx can be directly evaluated from the estimated curve of Q(L) using the linear interpolation, even for the measurement with a fairly large discrete level amplitude. This procedure is quite convenient to economize the calculation time, when compared to a usual numerical integration. Consequently, after obtaining the statistical information on the noise level fluctuation in the same manner as described in the previous section, one can calculate an arbitrary Lx using this method. Furthermore, based on this estimation method, it is not necessary to memorize the whole data of the noise level fluctuation, and one can solve the problem with a little memory capacity when using a microcomputer for the measurement system.
38
Y. Mitani, M. Ohta E X P E R I M E N T A L WORK
Confirmation of effectiveness of the proposed estimation method For the purpose of confirming the effectiveness of the proposed estimation method, a measurement system has been constructed using a digital sound level meter and a portable microcomputer. The block diagram of this measurement system is shown in Fig. 1. Two kinds of road traffic noise have been measured as a typical example of environmental noises. One of the road traffic noises was measured near a national main road in a large city. At that time, the traffic volume was 244 vehicles per ten minutes. Another type of road traffic noise was measured near a country road in a rural district. At that time, the traffic volume was 33 vehicles per ten minutes. The measurement time interval and its sampling period were selected to be 10 minutes and 1 second, respectively. Here, let us define the former case as 'Case A' and the latter one as 'Case B'. Figure 2 shows the actual location for measuring the road traffic noise tbr Case A. Figure 3 shows the actual location for measuring the road traffic noise fi~r Case B. Table 1 shows the estimated results of L,,q for Case A using the proposed estimation method. Here, let us define the expansion expression of eqn (1) from the first expansion term corresponding to the well known simplified estimation method, eqn (2), to the term due to the nth order cumulant, as the ( n - I)th approximation of the estimated L,,q. In addition, the measured value o f Leq in this table shows the value measured by a precision integrating sound level meter for Leq. According to these results, the calculated value. using a well known simplified estimation method derived under the assumption of a standard Gaussian distribution, agrees approximately with
Inr_er
f no("
(RS-2
Unit
L~(:)
M J (" F o C o m p t l t~! r
Fig. !.
Block diagram of measurement syslcm with the aid of portable n~icrocompulc~
Noise level fluctuation and noise indices
•
Car Park
Building
Factory
Htros|
Traffic Signal
Series of Stores
Route 2 To
:
39
7.7 m
• Car
Supermarket
mm
To Os~
lma
Building
Observation Point
Ser: Sto!
of
Park
50 m
Fig. 2. Actual location for measuring road traffic noise (Case A).
AI_
3
AI.
: Rice Field
.U_ ./I.
AL
To Higashimura
At.
Factory
I 5 m • 12 m
TO Matsunaga Observation Point .11-
.l.L
.It.
II
II
11..
Fig. 3. Actual location for measuring road traffic noise (Case B). TABLE 1 The Estimated Results of Leq Using the Proposed Method (Case A)
Measured value of Leq (dB(A)) 85-9
Estimated values using the proposed method (dB(A)) 86.4 (the 86.0 (the 86"1 (the 86.1 (the
first approximation) second approximation) third approximation) fourth approximation)
Y. Mitani, M. Ohta
40
TABLE 2 The Estimated Results of L,,~ Using the Proposed Method (Case B)
Measured value ~?[L~q {dB(A))
£~'thnated values using/tie proposed method (dB(A))
80-7
777 82-0 82-8 81.0
ithe (the {the (the
first approximation) second approximationt third approximation) lburth approximationl
the measured value. More specifically, however, the estimated values using the proposed method tend to be in good agreement with the successive addition of higher order expansion terms. Table 2 shows the estimated results for Case B. According to these estimated results, the estimation accuracy using the well known simplified estimation method is not sufficient for evaluation. On the other hand, it is obvious that the successive addition of higher order expansion terms moves the values estimated theoretically using the proposed method closer to the values measured experimentally. Figure 4 shows a comparison between the theoretically estimated curves using the proposed method and the experimentally sampled points in the I
I
I
I
I
1
"l
I 1.0
-
O. 5
-
- -
-~
o
o
I 50
60
70
80
l
1
I
90
] O0
110
__
1, ( d B A )
Fig. 4. A comparison between theoretically estimated curves using the proposed mctht)d and experimentally sampled points for cumulative probability distribution of noise level fluctuation (Case A). t , Experimentally sampled points. Theoretically estimated cur~es: • the first approximation ofeqn (5): , the second approximation (and the third and fourth approximations),
Noise level fluctuation and noise indices
41
cumulative distribution form of the noise level fluctuation for Case A (arbitrary Lx noise levels are directly extracted by this distribution form). From this figure, the theoretically estimated curve due to the first approximation (i.e. Gaussian distribution) agrees approximately with the experimentally sampled points. More precisely, however, the estimated values using the proposed method tend to be in good agreement with the measured values, with the successive addition of higher order expansion terms. Figure 5 shows the estimated results for Case B. From this figure, the theoretically estimated curve by using only the first expansion term corresponding to the Gaussian distribution does not agree well with the experimentally sampled points. It is obvious that the successive addition of higher order expansion terms moves the values estimated theoretically using the proposed method closer to the values measured experimentally. In each case, the estimation accuracy for both Lx and Le~ noise evaluation indices using the proposed estimation method is clearly sufficient. Construction of on-line measurement system for sequential measurement
In this section, an on-line measurement system for the sequential measurement of two noise evaluation indices, Lx and Leg, is specifically I
I
I
I
I
I
80
90
i00
1.0
-
0.5
S //
I
0
50
60
70
110
L (dBA)
Fig. 5. A comparison between theoretically estimated curves using the proposed method and experimentally sampled points for cumulative probability distribution of noise level fluctuation (Case B). O, Experimentally sampled points. Theoretically estimated curves: . . . . . , the first approximation of eqn (5); . . . . , the second approximation (and the third approximation); - - - , the fourth approximation.
42
Y. Mitani, M. Ohta
constructed based on the proposed estimation method. According to the estimated results mentioned above, in the actual estimation, the estimated values due to the fourth approximation have been employed. Since it is necessary to take the calculation time to estimate these two indices, the concurrent processing by two microcomputers can be used. The block diagram of this measurement system is shown in Fig. 6. Microcomputer A is used for the iterative process for extracting the moment statistics within each measurement time interval using eqn (3) and the control system for the communication lines between the digital sound level meter and Microcomputer B with the RS-232C type interface. Microcomputer B is used tbr the calculation of L~ and Leq using the proposed estimation method after transforming the moment statistics (L "> into the cumulant statistics h,. Each measurement time interval can be arbitrarily selected using the clock signal generator in Microcomputer A. In this measuremenL each Sound
Level
Mete1
[Interface
Unit
(R~;-232C)
1
'I
t. . . . .
Microcomputer
A
M u l t ip lexer J
Microcomputer
B
/ Microcomputer
A
: Iterative
calculation
of moment statistics and control system for c o m m u n i c a t i o n lines
Microcomputer B : C a l c u l a t i o n of L x and Leq using the proposed method
Fig. 6.
Block diagram of on-line measurement system for I.~ and L,,q using the proposed method.
Noise levei fluctuation and noise indices
43
~!~i~ii~i~,~.£v~¸¸¸i~? !! ~
Fig. 7.
Actual scene of road traffic noise measurements using the proposed on-line measurement system.
measurement time interval has been selected at 10 minutes. Two noise evaluation indices, L= and Leq, for the road traffic noise (Case A) have been measured for 24 hours. Figure 7 shows the actual scene of the road traffic noise measurement using the proposed on-line measurement system. Figure 8 shows the estimated results of L 5, L~0, Lso, L 9 o and Le~ using the proposed measurement system. The proposed system has good performance for a long-term measurement with each measurement time interval. CONCLUSION The conclusions of this study are characterized as follows: (1) A unified method for measuring two noise evaluation indices, L= and Leq, has been proposed using the statistical information on the noise level fluctuation.
Y. Mitani, M. Ohta
44 l
W
9O
1
,,z -1
i 8O
] c~q
70
60
50
4o 8:00
I 12:00
I
i
0:00
~ :,~(I Time
Fig. 8.
(h~u r)
The estimated results of Lx and L,,q using the proposed on-line measurement system.
(2) The proposed method is generally applicable to any kind of random
(3)
(4)
p h e n o m e n a of a non-Gaussian property. It has been derived in a generalized form including a well known simplified estimation method derived under the assumption of a standard Gaussian distribution. The statistical information used in the proposed method could be extracted successively by introducing an iterative calculation process. As the result, it is possible to economize the m e m o r y of a microcomputer. Based on the proposed method, it is necessary to use only a little a m o u n t of m e m o r y , even when changing the measurement time interval. In the case of evaluating Leq according to current practice, the problem o f the fine sampling period and the measurement errors due to an unequally quantized energy amplitude through the antilogarithmic transformation could be avoided using the proposed method. This is because of finding a reasonable theoretical relationship between the objective Leq value given by the sample mean operation with an equally quantized energy amplitude and the statistical information based on the level data with an equally quantized level amplitude.
Noise levelfluctuation and noise indices
(5)
45
By using this unified theory with the above advantages, an on-line measurement system for the sequential measurement t h r o u g h a longterm measurement time interval has been constructed using a digital sound level meter and a portable microcomputer.
Of course, this study is in its early stage and has been focused only on its fundamental aspects. Accordingly, there still remain m a n y future problems, as follows: (1) (2)
This m e t h o d must be applied to m a n y other actual cases to broaden and confirm its further effectiveness. In order to evaluate precisely the complicated situation of environmental noises, it is necessary to grasp systematically the general relationship between L x and Leq using this method. ACKNOWLEDGEMENTS
The authors would like to express their cordial thanks to M r H. Hosoya and Mr Y. Watanabe for their helpful assistance. The authors are also thankful for m a n y constructive discussions during the annual meeting of the Institute of Noise Control Engineering of Japan. 1° REFERENCES 1. Fisk, D. J., Statistical sampling in community noise measurement. J. Sound Vib., 30(2) (1973) 221-36. 2. Vaskor, J. G., Dickinson, S. M. & Bradley, J. S., Effect of sampling on the statistical descriptors of traffic noise. AppL Acoust., 12(2) (1979) l 11-24. 3. Don, C. G. & Rees, I. G., Road traffic sound level distributions. J. Sound Vib., 100(1) (1985) 41-53. 4. Ohta, M., Mitani, Y. & Sumimoto, T., A generalized theory for the mutual relationship among several type noise evaluation indices connected with Leq and Lx and its experiment. J. Acoust. Soc. Japan, 41(9) (1985) 598-607 (in Japanese). 5. U.S. Environmental Protection Agency, Information on levels of environmental noise requisite to protect public health and welfare with an adequate margin of safety. 550/9-74-004 (1974). 6. Lloyd, E., Handbook of Applicable Mathematics, Volume II: Probability. John Wiley, New York, 1980, pp. 251-2. 7. Ohta, M. & Koizumi, T., General statistical treatment of the response of a nonlinear rectifying device to a stationary random input. IEEE Trans. Inform. Theory, IT-14(4) (1968), 595-8. 8. Moriguchi, S., Utagawa, K. & Hitotsumatsu, S., Mathematical Formula IIl. lwanami, Tokyo, 1977, p. 93 (in Japanese). 9. Hastings, C., Approximations for Digital Computers. Princeton, New Jersey, Princeton University Press, 1955, p. 187.
46
Y. Mitani, M. Ohta
10. Mitani, Y. & Ohta, M., Practical estimation method of Leq and L x evaluation indices by the effective use of a microcomputer technique. Proc. 1986 Meeting Inst. Noise Control Engng Japan (1986), pp. 81--84 (in Japanese). A P P E N D I X A: D E R I V A T I O N O F E Q N (1) At the starting point o f the analysis, let us introduce the m o m e n t generating function mL(O) o f the noise level fluctuation L, as follows:
mL(O) L=(exp (OL)~
(A, I)
By using the relationship between the noise level fluctuation L and the noise energy fluctuation E, eqn (A.1) is rewritten as
mL(O) = ((E/Eo) °M)
(A.2)
On the other hand, based on the statistical theory, the m o m e n t generating function is generally expressed by using the cumulant statistics ~,, (n = 1, 2 .... ), as follows: 6 mL(O) = exp ~ ~, (A.31
IZ0 i n = 1
Therefore, by replacing 0 with 1/M in eqns (A.2) and (A.3) and using these equations and the definition of Leq, one can derive the following formula (see eqn (1)): ¢-.-.,
1 = 1 (E) ) Leq 2~ 10 og~o-~o ..... ~.a n!M'-
1 /x'n
(A.4)
n=l
A P P E N D I X B: D E R I V A T I O N O F EQN (5) Let us introduce the statistical Hermite series expansion-type expression ~'as a general explicit expression on the probability density function P(L) with respect to the noise level fluctuation L, as follows:
t Z
P(L)= N(L;p,a 2) 1+
A,H. - -
(B.t)
-It
(B.2)
n=3
with
1
H.
Noise level fluctuation and noise indices
47
At this time, by using the well known formula between the Gaussian distribution and the Hermite polynomial: 7 N(~; # , trZ)Hn
=
( - 1)"a"S-ff N(~; ~u,a 2)
(B.3)
the cumulative distribution form Q(LX~=~oP(~)d¢) is consequently obtained as follows: oo
n=3
After expressing specifically the expansion coefficient A, using eqn (B.2), eqn (5) can be derived.
A Calculation of L~ and Lea Noise Evaluation Indices by Use of Statistical Inforvhation on the Noise Level Fluctuation, and its Microcomputer-Aided On-Line Measurement Y. Mitani & M. Ohtaf Faculty of Engineering, Hiroshima University, Shitami, Saijo-cho, Higashi-Hiroshima City, 724, Japan (Received 29 May 1987; accepted 28 January 1988) ABSTRACT In this paper, a precise calculation method of Lx and Leq noise evaluation indices is theoretically proposed using the statistical information on the noise level fluctuation. More specifically, this method is given in a generalizedform universally applicable to the arbitrary non-Gaussian distribution, including a well known simplified expression derived under the assumption of a standard Gaussian distribution as the first approximation. Furthermore, based on this estimation method, a specific on-line measurement system is constructed with the aid of a microcomputer by introducing an iterative process for extracting the statistical information on the noise level fluctuation. Finally, the effectiveness of the proposed method has been confirmed experimentally by applying it to actual road traffic noise.
INTRODUCTION As is well known, two noise evaluation indices, Lx and Leq , play an important role in the field o f noise evaluation and regulation problems. In order to evaluate these two indices, the usual measurement methods are given respectively according to each original definition. That is, an Lx noise evaluation index is defined as the noise level exceeded by x percent throughout a total measurement time interval. In contrast, an Leq noise evaluation index is defined as a constant noise level whose noise energy value is equal to an averaged energy o f the noise level fluctuation over a total f To whom correspondence should be addressed. 33 Applied Acoustics 0003-682X/88/$03"50 © 1988 Elsevier Science Publishers Ltd, England. Printed in Great Britain
34
Y. Mitani, M. Ohta
measurement time interval. Nowadays, in an actual measurement, the noise level fluctuation is very often measured in a quantized amplitude form at every discrete time period using a digital-type instrument. In the case where these data are used to evaluate Lx and Leq, the following fundamental problems still remain as seen from the viewpoint of signal processing: (1)
In order to evaluate Leq, it is necessary to obtain many level data with a fairly fine sampling period, since the noise energy fluctuation after the anti-logarithmic transformation of the sampled level datum fluctuates with large excursions, as compared with the original decibel-scaled level fluctuation. (2) In principle, the mean value of the noise energy fluctuation should be given by the sample mean operation based on the noise energy data especially with an equally quantized energy amplitude. If one uses the noise energy fluctuation after transforming the measured noise level fluctuation with an equally quantized level amplitude into the energy scale through the anti-logarithmic transformation, a calculation error of the energy mean will occur because of the above level quantization. (3) In order to evaluate Lx, one must select the sampling period to ar,~ appropriate small value. The errors in L x due to this sampling period have already been investigated by many researchers (e.g. see Rel~ t and 2). Furthermore, the value of Lx when x is small (e.g. L~ and L 5) is statistically unstable in comparison with that of the median, since the number of data exceeding x percent is small.
In relation to the above problems, according to the current measurement techniques, the sampling period may greatly influence the accuracy of the measurement result. Especially in the case of the accuracy of L~q, a fine sampling period related to the time constant of the integration giving the noise level will generally give a good approximation to the results obtained with a true integration. Therefore, if one constructs the measurement system with the use of a microcomputer according to current practice, a huge memory capacity is needed for a long-term measurement with such a fine sampling period. From the above practical point of view, it is very convenient to utilize explicitly the statistical information on the decibelscaled level fluctuation itself with the aid of the theory for evaluating Lx and Leq (e.g. see Ref. 3). The above information is fairly stable and reliable based on the averaging operation supported by a large number of data. Thus, in order to extract the statistical information, the sampling period of the noise level fluctuation can be grosset than that of current practice mentioned above. At the same time, each order moment statistics can be successively obtained by introducing an iterative calculation process. Therefore, the
Noise level fluctuation and noise indices
35
problem with a huge memory capacity can be solved by using this procedure. In this report, a unified method for measuring two noise evaluation indices, Lx and L~, is theoretically proposed by using the statistical information directly on the decibel-scaled level fluctuation under consideration. It must be noticed that the proposed method is generally applicable to any kind of random phenomena of a non-Gaussian property. Furthermore, the proposed method is a generalized one containing a well known simplified estimation method derived under the assumption of a standard Gaussian distribution. Based on this estimation method, a specific on-line measurement system for the sequential measurement is constructed using a digital sound level meter and a microcomputer. Finally, the effectiveness of the proposed method has been confirmed experimentally by applying it to actual road traffic noise.
G E N E R A L A L G O R I T H M FOR ESTIMATING L~ Let us consider the noise level fluctuation L (L a= 10 loglo E/E 0 = M l n E/Eo, M ~ 10/ln 10, where E is the noise energy fluctuation, and E 0 is the reference noise energy usually taken as 10-'2 W/m 2) of an arbitrary non-Gaussian distribution type. According to the previous study, 4 a generalized expansion-type expression for estimating Le~ is given by using the cumulant statistics of L, as follows (see Appendix A):
Leq
lOloglo-
(E) -o =
x2
+
~¢a
+
x4
+
~cs + 120M
=/a + 0"l15a 2 + 8"84 x 10-3xa + 5"09 x 10-4x4 +2"34 x 10-sx 5 + ' "
(1)
where xn (n = 1, 2 .... ), la (= ~cl)and a 2 (= x2) denote the nth order cumulant, the mean value and the variance of L. Here, ( * ) is an averaging operation with respect to the random variable *. From eqn (1), it is possible to estimate Lea generally by reflecting not only lower order cumulants but also higher order cumulants in a hierarchical form. It should be noted that the above estimation formula agrees completely with a well known simplified estimation formula, s derived under the assumption of a standard Gaussian distribution as the first approximation: L~q =/a + 0.115o-2
(2)
since higher order cumulants xn (n = 3, 4,...) become zero for this special case. Thus, this evaluation method shows a generalized form including the well known simplified estimation method as a special case.
Y. Mitani, M. Ohta
36
To establish an evaluation algorithm with the aid of a microcomputer one must obtain the cumulant K, (n = 1,2 .... ) by spending a little a m o u n t of its memory. To achieve this purpose, one can first calculate the ruth order m o m e n t o f L, especially by means of the following iterative process:. (L')u-
N-l N
1 ( L " ) N ~ +---L"N u
~3}
F r o m eqn (3), one can obtain the mth order m o m e n t ( L " ) N at the Nth measurement time based on the memorized past value of the ruth order moment, ( L " ) N 1, at the ( N - 1 ) t h measurement time and the present datum, L N, at the Nth measurement time, in an iterative form. After obtaining the mth order m o m e n t within the specific measurement time interval using the above procedure, the resultant m o m e n t (L"'.:. (n = 1, 2,...,m), can be transformed into the cumulant he, (n = 1,2 ..... m~, as follows: 6 h-, : (LS,
h'2 = ( L 25 -- ( L ) h 1
h."3 = ( L 3) -- 2 ( L ) K 2 -- (L2)h'l 1,:4 = ( L 45 - 3 ( L ) K 3 -- 3(L2)h-2
--(L3)h-i
Ks = ( L 5) - 4 ( L ) K a
-- 4(L3)K2
-
6(L2)h'3
--
(L4)h-i. . . .
(4)
Thus one can estimate the objective Leq by substituting the calculated value of cumulant statistics into eqn (1).
G E N E R A L A L G O R I T H M F O R E S T I M A T I N G L,. In order to estimate the arbitrary L x noise evaluation indices (e.g. L~, L~;, Lso, L9o,...), let us introduce the statistical Hermite series expansion-type expression 7 of the cumulative probability distribution universally applicable to the noise level fluctuation, L, of an arbitrary non-Gaussian distribution type, as follows (see Appendix B):
Q(L)=
N ( ~ ; / 1 , O.2} d ~
-LK~32N(L;p,O.2)H2 DO"
2 f L -- fl~ h4 N ( L ; lt, O" )H3 { ........ J 24O"3 "\ a / • ~,~s 4 N ( L ; I~, ae}H4 1 ZUO.
....
i51
Noise level fluctuation and noise indices
37
where N(~; #, 0"2) denotes the Gaussian distribution function defined as: , o.2) =zx
x / ~1t r exp ( - (~ - g)2/2tr2)
(6)
Moreover, H.(.) denotes the nth order Hermite polynomial, and its value can be calculated by the following recursive formula suitable for computer processing:8 H.(0 = ~H._ 1(~) - (n - 1)H._ 2(~)
(7)
In addition, the calculation of eqn (6) can be easily performed by using the well known error function, as follows: fL
N(~;/t, tr2)d~ = 1
1 E r f c {L -/z'~
(8)
where Erfc (.) denotes the error function, defined as: Erfc (~) ~
exp -'2 dt
(9)
At this time, the value of Erfc(0 can be accurately calculated by the following approximation: 9 2
- - E r ~ ( ~ ) + 1/[1 +0.0705230784~ +0.0422820123¢ 2 + 0.0092705272~ 3 + 0-0001520143~ 4 + 0.0002765672~ s + 0.0000430638~6] 16
(10)
Since the relationship between Q(L) and Lx is given by: x
1 100- Q(Lx)
(11)
and Q(L) is a monotone-increasing function, the objective Lx can be directly evaluated from the estimated curve of Q(L) using the linear interpolation, even for the measurement with a fairly large discrete level amplitude. This procedure is quite convenient to economize the calculation time, when compared to a usual numerical integration. Consequently, after obtaining the statistical information on the noise level fluctuation in the same manner as described in the previous section, one can calculate an arbitrary Lx using this method. Furthermore, based on this estimation method, it is not necessary to memorize the whole data of the noise level fluctuation, and one can solve the problem with a little memory capacity when using a microcomputer for the measurement system.
38
Y. Mitani, M. Ohta E X P E R I M E N T A L WORK
Confirmation of effectiveness of the proposed estimation method For the purpose of confirming the effectiveness of the proposed estimation method, a measurement system has been constructed using a digital sound level meter and a portable microcomputer. The block diagram of this measurement system is shown in Fig. 1. Two kinds of road traffic noise have been measured as a typical example of environmental noises. One of the road traffic noises was measured near a national main road in a large city. At that time, the traffic volume was 244 vehicles per ten minutes. Another type of road traffic noise was measured near a country road in a rural district. At that time, the traffic volume was 33 vehicles per ten minutes. The measurement time interval and its sampling period were selected to be 10 minutes and 1 second, respectively. Here, let us define the former case as 'Case A' and the latter one as 'Case B'. Figure 2 shows the actual location for measuring the road traffic noise tbr Case A. Figure 3 shows the actual location for measuring the road traffic noise fi~r Case B. Table 1 shows the estimated results of L,,q for Case A using the proposed estimation method. Here, let us define the expansion expression of eqn (1) from the first expansion term corresponding to the well known simplified estimation method, eqn (2), to the term due to the nth order cumulant, as the ( n - I)th approximation of the estimated L,,q. In addition, the measured value o f Leq in this table shows the value measured by a precision integrating sound level meter for Leq. According to these results, the calculated value. using a well known simplified estimation method derived under the assumption of a standard Gaussian distribution, agrees approximately with
Inr_er
f no("
(RS-2
Unit
L~(:)
M J (" F o C o m p t l t~! r
Fig. !.
Block diagram of measurement syslcm with the aid of portable n~icrocompulc~
Noise level fluctuation and noise indices
•
Car Park
Building
Factory
Htros|
Traffic Signal
Series of Stores
Route 2 To
:
39
7.7 m
• Car
Supermarket
mm
To Os~
lma
Building
Observation Point
Ser: Sto!
of
Park
50 m
Fig. 2. Actual location for measuring road traffic noise (Case A).
AI_
3
AI.
: Rice Field
.U_ ./I.
AL
To Higashimura
At.
Factory
I 5 m • 12 m
TO Matsunaga Observation Point .11-
.l.L
.It.
II
II
11..
Fig. 3. Actual location for measuring road traffic noise (Case B). TABLE 1 The Estimated Results of Leq Using the Proposed Method (Case A)
Measured value of Leq (dB(A)) 85-9
Estimated values using the proposed method (dB(A)) 86.4 (the 86.0 (the 86"1 (the 86.1 (the
first approximation) second approximation) third approximation) fourth approximation)
Y. Mitani, M. Ohta
40
TABLE 2 The Estimated Results of L,,~ Using the Proposed Method (Case B)
Measured value ~?[L~q {dB(A))
£~'thnated values using/tie proposed method (dB(A))
80-7
777 82-0 82-8 81.0
ithe (the {the (the
first approximation) second approximationt third approximation) lburth approximationl
the measured value. More specifically, however, the estimated values using the proposed method tend to be in good agreement with the successive addition of higher order expansion terms. Table 2 shows the estimated results for Case B. According to these estimated results, the estimation accuracy using the well known simplified estimation method is not sufficient for evaluation. On the other hand, it is obvious that the successive addition of higher order expansion terms moves the values estimated theoretically using the proposed method closer to the values measured experimentally. Figure 4 shows a comparison between the theoretically estimated curves using the proposed method and the experimentally sampled points in the I
I
I
I
I
1
"l
I 1.0
-
O. 5
-
- -
-~
o
o
I 50
60
70
80
l
1
I
90
] O0
110
__
1, ( d B A )
Fig. 4. A comparison between theoretically estimated curves using the proposed mctht)d and experimentally sampled points for cumulative probability distribution of noise level fluctuation (Case A). t , Experimentally sampled points. Theoretically estimated cur~es: • the first approximation ofeqn (5): , the second approximation (and the third and fourth approximations),
Noise level fluctuation and noise indices
41
cumulative distribution form of the noise level fluctuation for Case A (arbitrary Lx noise levels are directly extracted by this distribution form). From this figure, the theoretically estimated curve due to the first approximation (i.e. Gaussian distribution) agrees approximately with the experimentally sampled points. More precisely, however, the estimated values using the proposed method tend to be in good agreement with the measured values, with the successive addition of higher order expansion terms. Figure 5 shows the estimated results for Case B. From this figure, the theoretically estimated curve by using only the first expansion term corresponding to the Gaussian distribution does not agree well with the experimentally sampled points. It is obvious that the successive addition of higher order expansion terms moves the values estimated theoretically using the proposed method closer to the values measured experimentally. In each case, the estimation accuracy for both Lx and Le~ noise evaluation indices using the proposed estimation method is clearly sufficient. Construction of on-line measurement system for sequential measurement
In this section, an on-line measurement system for the sequential measurement of two noise evaluation indices, Lx and Leg, is specifically I
I
I
I
I
I
80
90
i00
1.0
-
0.5
S //
I
0
50
60
70
110
L (dBA)
Fig. 5. A comparison between theoretically estimated curves using the proposed method and experimentally sampled points for cumulative probability distribution of noise level fluctuation (Case B). O, Experimentally sampled points. Theoretically estimated curves: . . . . . , the first approximation of eqn (5); . . . . , the second approximation (and the third approximation); - - - , the fourth approximation.
42
Y. Mitani, M. Ohta
constructed based on the proposed estimation method. According to the estimated results mentioned above, in the actual estimation, the estimated values due to the fourth approximation have been employed. Since it is necessary to take the calculation time to estimate these two indices, the concurrent processing by two microcomputers can be used. The block diagram of this measurement system is shown in Fig. 6. Microcomputer A is used for the iterative process for extracting the moment statistics within each measurement time interval using eqn (3) and the control system for the communication lines between the digital sound level meter and Microcomputer B with the RS-232C type interface. Microcomputer B is used tbr the calculation of L~ and Leq using the proposed estimation method after transforming the moment statistics (L "> into the cumulant statistics h,. Each measurement time interval can be arbitrarily selected using the clock signal generator in Microcomputer A. In this measuremenL each Sound
Level
Mete1
[Interface
Unit
(R~;-232C)
1
'I
t. . . . .
Microcomputer
A
M u l t ip lexer J
Microcomputer
B
/ Microcomputer
A
: Iterative
calculation
of moment statistics and control system for c o m m u n i c a t i o n lines
Microcomputer B : C a l c u l a t i o n of L x and Leq using the proposed method
Fig. 6.
Block diagram of on-line measurement system for I.~ and L,,q using the proposed method.
Noise levei fluctuation and noise indices
43
~!~i~ii~i~,~.£v~¸¸¸i~? !! ~
Fig. 7.
Actual scene of road traffic noise measurements using the proposed on-line measurement system.
measurement time interval has been selected at 10 minutes. Two noise evaluation indices, L= and Leq, for the road traffic noise (Case A) have been measured for 24 hours. Figure 7 shows the actual scene of the road traffic noise measurement using the proposed on-line measurement system. Figure 8 shows the estimated results of L 5, L~0, Lso, L 9 o and Le~ using the proposed measurement system. The proposed system has good performance for a long-term measurement with each measurement time interval. CONCLUSION The conclusions of this study are characterized as follows: (1) A unified method for measuring two noise evaluation indices, L= and Leq, has been proposed using the statistical information on the noise level fluctuation.
Y. Mitani, M. Ohta
44 l
W
9O
1
,,z -1
i 8O
] c~q
70
60
50
4o 8:00
I 12:00
I
i
0:00
~ :,~(I Time
Fig. 8.
(h~u r)
The estimated results of Lx and L,,q using the proposed on-line measurement system.
(2) The proposed method is generally applicable to any kind of random
(3)
(4)
p h e n o m e n a of a non-Gaussian property. It has been derived in a generalized form including a well known simplified estimation method derived under the assumption of a standard Gaussian distribution. The statistical information used in the proposed method could be extracted successively by introducing an iterative calculation process. As the result, it is possible to economize the m e m o r y of a microcomputer. Based on the proposed method, it is necessary to use only a little a m o u n t of m e m o r y , even when changing the measurement time interval. In the case of evaluating Leq according to current practice, the problem o f the fine sampling period and the measurement errors due to an unequally quantized energy amplitude through the antilogarithmic transformation could be avoided using the proposed method. This is because of finding a reasonable theoretical relationship between the objective Leq value given by the sample mean operation with an equally quantized energy amplitude and the statistical information based on the level data with an equally quantized level amplitude.
Noise levelfluctuation and noise indices
(5)
45
By using this unified theory with the above advantages, an on-line measurement system for the sequential measurement t h r o u g h a longterm measurement time interval has been constructed using a digital sound level meter and a portable microcomputer.
Of course, this study is in its early stage and has been focused only on its fundamental aspects. Accordingly, there still remain m a n y future problems, as follows: (1) (2)
This m e t h o d must be applied to m a n y other actual cases to broaden and confirm its further effectiveness. In order to evaluate precisely the complicated situation of environmental noises, it is necessary to grasp systematically the general relationship between L x and Leq using this method. ACKNOWLEDGEMENTS
The authors would like to express their cordial thanks to M r H. Hosoya and Mr Y. Watanabe for their helpful assistance. The authors are also thankful for m a n y constructive discussions during the annual meeting of the Institute of Noise Control Engineering of Japan. 1° REFERENCES 1. Fisk, D. J., Statistical sampling in community noise measurement. J. Sound Vib., 30(2) (1973) 221-36. 2. Vaskor, J. G., Dickinson, S. M. & Bradley, J. S., Effect of sampling on the statistical descriptors of traffic noise. AppL Acoust., 12(2) (1979) l 11-24. 3. Don, C. G. & Rees, I. G., Road traffic sound level distributions. J. Sound Vib., 100(1) (1985) 41-53. 4. Ohta, M., Mitani, Y. & Sumimoto, T., A generalized theory for the mutual relationship among several type noise evaluation indices connected with Leq and Lx and its experiment. J. Acoust. Soc. Japan, 41(9) (1985) 598-607 (in Japanese). 5. U.S. Environmental Protection Agency, Information on levels of environmental noise requisite to protect public health and welfare with an adequate margin of safety. 550/9-74-004 (1974). 6. Lloyd, E., Handbook of Applicable Mathematics, Volume II: Probability. John Wiley, New York, 1980, pp. 251-2. 7. Ohta, M. & Koizumi, T., General statistical treatment of the response of a nonlinear rectifying device to a stationary random input. IEEE Trans. Inform. Theory, IT-14(4) (1968), 595-8. 8. Moriguchi, S., Utagawa, K. & Hitotsumatsu, S., Mathematical Formula IIl. lwanami, Tokyo, 1977, p. 93 (in Japanese). 9. Hastings, C., Approximations for Digital Computers. Princeton, New Jersey, Princeton University Press, 1955, p. 187.
46
Y. Mitani, M. Ohta
10. Mitani, Y. & Ohta, M., Practical estimation method of Leq and L x evaluation indices by the effective use of a microcomputer technique. Proc. 1986 Meeting Inst. Noise Control Engng Japan (1986), pp. 81--84 (in Japanese). A P P E N D I X A: D E R I V A T I O N O F E Q N (1) At the starting point o f the analysis, let us introduce the m o m e n t generating function mL(O) o f the noise level fluctuation L, as follows:
mL(O) L=(exp (OL)~
(A, I)
By using the relationship between the noise level fluctuation L and the noise energy fluctuation E, eqn (A.1) is rewritten as
mL(O) = ((E/Eo) °M)
(A.2)
On the other hand, based on the statistical theory, the m o m e n t generating function is generally expressed by using the cumulant statistics ~,, (n = 1, 2 .... ), as follows: 6 mL(O) = exp ~ ~, (A.31
IZ0 i n = 1
Therefore, by replacing 0 with 1/M in eqns (A.2) and (A.3) and using these equations and the definition of Leq, one can derive the following formula (see eqn (1)): ¢-.-.,
1 = 1 (E) ) Leq 2~ 10 og~o-~o ..... ~.a n!M'-
1 /x'n
(A.4)
n=l
A P P E N D I X B: D E R I V A T I O N O F EQN (5) Let us introduce the statistical Hermite series expansion-type expression ~'as a general explicit expression on the probability density function P(L) with respect to the noise level fluctuation L, as follows:
t Z
P(L)= N(L;p,a 2) 1+
A,H. - -
(B.t)
-It
(B.2)
n=3
with
1
H.
Noise level fluctuation and noise indices
47
At this time, by using the well known formula between the Gaussian distribution and the Hermite polynomial: 7 N(~; # , trZ)Hn
=
( - 1)"a"S-ff N(~; ~u,a 2)
(B.3)
the cumulative distribution form Q(LX~=~oP(~)d¢) is consequently obtained as follows: oo
n=3
After expressing specifically the expansion coefficient A, using eqn (B.2), eqn (5) can be derived.