An Estimation Method Matched to Microcomputer-Aided On-Line Measurement of L eq by Use of Statistical Information on the Noise Level Fluctuation

Transcription

1 GESTS It l Tras. Coputer Sciece ad Egr., Vol.8, o. 9 A Estiatio Method Matched to Microcoputer-Aided O-Lie Measureet of L eq by Use of Statistical Iforatio o the oise Level Fluctuatio Yasuo Mitai, oboru akasako, ad Kazuhiro Tsutsuoto Faculty of Egieerig, Fukuyaa Uiversity, Hiroshia, Japa itai@fuee.fukuyaa-u.ac.jp School of Biology-Orieted Sciece ad Techology, Kiki Uiversity, Wakayaa, Japa akasako@ifo.waka.kidai.ac.jp Faculty of Hua Cultures ad Scieces, Fukuyaa Uiversity, Hiroshia, Japa tsutsu@fucc.fukuyaa-u.ac.jp Abstract. I this paper, a geeral estiatio ethod of L eq oise evaluatio ide is proposed usig the statistical iforatio o the oise level fluctuatio. This ethod is give i a epasio type estiatio forula uiversally applicable to the arbitrary o-gaussia distributio, icludig a well-kow siplified epressio derived uder the assuptio of a stadard Gaussia distributio as the first approiatio. At this tie, for etractig the statistical iforatio o the oise level fluctuatio, we propose a iterative processig algorith atched to the icrocoputer-aided o-lie easureet. et, a selectio ethod of the optiu order of this epasio type L eq estiatio forula is proposed by itroducig Akaike s iforatio criterio (AIC). Fially, the effectiveess of the proposed ethod has bee cofired eperietally by applyig it to actual road traffic oise. Itroductio As is well kow, the oise evaluatio ide, L eq, plays a iportat role i the field of oise evaluatio ad regulatio probles. I order to evaluate this ide, the usual easureet ethod is give accordig to the origial defiitio. That is, the L eq oise evaluatio ide is defied as a costat oise level whose oise eergy value is equal to a averaged eergy of the oise level fluctuatio over a total easureet tie iterval. owadays, i a actual easureet, the oise level fluctuatio is very ofte easured i a quatized aplitude for at every discrete tie period usig a digital-type istruet []. I the case where these data are used to evaluate L eq, the followig fudaetal probles still reai as see fro the viewpoit of sigal processig: () I order to evaluate L eq, it is ecessary to obtai ay level data with a fairly fie saplig period, sice the oise eergy fluctuatio after the ati-logarithic trasforatio of the sapled level datu fluctuates with large ecursios, as GESTS-Oct.005

2 0 A Estiatio Method Matched to Microcoputer-Aided copared with the origial decibel-scaled level fluctuatio. () I priciple, the ea value of the oise eergy fluctuatio should be give by the saple ea operatio based o the oise eergy data especially with a equally quatized eergy aplitude. If we use the oise eergy fluctuatio after trasforig the easured oise level fluctuatio with a equally quatized level aplitude ito the eergy scale through the ati-logarithic trasforatio, a calculatio error of the eergy ea will occur because of the above level quatizatio. I relatio to the above probles, accordig to the curret easureet techiques, the saplig period ay greatly ifluece the accuracy of the easureet result. aely, a fie saplig period related to the tie costat of the itegratio givig the oise level will geerally give a good approiatio to the results obtaied with a true itegratio. Therefore, if we costruct the easureet syste with the use of a icrocoputer accordig to curret practice, a huge eory capacity is eeded for a log-ter easureet with such a fie saplig period. Fro the above practical poit of view, it is very coveiet to utilize eplicitly the statistical iforatio o the decibel-scaled level fluctuatio itself with the aid of the theory for evaluatig L eq. The above iforatio is fairly stable ad reliable based o the averagig operatio supported by a large uber of data. Thus, i order to etract the statistical iforatio, the saplig period of the oise level fluctuatio ca be grosser tha that of curret practice etioed above. At the sae tie, each order oet statistics ca be successively obtaied by itroducig a iterative calculatio process. Therefore, the proble with a huge eory capacity ca be solved by usig this procedure. I this paper, a geeral estiatio ethod of L eq oise evaluatio ide is proposed usig the statistical iforatio o the oise level fluctuatio. This ethod is give i a epasio type estiatio forula uiversally applicable to the arbitrary o-gaussia distributio, icludig a well-kow siplified epressio derived uder the assuptio of a stadard Gaussia distributio as the first approiatio. At this tie, for etractig the statistical iforatio o the oise level fluctuatio, we propose a iterative processig algorith atched to the icrocoputer-aided o-lie easureet. et, a selectio ethod of the optiu order of this epasio type L eq estiatio forula is proposed by itroducig AIC []. Fially, the effectiveess of the proposed ethod has bee cofired eperietally by applyig it to actual road traffic oise. Geeral Method for Estiatig L eq Let us cosider the oise level fluctuatio of a arbitrary o-gaussia distributio type. As is well kow, the relatioship betwee the oise level fluctuatio ad the oise eergy fluctuatio E is give as follows: E E 0 = 0log 0 = M l M, () E E l0 0 0 cgests-oct.005

3 GESTS It l Tras. Coputer Sciece ad Egr., Vol.8, o. where E 0 is the referece oise eergy usually take as ( W ) itroduce the oet geeratig fuctio ( θ ) fluctuatio, as follows: 0. Here, we M with respect to the oise level ( θ ) = ep( ) M θ = E ep θ M l, () E0 where <*> deotes a averagig operatio with respect to the rado variable *. The atheatical relatioship betwee the -th order cuulat κ ( =,,L) with M θ is give by respect to ad the oet geeratig fuctio ( ) M By replacig the paraeter θ to be easily obtaied as follows: ( ) ep θ = κ θ. () =! M i Eqs.() ad (), the ea value of E ca E = κ E0 ep =! M. () Thus, a substitutio of Eq.() ito the defiitio of L eq yields a geeral epasio type epressio for estiatig L eq, as follows []: L eq = 0log 0 E E 0 = κ κ κ κ 5 κ M 6M M 0M + L 5 = µ + 0.5σ κ κ +. 0 κ +L, (5) 5 where ( ) κ µ = ad ( κ ) σ = deote the ea value ad the variace of. Fro Eq.(5), it is possible to estiate L eq geerally by reflectig ot oly lower order cuulats but also higher order cuulats i a hierarchical for. It should be oted that the above estiatio forula agrees copletely with a well-kow siplified estiatio forula [] derived uder the assuptio of a stadard Gaussia distributio as the first approiatio: sice higher order cuulats ( =,,L) L eq = µ + 0.5σ, (6) κ becoe zero for this special case. Thus, this estiatio ethod shows a geeralized for icludig the well-kow GESTS-Oct.005

4 A Estiatio Method Matched to Microcoputer-Aided siplified estiatio ethod as a special case. To establish a estiatio ethod of L eq with the aid of a icrocoputer, we ust κ =,,L by spedig a little aout of its eory. To obtai the cuulat ( ) achieve this purpose, we ca first calculate the -th order oet of, especially by eas of the followig iterative process: = +. (7) Fro Eq.(7), we ca obtai the -th order oet tie based o the eorized past value of the -th order oet ( ) -th easureet tie ad the preset datu at the -th easureet at the at the -th easureet tie, i a iterative for. After obtaiig the -th order oet withi the specific easureet tie iterval usig the above procedure, the resultat oet ( =,, L ), ca be trasfored ito the cuulat κ ( =,, L ), follows [5]:,, as κ =, κ = κ, κ = κ κ, κ = κ κ κ, 5 κ = κ 6 κ κ κ,. (8) 5 L Thus, we ca estiate the objective L eq by substitutig the calculated values of cuulat statistics ito Eq.(5). Selectio Method for the Optiu Order of Epasio Ter Here, it should be oticed that the saple uber of actually obtaied data is fiite. Moreover, sice it is ipossible to calculate the ifiite uber of epasio ters i the actual data processig, the fiite uber of epasio ters ust be ievitably eployed. Therefore, we ust establish a reasoable ethod for selectig the uber of higher order correctio ters i Eq.(5). Fro the practical poit of view, we regard the reaiig error after eployig a appropriate epasio ter as the eaigless error iforatio. Therefore, i a case whe the fiite uber of correctio ters is eployed, it is possible to eploy AIC as a evaluatio criterio for deteriig the optiu order of epasio epressio. I this case, each epasio coefficiet of higher order cuulats i Eq.(5) should be estiated i cgests-oct.005

5 GESTS It l Tras. Coputer Sciece ad Egr., Vol.8, o. advace by usig least squares ethod, as follows: L ˆ + ˆ + ˆ + ˆ eq µ σ aκ bκ cκ dκ +L. (9) = 5 6 Thus, we ca obtai the optiu order so as to iiize the value of AIC [], as follows: ˆ σ (0) AIC = l + ( uber of correctio ters) i. e Here, deotes the uber of data ad ˆe σ deotes the variace of estiatio errors betwee the true values easured actually ad the estiated values by usig the estiatio forula with the arbitrary order. Eperietal Work For the purpose of cofirig the effectiveess of the proposed estiatio ethod, a easureet syste has bee costructed usig a digital soud level eter ad a portable icrocoputer. The road traffic oise data of 00 kids have bee easured at various observatio poits i Fukuyaa City by use of this easureet syste. The easureet iterval ad its saplig period were selected to be 0 iutes ad 0. secods, respectively. Table shows the estiated values of epasio coefficiets by use of least squares ethod. We calculated the variaces of estiatio errors betwee the true values easured actually ad the estiated values by usig the estiatio forulae with fiite epasio ters with the above estiated coefficiets. The estiated values of AIC versus the uber of correctio ters for the easured road traffic oise are show i Fig.. Fro this figure, the optiu order is (i.e., the optiu uber of correctio ters is ). The optiu estiatio forula of L eq is obtaied as follows: L = µ +. () eq 0.5σ κ κ Table. The estiated values of epasio coefficiets by use of least squares ethod Eployet of correctio Estiated epasio coefficiets ters up to â bˆ ĉ dˆ st ter.09 0 d ter rd ter th ter GESTS-Oct.005

6 A Estiatio Method Matched to Microcoputer-Aided It is oticeable that the coefficiets i the above optiu estiatio forula are approiately equal to the epasio coefficiets give i Eq.(5). This is because of the reasoable cosideratio of o-gaussia property based o the fairly stable iforatio fro the first order cuulat to fourth order cuulat. We apply the proposed ethod to kids of data selected radoly i the above 00 kids of easured data (these are defied as Case A, Case B ad Case C). The estiated results of L eq by usig the proposed estiatio ethod are show i Table. I order to cofir the practical effectiveess of the proposed ethod, it is applied to the other kids of data easured at the other observatio poits (these are defied as Case D, Case E ad Case F). The estiated results for these cases are show i Table. Accordig to these tables, the estiated values by usig the proposed ethod are i good agreeet with the eperietal values. uber of Correctio Ters AIC Fig.. The values of AIC versus the uber of correctio ters for the easured road traffic oise data Table. The estiated results for L eq by use of the proposed ethod Eperietal values (db) Estiated values (db) Estiatio errors (db) Case A Case B Case C Table. The estiated results for L eq by use of the proposed ethod Eperietal values (db) Estiated values (db) Estiatio errors (db) Case D Case E Case F cgests-oct.005

7 GESTS It l Tras. Coputer Sciece ad Egr., Vol.8, o. 5 5 Coclusio I this paper, a geeral ethod of estiatig L eq oise evaluatio ide has bee proposed usig the statistical iforatio o the oise level fluctuatio. This ethod has bee give i a epasio type estiatio forula uiversally applicable to the arbitrary o-gaussia distributio, icludig a well-kow siplified epressio derived uder the assuptio of a stadard Gaussia distributio as the first approiatio. At this tie, for etractig the statistical iforatio o the oise level fluctuatio, we proposed a iterative processig algorith atched to the icrocoputer-aided o-lie easureet. et, a selectio ethod of the optiu order of this epasio type L eq estiatio forula has bee proposed by itroducig AIC. Fially, the effectiveess of the proposed ethod has bee cofired eperietally by applyig it to actual road traffic oise. Of course, this study is i its early stage ad has bee focused oly o its fudaetal aspects. Accordigly, there still reai future probles, as follows: () This ethod ust be applied to ay other actual cases to broade ad cofir its further effectiveess. () We ust propose a estiatio ethod for the arbitrary L oise evaluatio idices (e.g., L 5, L 0, L 50, L 90, L 95,...). Ackowledgeets The authors would like to epress their cordial thaks to Dr. M. Ohta, Mr. M. Shigeawa ad Mr. S. Shiizu for helpful discussios. Refereces [] M. Ohta, S. Yaaguchi ad Y. Mitai, A ew O-lie Estiatio Method of Represetative Statistics of Eviroetal Rado oise ad Vibratio Based o Actual Digital Measureet, J. Acoust. Soc. Jp., Vol. 9, pp.6-66, 98 (i Japaese). [] H. Akaike, A ew Look at the Statistical Model Idetificatio, IEEE Tras. Auto. Cot., Vol. AC-9, pp.76-7, 97. [] M. Ohta, Y. Mitai ad T. Suioto, A Geeralized Theory for the Mutual Relatioship aog Several Type oise Evaluatio Idices Coected with L eq ad L ad Its Eperiet, J. Acoust Soc. Jp., Vol., pp , 985 (i Japaese). [] U.S. Eviroetal Projectio Agecy, Iforatio o Levels of Eviroetal oise Requisite to Protect Public Health ad Welfare with a Adequate Margi of Safety, 550/9-7-00, 97. [5] E. Lloyd, Hadbook of Applicable Matheatics, Volue II: Probability, Joh Wiley, ew York, pp.5-5, 980. GESTS-Oct.005

8 6 A Estiatio Method Matched to Microcoputer-Aided Biography ae: Yasuo Mitai Address: Faculty of Egieerig, Fukuyaa Uiversity, Sazo, Gakue-cho, Fukuyaa City, Hiroshia, Japa Educatio & Work eperiece: He received the B.E. ad M.E. degrees fro Fukuyaa Uiversity i 979 ad 98, respectively. He received the Dr. Eg. degree fro Hiroshia Uiversity i 990. He is ow a professor at Faculty of Egieerig, Fukuyaa Uiversity. His research iterest icludes acoustic sigal processig, soud ad vibratio cotrol, ad their applicatio to acoustics. Tel: E-ail: itai@fuee.fukuyaa-u.ac.jp ae: oboru akasako Address: Faculty of Biology-Orieted Sciece ad Techology, Kiki Uiversity, ishi-mitai 90, Uchita-cho, aga-gu, Wakayaa, Japa Educatio & Work eperiece: He received the B.E., M.E., ad Dr. Eg. degrees fro Hiroshia Uiversity i 98, 98, ad 990, respectively. He is ow a professor at Faculty of Biology-Orieted Sciece ad Techology, Kiki Uiversity. His research iterest icludes acoustic sigal processig, soud ad vibratio cotrol, idepedet copoet aalysis ad their applicatio to acoustics. Tel: E-ail: akasako@ifo.waka.kidai.ac.jp ae: Kazuhiro Tsutsuoto Address: Faculty of Hua Cultures ad Scieces, Fukuyaa Uiversity, Sazo, Gakue-cho, Fukuyaa City, Hiroshia, Japa Educatio & Work eperiece: He received the B.E. degree fro Fukuyaa Uiversity i 979. He is ow a associate professor at Faculty of Hua Cultures ad Scieces, Fukuyaa Uiversity. His research iterest icludes acoustic sigal processig, soud ad vibratio cotrol, ad their applicatio to acoustics. Tel: E-ail: tsutsu@fucc.fukuyaa-u.ac.jp cgests-oct.005