 TITLE:

A Generalized FFT for Many Simultaneous Receive Beams
 AUTHORS:
 J. O. Coleman
 ABSTRACT:

It is well known that when the identical elements of a
planar receive array are laid out in horizontal rows
and vertical columns, a fast Fourier transform or FFT
can be used to efficiently realize simultaneous beams
laid out in rows and columns in the direction cosines
associated with the azimuth and elevation directions.
Here a more general formulation and an associated
design discipline is developed. Identical elements
are laid out on an arbitrary planar lattice  it could
be square, rectangular, diamond, or triangular and
might display tremendous symmetry or very little and
the beams in directioncosine space are laid out on an
arbitrary superlattice of the dual of the
elementlayout lattice.
The generality of these two arbitrary lattices can
yield significant cost reductions for large, manybeam
arrays and arises from, first, formulating the desired
beam outputs using a discrete Fourier transform or DFT
generalized to use an integermatrix size parameter
and, second, efficiently realizing the required
realtime computations with the generalized FFT based
on a matrix factorization of that size parameter that
is developed here. This generalized FFT includes as
special cases the usual 1D and 2D FFT's in radix2
and mixedradix forms but offers many more
possibilities as well. The approach cannot outperform
but does match, when the matrix size parameter factors
well, the N log N computational efficiency of the
usual FFT.
Examples illustrate a design discipline for the two
lattices that involves jointly determining element
spacing, steering range and beamlayout geometry,
gratinglobe behavior, and FFT factorability and
therefore computational efficiency.
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 STATUS:
 Published June 29, 2007 as NRL Memo Report 9029.
Approved for public release; distribution is unlimited.
 DATE OF ENTRY:

September 2007.