General Optimized Design of the E-Plane Waveguide Butler Matrix With Non-2 <i> ⁿ </i> Beams Based on the FFT

The Butler matrix is a well-known beamforming network (BFN) for antenna arrays that can achieve multiorthogonal beams with the number of ports typically limited to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{n}$ </tex-math></inline-formula>. Current methods to achieve non-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{n}$ </tex-math></inline-formula> ports are mainly based on additional power distribution networks or modifications of the directional coupler, resulting in larger sizes and higher insertion loss. This article presents a generalized design procedure of the Butler matrix with arbitrary positive integer beams, including arbitrary non-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{n}$ </tex-math></inline-formula> beams, based on the fast Fourier transform (FFT) algorithm. By making the transmission matrix of the Butler matrix equal to the target matrix, the parameters of each device in the network structure can be determined. Thanks to the feature of the FFT, the proposed Butler matrix structure is compact, and the design procedure is general. The design procedure of Butler matrices for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N = 5$ </tex-math></inline-formula>, 6, and 7 is presented as examples. To show the effectiveness of the proposed procedure, a prototype of the Butler matrix operating at the Ku-band is designed and fabricated to feed an array. The E-plane waveguide feed network is used to achieve low insertion loss and small size. The measured S-parameters are better than −10 dB, the insertion loss is less than 1.2 dB, the phase imbalance is ±8.3°, and the steering angles are 0°, ±22°, and ±46°, respectively. The measured results are in good agreement with the simulated results.