Approximate synthesis of nonseparable design responses for rectangular arrays

A method is presented for the approximation of any real, symmetrical, nonseparable array design response polynomial of orderM - 1-by-N - 1for rectangular arrays withMbyNuniformly spaced elements (M ≥ N). Previously a sampling method has been used to exactly synthesize response functions of a single variable byNelement line arrays [1] and nonseparable functions of two variables byN × N(square) arrays. The sampling method is extended herein to apply to rectangular arrays by deriving the element weights for anMbyNrectangular array that exactly yield the values of any desired real symmetrical array response function on anMbyNgrid of the interelement phase shifts. This would yield the designed array response exactly everywhere if the design equation were of orderM - 1with respect to one dimension and of orderN - 1with respect to the other dimension. To approximate this condition, the original design function of orderN - 1in both dimensions is modified to make it of orderM - 1in one dimension and of orderN - 1along a single grid line in the other dimension. (A second function is derived with these roles reversed.) The resulting function has the form of the original design function, exactly equaling it along theNgrid lines ofMpoints each and along the single perpendicular chosen grid line ofNpoints, and closely approximating it throughout the entire range of the variables. This technique is illustrated for Chebyshev response functions, for which the two approximating functions are derived and evaluated for several combinations ofMandN. The responses are shown to have generally uniform sidelobe regions, with all but a small percentage of the sidelobes within 1 dB of the designed sidelobe value.