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The paper presents a new theory of coupled resonator bandpass microwave filters of arbitrary bandwidth. Constraints on the scattering parameters of lossless, time-invariant, and reciprocal two-port networks, expressed as rational functions of the physical frequency, are presented. Filtering functions exhibiting arbitrarily positioned transmission zeros and locally symmetric or asymmetric responses with respect to the center of the passband, and satisfying these constraints are then introduced. A direct synthesis technique to extract the elements of the transversal equivalent circuit to yield a prescribed response is presented. Response-preserving similarity transformations are then applied to the circuit to force a desired topology. As opposed to the prevailing narrowband approximation, any topology other than the transversal will have off-diagonal elements that depend on frequency. Interesting features that are not found in the narrowband approximation are reported. Examples demonstrating the soundness of the theory are presented.