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A new full-wave electromagnetic analysis method based on the Boundary Integral Resonant Mode Expansion (BI-RME) is presented in this paper. The technique allows the computation of the broad band circuital response of arbitrarily shaped cavity filters. The coaxial excitation of this kind of filters is rigorously modeled by taking into account the high order modes on the coaxial aperture. A singular decomposition algebraic procedure has been implemented in order to obtain the solenoidal and non-solenoidal counterparts contributions of the electric current density inside the cavity needed by the BI-RME formulation. This method is simple and avoids some issues associated to star and loop based functions which are typically used for the same purpose.