Theory and design of broadband sensor arrays with frequency invariant far-field beam patterns

The theory and design of a broadband array of sensors with a frequency invariant far-field beam pattern over an arbitrarily wide design bandwidth is presented. The frequency invariant beam pattern property is defined 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 filtered broadband omnidirectional 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 the results is that the frequency response of the filter applied to the output of each sensor can be factored into two components: One component is 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 it is shown 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 the 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.