In this article, an efficient algorithm implementation is presented for the synthesis of microwave circuits with arbitrary frequency responses. A comprehensive theory development is included starting from the formalism of the coupled-mode theory to obtain an exact series solution for the inverse scattering problem. A final algorithm in pseudocode is available in this article together with graphical diagrams to provide a straightforward implementation for the interested reader showing its feasibility for parallel computing. As an example, a design of a matched filter for broadband communications is presented along with the operational insight of the applied method, the influence of the auxiliary parameters, and the comparison between target frequency responses, simulations, and measurements. This application in microstrip technology is enclosed as an instance of the versatility of our synthesis technique and the accuracy of the presented algorithm.