We develop equations describing the capability of arbitrary three-dimensional arrays to resolve and detect two narrowband sources using conventional (CONV) beamforming, minimum energy (ME) adaptive beamforming, and linear predictive (LP) processing. In addition to allowing one to evaluate the performance of each algorithm with arbitrary array geometries, these equations contain geometry-dependent terms that allow us to isolate analytically the effect of array geometry on the ideal resolution and detection capabilities of each algorithm. We show that the sensitivity of each processing algorithm to finite amounts of data depends on the number of sensors in an array but is independent of array geometry: consequently this sensitivity should not influence choice of array geometry. Based on the resolution and detection equations, we develop analytically a class of linear arrays which, for a fixed number of sensors, offer simultaneously optimum resolution capability and sidelobe suppression when either CONV or ME beamforming is employed. Signal processing considerations indicate that optimum CONV and ME performance is obtained when an array samples the spatial correlation function at as many consecutive multiples of a half-wavelength as possible. Minimum redundancy and minimum hole arrays represent two means of achieving such spatial sampling.
CONV, ME, and LP yield asymptotically biased estimates of source bearing when multiple sources are present. Through Taylor series analysis of the array factors of arbitrary linear arrays, we develop approximations for the asymptotic bias inherent in the estimates of source bearing produced by each algorithm.
It is possible to extend the array optimization problem to the design of two-dimensional arrays; unfortunately, the 2-D problem is intractable, and remains unsolved theoretically. However, we do establish the importance of the 2-D coarray in determining 2-D array performance. Consideration of the coarray leads to the conclusion that arrays with three-fold rotational symmetry and elements lying on an isometric grid having a half-wavelength cell size are probably nearly optimum. Further, equilateral Y configurations of linear arrays exhibit coarrays with six-fold rotational symmetry, thus providing good omni-directional capability. Use of linear minimum hole arrays along each leg of the Y results in extremely good resolution capability and sidelobe suppression in conjunction with ME beamforming.