騒音変動波形の激しさに対する統計的評価の実験的一考察

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It is necessary to consider the first and higher order correlations between the level and the slope of an arbitrary random street noise in order to find quantitatively the acuity of its fluctuation. The expected number of level-crossing is known as one of the physical quantities determined by only two values of the level and the slope, and the investigation on the number of level-crossing at each level gives, therefore, some significant information about the acuity of the street noise fluctuation. We have shown previously that the probability density function (hereafter, abbreviated as p. d. f. ) of level-crossing can be expressed in terms of orthonormal expansion of the statistical Lauergre series. However, the problem how to choose the origin of level axis remains unsettled in above method. Namely, as the result that null phon which is not measured practically in any street noise should be taken as the level origin, the above expansion expression of the statistical Laguerre series may lack the reliability. If we choose the mean of level calculated from many sample points of the street noise(measured by phon) as the level origin, the reliability will be raised. Then, we must consider that even the street noise fluctuating only in positive region fluctuates in both positive and negative sides of the origin thus taken. From a theoretical viewpoint, the statistical treatment shown previously in the field of surface roughness can be used. In this paper, the application of the theory to street noise, its experimental consideration and the comparison between experiment and theory are reported newly. First, a joint p. d. f. P(X, Y) of the instantaneous value X and the slope Y in an arbitrary random wave fluctuating in positive and negative regions are expressed by Eqs. (1) and (2) in terms of the statistical Hermite orthonormal expansion series. If we take <X>=<Y>=0, σ_X^2=<X^2> and σ_Y^2=<Y^2>, it is shown that β(0, 0)=β(0, 1)=β(2, 0)=β(0, 2)=0 and that β(1, 1) corresponds to linear correlation coefficient (ρ(X, Y)). The higher order correlations are reflected into the expansion coefficients as shown in Eq. (4). The relations (1) and (2) can be applied to the street noise, whose mean value is chosen as the origin, fluctuating on both positive and negative sides of the origin chosen. The expected number M(X) of X per second passing through a level X with positive and negative slope is given by M(X)=∫^∽_<-∽>│Y│P(X, Y)dY. the p. d. f. M_0(Z) of level-crossing at level Z is expressed by Eq. (5), where Z=X/σx and the expansion coefficients are given in Eq. (6). In the same manner, the expected number N(X) of level-crossing with only positive slope is given by N(X)=∫^∽_0YP(X, Y)dY and can be led to the expressions (8) and (9) as the p. d. f. N_0(Z) of level-crossing at level Z by the aid of integral formulas(7). On the other hand, the p. d. f. P(Z) of the instantaneous value of level fluctuation can be obtained by Eq. (10). Thus, the difference ε_M(Z) or ε_N(Z) between the cumulative probability distribution of the instantaneous level fluctuation and that of level-crossing is expressed as Eq. (11) or Eq. (12) in the similar expansion form. It is worthy to note that the expansion coefficients of Eq. (11) or Eq. (12) do not contain β(3, 0)(denoting the skewness) and β(4, 0)(denoting the kurtosis), but contain only the linear and nonlinear correlations between X and Y. An information about the acuity of the street noise fluctuation is given in each expansion coefficient. Then, the expansion coefficients can be estimated experimentally by method of moment as shown in Eqs. (14) and (16), where< >, < >' and < >" denote the averaging treatment about P(Z), M_0(Z) and N_0(Z) respectively. Finally, it is proved from experiment as shown in Figs. 1 to 3 that the consideration on the first and higher order correlation is indispensable in order to elucidate the acuity of the street noise fluctuation.
社団法人日本音響学会の論文
1970-05-10

騒音変動波形の激しさに対する統計的評価の実験的一考察

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