不規則騒音のレベル変動に関する統計的研究(VII) : 有限メモリ整流形非線形変換器を通った任意不規則騒音波の統一理論

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When random signals (e. g. , street noise, voice) of both Gaussian and non-Gaussian type are passed through an arbitrary rectifying nonlinear transducer with finite memory, it is a very important problem in the engineering field of noise control to find out a unified method of treating statistically the output random fluctuation. We are well aware of the fact that without the finite memory effect of transducer the nonlinear system cannot perform effectively its nonlinear action on the output. The extent of this difficulty, however, can be diminished by redefining approximately the function of a given nonlinear transducer as "zero-memory nonlinear element" plus "linear finite memory part". Since a rectifying nonlinear element of zero-memory type, whatever it is, always produces an output fluctuation only in positive region, a joint probability density function for a few arbitrarily chosen samples X_i (i=l, 2, . . . , k) of the output random signal can be firstly introduced in terms of an orthonormal expansion of the statistical Laguerre series as seen in Eqs. (1) and (2). It must be noticed that the first and higher order correlations among sampled values are reflected in each of the expansion coefficients β(n_l, n_2, . . . , n_k) (n_i≠0, A_i). If we notice the output fluctuation Z=Σ^^^k___<i=1> a_iX_i in the form of the weighted mean given as the memory effect after zero-memory nonlinear transformation, it is convinient to start our analysis from the joint moment generating function m(t_1, t_2, . . . , t_k)=<expΣ^^^k___<i=1>t_iX_i> (cf. Eqs. (3) to (6)) in the light of Levy's continuity theorem and uniqueness theorem for the characteristic function. Thus a moment generating function of Z can be expressed by m(a_1t, a_2t, . . . , a_kt) and therefore it must be in principle able to derive an expression of probability density function P(Z) of Z in the form of expansion. Particularly, when we take our interest in the mean operation Z =Σ^^^k___<i=1>X_i/K as a special form and the stationary random output process, we can obtain an expression of P(Z) (cf. Eq. (9)) from the solution of an integral equation (cf. Eq. (7)). Then, the universal expressions of cumulative probability and probability density functions for the output Z of nonlinear transducer have been explicitly derived in the general form of expansion series by introducing a nondimensional variable u into the above expression (cf. Eq. (14)). Each of the expansion coefficients A_l (l = 2, 3, 4, . . . ) expresses the effect of general correlations among sampled values, the nonlinear characteristic and the finite memory on the distribution. Finally, it has been shown that the above expansion coefficients A_l can be estimated from the experimental measurement of the moment with respect to P(Z) by the method of moment (cf. Eqs. (22), (23) and (24)). Because of the arbitrariness of input distribution, correlations, kind of rectifying nonlinear transducer and time interval of mean operation, the general method described in this paper is also applicable to the other fields of random phenomena.
社団法人日本音響学会の論文
1969-11-10

不規則騒音のレベル変動に関する統計的研究(VII) : 有限メモリ整流形非線形変換器を通った任意不規則騒音波の統一理論

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