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 order
M - 1
-by-
N - 1
for rectangular arrays with
M
by
N
uniformly spaced elements (
M \geq N
). Previously a sampling method has been used to exactly synthesize response functions of a single variable by
N
element line arrays [1] and nonseparable functions of two variables by
N \times N
(square) arrays. The sampling method is extended herein to apply to rectangular arrays by deriving the element weights for an
M
by
N
rectangular array that exactly yield the values of any desired real symmetrical array response function on an
M
by
N
grid of the interelement phase shifts. This would yield the designed array response exactly everywhere if the design equation were of order
M - 1
with respect to one dimension and of order
N - 1
with respect to the other dimension. To approximate this condition, the original design function of order
N - 1
in both dimensions is modified to make it of order
M - 1
in one dimension and of order
N - 1
along 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 the
N
grid lines of
M
points each and along the single perpendicular chosen grid line of
N
points, 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 of
M
and
N
. 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.