A systematic design formulation for Butler matrix applied FFT algorithm

A systematic general design formulation for a Butler matrix (
B
matrix) is described. The
B
matrix design problem discussed is used to determine phase shift location and value in a matrix, when the number of beam (elements of array)
M = 2^{N}
and the scattering matrix for the hybrid couplers are specified. The design formulation presented is based on the fact that a
B
matrix design procedure and an FFT algorithm are equivalent in fundamental concepts. It is shown that the
B
matrix design procedure can be systematically formulated by the FFT algorithm modifications, which preserves the topological properties of the original signal flow diagram. A simple design formula has been established by this formulation.