量子化レベルでの実測騒音・振動レベル分布に適合した統計的基礎理論とその実験

抄録

In previous papers, we have reported some trial of the statistical treatment for the continuous level fluctuation of arbitrary random noise or vibration. However, it is necessary to measure the actual noise level data (e. g. , by a sound level meter) in a form of digital level at descrete times. For this digital level data the use of digital computer is essential for the various statistical evaluations and the extraction of statistical information (e. g. , median, mean, variance, higher order moments, 90% range, etc. ) of random noise. From these points of view, in this paper we give theoretical consideration of the statistical treatment of random noise or vibration level distribution suitable to the actual situation, on which the real experimental data are based, in the form of digital level and finite number. Specifically, when a random noise or vibration with the digital level Z of arbitrary distribution type is considered to be the sum of two different random processes X and U with digital levels resulting from the natural internal structure of the fluctuation or the analytically artificial classification of the fluctuation, a unified statistical treatment for the probability distribution of the resultant random fluctuation Z(=X+U) is introduced exactly in a new form of expansion terms (here, X and U may be mutually correlated). Let us now introduce an arbitrary function φ(Z) and consider its expectation value Eq. (1) . Eqs. (3) and (4) can be obtained by use of the Newton's interpolation formula. Our main problem is how to derive the probability function P(Z) by the difference form of expansion terms based on the statistical information of X and U. After a somewhat complicated derivation, we obtain the two expansion expressions, Eqs. (11) and (13) when X is statistically correlated with U and Eqs. (14) and (15) when X is statistically independent of U. Compared with theories regarding a continuous level distribution, the above theoretical result is characterized by some specific features: (1) This result has a form of difference type instead of differential type in its expression. Therefore, the experimental frequency distribution P_X(X) can be directly used by keeping its crude numerical form . i. e. there is no necessity for previously approximating P_X(X) with an appropriate function form. (2) When the difference operation is actually done in practice, the above infinite series type expansion expression is exactly truncated with a finite number of terms. (3) In the special case of taking P_X(X) as Poisson distribution, the above theoretical result agrees with the well-known Charlier B type expansion series. (4) As another special case of letting the level width tend to 0, the above theory includes the well-known expansion series distribution in the continuous level form. Finally, we have experimentally confirmed the validity of our theory not only by means of digital simulation but also by experimentally obtained road traffic noise data in Hiroshima City.