4. Spherical Array Configurations

Motivated by the problem of spatial sampling of a sound field by a spherical array, Chap. 3 presented methods for sampling functions on a sphere, followed by methods for reconstructing a function from its samples. These could form the basis for computing the sound pressure on the surface of a sphere, given measurements by an array of microphones. However, in spherical microphone array processing one may also be interested in computing the sound field around the array by decomposing the sound field into plane-wave components, for example. In this case, placing pressure or omni-directional microphones on the surface of a single sphere in free-field may not allow accurate plane-wave decomposition, due to zeros of the spherical Bessel function. This problem is presented at the beginning of the chapter. One possible solution is to place microphones on the surface of a rigid sphere. This configuration offers a practical advantage—the rigid sphere provides an ideal housing for all microphone wiring and conditioning electronics. However, one drawback of the rigid sphere is that sound scattered from the sphere can be reflected back by surrounding objects, thereby modifying the sound field it measures. This is particularly important for arrays used for sound field analysis in room acoustics, for example, in which case placing microphones in a free field, in an open-sphere configuration, may be preferable. Open spherical array configurations that avoid the problem of the zeros of the spherical Bessel function are therefore presented next. The array configuration may also affect other aspects of array performance related to the frequency range of operation and to the sensitivity to sensor noise and to other uncertainties. A general framework for array design that considers a range of objectives is introduced, followed by example designs. The chapter concludes with a description of an open spherical array configuration in which the microphones are placed within the volume of a shell. Other array configurations, including the hemispherical array, another array comprised of concentric rigid and open spheres , and an array incorporating non-spherical sampling surfaces, are also discussed.