Since the fundamental operation of Surface Acoustic Wave (SAW) filters is about the same as that of transversal filters, the impulse response model has been used in designing the SAW filters. However, the SAW filter having responses which satisfy the specification of several filters used in communication systems, were not obtainable by the simple model, due to the Gibbs oscillation which is caused by the truncation of the Fourier series after a finite number of terms. For this reason, the design methods, which enables reduction of such ripples, have been reported by several authors. However, these design methods were not optimum one because the filters designed by these methods did not always satisfy the specification, or consisted of a structure with either input or output tranducer must be with a single electrodes pair, which causes large insertion loss. In this paper, we present a design of SAW filters using a linear programming technique (LP), which enables optimum design so that the designed filter always satisfies the passband or stopband specification, and both input and output transducers consist of multi-paired electrodes as shown in Fig. 1. The design methods of SAW filters using LP contain both a time domain and a frequency domain. The former is suitable for the optimum design of SAW filters, and the latter is suited for the simplest design of narrow bandpass filters. The reason why LP is used in the design, is that the frequency response of SAW filters is a linear function with regard to the design parameter W_n or H_n, where W_n is the weighting function of apodized electrodes and H_n is the discrete fourier transform coefficient as noted in Eq. (2). The design model of the time domain is shown in Fig. 2. The design problem; "Determine W_n to minimize δ_2 (>0) under constraint that δ_1 is a constant specification value" is defined as LP in Eq. (6). The results of calculation are shown in Figs. 4〜6. and show the effectiveness of the design method. The design model of the frequency domain is shown in Fig. 7 and Fig. 10, the former is the design model with a narrow bandpass filter, and the latter is that with an ideal filter. The design problem; "Determine H_n to minimize δ_2 (>0)" are defined as LP in Eqs. (7) and (8). The results of calculation are shown in Fig. 8 and Fig. 9 for the former, and Fig. 11 for the latter. These results indicate that the designed filters are optimal inspite of their design simplicity.
In analysis of road traffic noise, the width of the road is usually neglected. The road span and distribution of vehicles on it, however, have a great influence on the sound fields especially in the vicinity of the road. To investigate these problems we statistically studied the noise fields emitted by acoustic point sources randomly distributed on a strip shown in Fig. 1. Mean, variance and the other cumulants of sound intensity are derived and expressed in Eqs. (2・7)〜(2・8) respectively. Attenuation characteristics of mean sound intensity with distance along various directions are shown in Fig. 2. These attenuation curves have an identical asymptotic straight line corresponding to the mean sound intensity around a random distribution of point sources on a straight line. Similar attenuation curves for standard deviation and coefficient of variation of sound intensity are also illustrated in Fig. 3 and 4 respectively. We also consider the time and space correlation of the sound intensity fields emitted by moving point sources on the strip. When the velocity of moving sources is equal and constant a simple expression for the time correlation of sound intensity is derived as in (3・6) and shown in Fig. 6. The time correlation at the fixed observation point is a monotonically decreasing function of the time difference τ and usually becomes less than 10% for τ≧3. Furthermore the correlation curves for ρ_0≧4 tend towards the limiting one labeled as ρ=∞ and expressed by Eq. (3・10) which corresponds to the equation derived from moving point sources on a straight line.
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.