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.
The Loudness Level of S. S. Stevens, well-known as a subjective noise rating number. cannot be simply obtained because of the need of indispensable definit procedures by Octave Band Sound Pressure Level. It has already been clarified that there exists a close correlation between Sound Levels (A, B and C) and Loudness Level owing to the studies of S. Morita, A. Nakano and P. H. Parkin. But in case the Sound Level (for example, dB(A)) is used in place of the Loudness Level, it cannot be free of some error. Figs. 9, 10 and 11 show a very intimate correlation between Sound Level and Loudness Level in our investigation, using (C-A) value as a parameter on the basis of noise measured on various parts of motor cars. The results of regression analysis are given in Table 6. Furthermore, in case the formula: LL = 1/2(C+A)+10 (phon) is assumed for a simpler calculating method, the error becomes very small and no problem in operation occurs. This means that the Loudness Level can be simply measured by using only Sound Level Meter.
The piezoelectric ceramic vibrator of thin disc type with surrounding electrodes on both surfaces as shown in Fig. 1 presents a unique phenomenon that the first higher order vibration in the radial direction can be excited more strongly than the fundamental vibration. The fundamental equations of stress components and electric flux density for the symmetrically extended vibration of the above vibrator are expressed by Eq. (1) in cylindrical coordinates. In order to carry out the analytical calculation, the differences of sound velocities in the parts of ceramics with and without electrodes were considered, as Tachibana had already tried in the analysis of ceramic vibrator with dot-shaped electrodes. Young's modulus and Poison's ratio in the part with electrodes are denoted by Y_0 and σ_0, and those in the part without electrodes by Y and σ, respectively. By considering the boundary conditions that (1) displacement component [U_r]_<r=0> = 0, (2) stress component [T_<rr>]_<r=a> = 0 and (3) U_r and T_<rr> are continuous at r=b, the constants A, B and D are determined by Eq. (15), (16) and (17). Two-terminal admittance of the vibrator is calculated by Eq. (23). Furthermore, resonant frequency fR, antiresonant frequency f_A and Δf/f_R are expressed by Eqs. (25), (27) and (29), respectively. Fig. 2 shows the equivalent circuit of the vibrator in the neighborhood of resonant frequency. Equivalent constants C_0, L_1 and C_1 are calculated by Eqs. (30), (31) and (33), respectively. In order to ascertain whether or not the above calculated results coincide with the measured results, experiments were made for PZT disc ceramics by varying the diameter of surrounding electrodes. σ_0 was 0. 295 which was measured for the unsymmetrical vibration. σ was 0. 36 which was determined by extrapolating the curve of the ratio of the fundamental resonant frequency to the first higher order resonant frequency vs. p=b/a to p=1 (which corresponds to the part without electrodes). As the ratio of sound velocities λ a value of 0. 982 was used which was calculated by Eq. (34). The electromechanical coupling coefficient k_p of the used specimens was about 28. 15%. Measured results of resonant and antiresonant frequencies for several specimens are shown in Table 3. Fig. 3 shows the roots of resonant frequency equation H(R, λ, p, σ_0, σ)=O (Eq. (24)) as a function of p by varying the parameter λ under the condition of σ_0=σ=0. 295. Fig. 4 gives the calculated curves of X_R vs. p and X_A vs. p where X_R and X_A are the roots of Eq. (24), and Eq. (26), respectively. Calculated and measured results of resonant frequency are shown in Fig. 5 as a function of p. In the same way, antiresonant frequency, Δf/f_R, equivalent inductance and capacitance are shown in Figs. 6, 7, 8 and 10, respectively. From these results it was concluded that the calculated values coincide with the measured ones considerably well. From the relation of Eq. (32), it is understood that the equivalent inductance L_1 is, proportional to the thickness of the ceramic disc (t). Calculated and measured results of L_1 are shown in Fig. 9 as a function of t. Though the gradient of the measured L_l vs. t curves coincided with the calculated one, fairly large differences was found in the absolute values. Measured results of equivalent resistance at resonant frequency for various surrounding electrodes are shown in Table 4. As mentioned in the first place of this paper, this result shows that the first higher order vibration can be excited more strongly compared with the fundamental vibration. Consequently, ceramic disc vibrators of several MHz class can be obtained easily by applying the above-mentioned fact. Such vibrators are interesting from the view point of practical applications for filter element and so on.