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In this paper, we present an exact analytical solution for the inverse scattering problem formulated by using the coupled-mode theory in the microwave range, assuming single-mode operation. The solution is a series that gives the required coupling coefficient for the microwave device as a function of the target frequency response. A clear physical meaning for the terms of the series is discussed, and several useful properties for the synthesis are given, thus providing a deep understanding of the synthesis mechanics. The method allows for the design of microwave devices with arbitrary frequency response only constrained by principles of causality, passivity, and stability. The resulting device is, in general, a nonuniform waveguide that features a continuously varying profile, e.g., a nonuniform microstrip line with continuously varying strip width. As an example of the potential of the synthesis method, a matched filter for ultra-wideband applications is finally designed in microstrip technology and successfully tested.