A Generalized FFT for Many Simultaneous Receive Beams

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 vary little -- and the beams in direction-cosine space are laid out on an arbitrary superlattice of the dual of the element-layout lattice. The generality of these two arbitrary lattice can yield significant cost reductions for large, many-beam arrays and arises from, first, formulating the desired beam outputs using a discrete Fourier transform or DFT generalized to use an integer-matrix size parameter, and second, efficiently realizing the required real-time 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 FFTs in radix-2 and mixed-radix 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 beam-layout geometry, grating-lobe behavior, and FFT factorability and therefore computational efficiency.