Fundamentals of Spherical Array Processing

This chapter provides the mathematical background necessary for studying spherical array processing. Spherical arrays typically sample functions on a sphere (e.g. sound pressure); therefore, this chapter begins by presenting the spherical coordinate system, as well as some examples of functions on the sphere. Spherical harmonics are a central theme of this book as they form a basis for representing functions on the sphere. Therefore, spherical harmonics are first defined and illustrated, and then an introduction to the spherical Fourier transform and a description of functions on the sphere in Hilbert space follows. The chapter concludes with a presentation of the topics of rotation, convolution, and correlation defined for functions on the sphere.

The mathematical background for functions defined on the unit sphere was presented in Chap. 1. Spherical harmonics played an important role in presenting and manipulating these functions. In this chapter, functions on the sphere are defined through the formulations of fields in three dimensions. Although sound fields are of primary concern in this book, which is oriented towards microphone arrays, the material presented here can be applied to scalar fields in general. This chapter begins by presenting the acoustic wave equation in Cartesian and spherical coordinates, with possible solutions. Solutions to the wave equation in spherical coordinates are shown to involve spherical harmonics and spherical Bessel and Hankel functions. Having formulated the fundamental solutions, sound fields due to a plane wave and a point source are presented, including an analysis of the effect of a rigid sphere introduced into the sound field. The latter is useful for describing the sound field around a microphone array configured over a rigid sphere, for example. The chapter concludes with a formulation of the three-dimensional translation of sound fields.

Spherical microphone arrays are realized by placing microphones in three-dimensional space and recording the signals at the microphone locations. When the microphones are placed on the surface of a sphere, they sample the sound pressure at the sphere surface. Estimation of the sound pressure function on the measurement sphere may depend on the sampling configuration and, therefore, methods for sampling functions on the sphere, such as equal-angle sampling, Gaussian sampling, and uniform sampling, are presented in this chapter. An important feature of the sampling methods is their capacity to facilitate computation of the spherical Fourier transform of the function on the sphere in the case of order-limited functions. When this capacity is not fully achieved, sampling errors occur and the function cannot be reconstructed from its samples. The sampling methods mentioned above have closed-form expressions for computing the spherical harmonic coefficients from the samples, using a summation rather than integration. Computation of the spherical harmonic coefficients can also be realized for arbitrary sampling configurations, using an inversion of the sampled spherical harmonics matrix, as detailed in this chapter. The methods presented here will provide the basis for selecting microphone locations in the process of spherical microphone array design.

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 yield 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.

Chapter 4 presented various ways to configure a spherical microphone array and discussed the advantages of each configuration. Once microphones are positioned in space in a desired configuration, e.g. on the surface of a rigid sphere, they can be connected to conditioning equipment, and the signal at each microphone can be recorded. In this chapter, the signals at the microphones are defined as the inputs to an array processor, producing a single processed output with some desired characteristics. One possible desired characteristic is to enhance signals from a sound source that is located in a specific direction and to attenuate signals from sources located in other directions, therefore forming a spatial, or directional filter. Such a filter is called a beamformer, because the beam it forms looks at a desired direction, and is probably the simplest form of array processing. The first section of this chapter presents array equations, with array input, spatial filter, and array output formulated in the space domain. This is followed by the derivation of the same equations in the spherical harmonics domain, where the benefits of processing in this domain are emphasized. Two important measures of array performance, namely directivity index and white noise gain (WNG), are presented in the following sections. These are derived both in the space and in the spherical harmonics domains. A simplified beamforming structure that produces axis-symmetric beam patterns and that decouples the shaping and the steering of a beam pattern is also introduced. The chapter continues with a presentation of two common beamformers, namely delay-and-sum and plane-wave decomposition. Finally, steering of non axis-symmetric beamformers is presented, and the chapter concludes with a beamforming example.

Beamforming with spherical microphone arrays was presented in Chap. 5 as an instrument to achieve directional filtering, characterized by the beam pattern of the array. It may be desired to control the beam pattern in a more explicit manner to achieve specific properties. For example, beamformers that achieve maximum directivity index may be useful to enhance a desired plane wave relative to undesired plane waves arriving from the entire range of directions. Beamformers that achieve maximum white noise gain (WNG) may be desired if robustness to system uncertainty is important. We may also be interested in enhancing a desired plane wave while guaranteeing a specific reduction level for undesired plane waves in other directions. This can be achieved by restricting the side-lobe level in the beam pattern using the Dolph-Chebyshev design. Design objectives can also be combined into a single objective, or integrated into a more complex constrained optimization formulation. In summary, this chapter presents methods for beam pattern design formulated explicitly for spherical arrays, with the aim of providing tools for matching the properties of the array to specific performance aspects.

Optimal beamformer design, as presented in Chap. 6, may be very useful, but does not take into account the properties of the specific sound field producing the signals at the microphones. In this chapter, beamforming in which the beam pattern is tailored to the actual sound field is presented. This beamforming distinguishes between the desired signal and the noise and, therefore, potentially achieves improved performance in real, noisy sound fields. The measured sound field is characterized by spatial cross-spectrum matrices, typically divided into matrices representing the desired signals and matrices representing the unwanted noise. Therefore, the first part of this chapter extends the array equations, in both the space and the spherical harmonics domains (as presented in Chap. 5) to include noise. In particular, explicit expressions are developed for designs that consider noise fields that are spatially white and noise fields that are acoustically diffuse. The second part of the chapter employs the new models in the development of popular beamformers, such as the minimum variance distortionless response (MVDR) and the linearly constrained minimum variance (LCMV). These beamformers are developed for spherical arrays with explicit formulations in the spherical harmonics domain, emphasizing their advantages when formulated in this domain. The chapter concludes with design examples to illustrate the performance of the beamformers under various conditions.