Unification of various type expressions for probability distribution of arbitrary random noise and vibration waves based on their actual fluctuation ranges. Theory and experiment

In the measurement of actual random phenomena, the observed data often result in a loss or a distortion of information due to the existence of a definite dynamic range of measurement equipments. In this paper, a unified expression of the fluctuation probability distribution for an environmental noise or vibration wave is proposed in an actuc:il case when this wave has a finite range of amplitude fluctuation in itself or is measured through the usual instruments (e.g., sound level meter, level recorde, etc.) with a finite dynamic range. The resultant expression of the probability distribution function has been derived in a form of the statistical Jacobi series type expansion taking a Beta distribution as the Ist expansion term and Jacobi polynomial as the orthogonal polynomial. This unified probability expression contains the well-known statistical Gegenbauer series type probability expansion as a special case, and the statistical Laguerre and Hermite series type probability expansions as two special limiting cases. Finally, the validity of the proposed theory has been experimentaly confirmed by applying to the actually observed data of a road traffic noise. This statistical Jacobi series expression shows good agreement with experimentally sampled points as compared with other types of statistical series expression.