The Effect of General Correlations of High Degree on the Expected Number of Level Crossing in the Arbitrary Random Waves Fluctuating only in Positive Region

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概要

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• A joint probability density distribution of the instantaneous value X and the slope Y in the arbitrary random wave fluctuating only in positive region can be expressed in the form of a mixed orthonormal expansion series in the statistical Laguerre series and the statistical Hermite series as follows:P(X, Y)=Xm-1/Γ(m)Sme-X/S·1/√2πλe-(Y-μY)2/2λ{1+∞Σn=1∞ΣK=1β(n, K)Ln(m-1)(X/S)HK(Y-μY/√λ)}.It must be noticed that the general correlation of high degree between level X and slope Y is reflected in each expansion coefficient β(n, k) (n≠0, k≠0).While, the expected number M(X) of X per second passing through a level X with positive and negative slopes is given by M(X)=∫∞-∞|Y|P(X, Y)dY.Therefore, by using an expression of the expected number in the normalized form: M0(X)=M(x)/(∫∞0M(X)dX), the probability density distribution function of crossing at level X with positive and negative slopes can be expressed as M0(Z)=Pr(Z;m)+AmPr(1)(Z;m+1)+Bm(m+1)/2!Pr(2)(Z;m+2)+Cm(m+1)(m+2)/3!Pr(3)Z;m+3)+Dm(m+1)(m+2)(m+3)/4!Pr(4)Z;m+4……, where Z=X/S, S=μx/m, m=μx2/δx2 (μx: mean, δx2: variance) and Pr(Z;m)≡Zm-1/Γ(m)e-Z is a gamma probability density distribution function. In the same way, we can obtain an expression of the probability density distribution function N0(Z) of crossing at level X with a positive slope in the similar expansion form.Finally, the detailed experimental considerations to corroborate the above theories are given in the following two cases:(a) relation between the cumulative probability distribution ∫Z0P(Z)dZ of the instantaneous value and the cumulative probability distribution ∫Z0M0(Z)dZ(or∫Z0N0(Z)dZ) of crossing, (b) high correlations β(n, k) between level X and slope Y, by taking the street noise measured in the city of Fukui as an example of the above random wave fluctuating only in positive region.

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公益社団法人 計測自動制御学会 | 論文

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