A systematic design formulation for Butler matrix applied FFT algorithm

A systematic general design formulation for a Butler matrix (Bmatrix) is described. TheBmatrix 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 aBmatrix design procedure and an FFT algorithm are equivalent in fundamental concepts. It is shown that theBmatrix 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.