Simplified design procedures for Butler matrices incorporating 90???? hybrids or 180???? hybrids

This paper gives detailed procedures for designing Butler matrices up to 128 ¿¿ 128 using 90¿¿ or 180¿¿ hybrids. A formula is given for calculating the number of fixed phase shifters in each row for matrices using 180¿¿ hybrids and the positions and magnitudes of these phase shifters are shown. The phase shifts obtained across the output ports are specified, together with the input ports that produce these phase shifts. Design procedures for larger Butler matrices are also indicated