On optimal shading for arrays of irregularly-spaced or noncoplanar elements

The majority of optimal shading methods for arrays of irregularly spaced or noncoplanar elements rely on numerical optimizations and iterative techniques to compute the desired weighting function because analytic solutions generally do not exist. Optimality is meant here in the Dolph-Chebyshev sense to provide the narrowest mainlobe width for a given sidelobe level. We present a simple and efficient technique to compute real shading coefficients for nonuniform-line, curved-line, and noncoplanar arrays by resampling the optimal Dolph-Chebyshev window computed for a uniform line or plane array of equivalent aperture at the element position of the irregular array. Computer simulation examples of narrowband plane-wave beamforming with irregular arrays, in which phase compensation is achieved by projecting the elements on a line or plane tangent to the array, show peak sidelobe levels close to those obtainable for optimally shaded uniform arrays of equal aperture sizes and numbers of elements, where the differences depend upon the spacing variations and numbers of elements. This resampling technique is applied to seafloor acoustic backscatter data collected at sea with the 68-kHz Toroidal Volume Search Sonar to highlight a tradeoff between peak and outer sidelobe levels and illustrate the requirement for element pattern when processing data from irregular arrays.