Phase characteristics of scattering transfer functions of a sphere

It is important to investigate the sound field scattered by a scatterer in order to estimate: the transfer function around the head or to predict the variation of the indoor transfer function. R. Lyon conjectured that the transfer characteristics of the sound field scattered by a sphere have the minimum-phase property. This paper analyzes numerically the phase characteristics of the sound field scattered by a hard or a soft sphere in order to verify Lyon's conjecture. The following results are obtained. (1) The scattered wave in the scattering field produced by a hard sphere has the minimum-phase property in the forward direction from the sphere, but is a nonminimum phase signal in the backward direction. (2) In the backward direction from the sphere, the all-pass component decays as the wave progresses, approaching gradually the minimum-phase characteristics. (3) The wave scattered by a soft sphere exhibits the minimum-phase property wherever it is observed. The scattered wave from the sphere is approximated by an all-zero model, and the variation of the phase characteristics of the scattered wave is analyzed in terms of the locus of the zero. The scattered wave is composed of reflected and refracted waves. In the backward direction from the sphere, the refracted wave has a larger energy than the reflected wave, and the energy of the reflected wave gradually increases as one moves toward the forward direction of the sphere. By the presented analysis, it is seen that the zero on the imaginary axis of the complex frequency plane shifts from the nonminimum-phase to the minimum-phase position, as one moves from the backward to the forward direction of the sphere. It is verified that the scattered wave has the minimum-phase property, except for the low-frequency region; this is also demonstrated by measurement of waves scattered by a sphere. (C) 1999 Scripta Technica.