The idea behind the coupling matrix identification is to find the coupling matrix corresponding to the measured or designed scattering characteristics of the microwave filter. The typical attitude towards coupling matrix parameter extraction is to use some optimization methods to minimize the appropriate cost function. In this paper we concentrate on the analytic solutions - how they may be found and their application in further optimization processes. In general case the suggested method generates complex-valued coupling matrix. For a special case of the filter without cross-couplings we give fast and simple recursive method of finding such complex-valued coupling matrix. The method is based on Laplace's formula for expanding the determinant. The complex-valued coupling matrix is used as a good starting point for the optimization methods to find the regular coupling matrix. The examples are presented showing that the optimization arrives to global minimum starting from real parts of complex-valued entries considerably more often than when the starting point is selected randomly.
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