1965 年 21 巻 6 号 p. 333-346
In the present paper, our attention is focused on the actual random variation itself in time of the noise level, such as the street noise, and the statistical treatment from such a dynamic viewpoint is in striking contrast to the statistical method of treating treating the noise fluctuation in the form of one-variate level distribution as given in the previous paper. When we observe simultaneously one or more noise fluctuations having rondom phases at two or more different observation points with differences in time, position and frequency, we need to consider the joint probability distribution of the multiple correlative random noise variates. For instance, it corresponds to the case where we consider the general correlation among many instantaneous of noise level (phon) fluctuating only in positive region and their variations (speed or gradients of degrees 2, 3, 4, ・・・・) for the purpose of analysing quantitatively the variety of random noise fluctuation. First, we define the range of the fluctuation of the multiple correlative random noise variates (e. g. , the noise level is defined in 「(0,∞) and its gradient is defined in (-∞, ∞)」 and then express the joint probability density distribution which governs the noise variates in the form of orthonormal series within the range of its definition - a mixed expansion in statistical Laguerre series and statistical Hermite series. In practical application it is necessary to truncate the above statistical series expansion to an appropriate number of terms according to the stability of the distribution. We must call our attention to the fact that the statistical meaning (i. e. , the randomness property) is reflected in each expansion coefficient. More explicitly, each coefficient gives the general correlation of high degree among the multiple random noise variates, of which the usual linear correlation is a first-order approximation. Finally, the detailed experimental considerations of the detection of weak signal buried in the random noise by means of a certain linear cross-correlation are given in the following five cases: (a) the periodicity of a crossing signal buried in the traffic noise, (b) the period of a sinusoidal or a square wave buried in the white noise, (c) the velocity of motor rotation buried in the factory noise, (d) the commercial frequency buried in the noise of transformer room, (e) the klirrfactor of a distorted wave buried in the white noise. The statistical method described in this paper seems to be applicable also to other wide fields of measurement on random phenomena because of its generality.