A Study Of Non-2~n Butler Matrices

With the rapid development of modern communication technology,multi-beam antenna has attracted more and more attention due to its wide coverage and high communication rate.As a multi-beam forming network,Butler matrix is widely used in the design of multi-beam array antenna due to its advantages of simple structure and low cost.In recent years,the researches on Butler matrix are mostly based on the traditional even-order matrices,such as4×4,8×8,etc,which can only form beams with normal symmetry and have some limitations in application.In this paper,the design method of non-2ⁿ Butler matrix is studied in detail based on non-2ⁿ point FFT and the theory of matrix decomposition.The specific work is as follows:(1)Based on FFT(Fast Fourier Transform),a generalized method for designing non-2ⁿButler matrix structure is proposed.Firstly,the structure of Butler matrix is obtained according to the signal flow diagram of FFT,and then the transmission matrix of Butler matrix can be obtained.Then the target matrix of Butler matrix is derived from the antenna array pattern formula.The amplitude and phase relationship of the coupler in the Butler matrix are calculated by equalizing the transmission matrix and the target matrix.Finally,the final Butler matrix structure can be obtained by designing the coupler,crossover and phase shifter according to the amplitude and phase relation.(2)Based on the matrix decomposition,another method for designing non-2ⁿ Butler matrix structure is presented.Firstly,the transmission matrix T of Butler matrix,which is unitary matrix,is given according to the existing theory.T can be decomposed into T=YQY.By solving the matrix Y and Q,the transmission matrix T can be obtained.According to the form of matrix Y,it can be seen that Y is composed of several couplers with a coupling of 3d B,while the Q matrix is a block-diagonal matrix composed of the matrices Q₁ and Q₂.Therefore,the required Butler matrix structure can be obtained by decomposition of Q₁ and Q₂.The decomposition results show that the structure of Butler matrix is consistent with that obtained by FFT.(3)A 7×7 Butler matrix with 5GHz microstrip is designed.The network structure can be derived from FFT and matrix decomposition.And the two methods can be verified by each other.According to the deduced Butler matrix structure,the simulation design was carried out in Ansoft HFSS software to realize the equal-amplitude output signal of 7 output ports,and the phase difference was-154°?-102°?-51°?0°?+51°?+102°?+154°,respectively.To verify the performance of the Butler matrix,a 1×7 microstrip patch antenna array was designed.And the Butler matrix is used to feed the antenna array to obtain 7 beams with different pointing directions:0.6°?18°?-15.6°?34.8°?-31.8°?55.5°?-55.5°.The Butler matrix is processed and tested,and the test results show that its performance is consistent with the simulation results.