Reproduction of a sound field is a fundamental problem in acoustic signal processing. In this paper, we use a spherical harmonics analysis to derive performance bounds on how well an array of loudspeakers can recreate a three-dimensional (3-D) plane-wave sound field within a spherical region of space. Specifically, we develop a relationship between the number of loudspeakers, the size of the reproduction sphere, the frequency range, and the desired accuracy. We also provide analogous results for the special case of reproduction of a two-dimensional (2-D) sound field. Results are verified through computer simulations.

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The theory and design of a broadband array of sensors with a frequency invariant far-eld beam pattern over an arbitrarily wide design bandwidth is presented. The frequency invariant beam pattern property i s de ned in terms of a continuously distributed sensor, and the problem of designing a practical sensor array is then treated as an approximation to this continuous sensor using a discrete set of ltered broadband omni-directional array elements. The design methodology is suitable for one, two, and three dimensional sensor arrays it imposes no restrictions on the desired aperture distribution (beam shape), and can cope with arbitrarily wide bandwidths. An important consequence of our results is that the frequency response of the lter applied to the output of each sensor can be factored into two components: one component i s related to a slice of the desired aperture distribution, and the other is sensor independent. The results also indicate that the locations of the sensors are not a crucial design consideration, although we show that nonuniform spacings simultaneously avoid spatial aliasing and minimize the number of sensors. An example design which covers a 10:1 frequency range (which is suitable for speech acquisition using a microphone array) illustrates the utility of our method. Finally, the theory is generalized to cover a parameterized class of arrays in which the frequency dependence of the beam pattern can be controlled in a continuous manner from a classical single frequency design to a frequency invariant design.