In this paper, we propose a novel synthesis strategy for the design of one dimensional electromagnetic bandgap (1-D-EBG) structures where all the performance parameters of these devices can fully be controlled, i.e., the central frequency of the forbidden band, its attenuation level and bandwidth, and the ripple level at the passbands. The novel synthesis strategy employs a new inverse-scattering technique to accurately synthesize the 1-D-EBG structure, targeting a properly interpolated version of a classical periodic filter fulfilling the required frequency specifications. The new inverse-scattering technique follows a continuous layer peeling approach and relies on the coupled-mode theory to precisely model the microwave structures. Telecommunication and radar systems, as well as material characterization devices, will be profited by this proposal with which enhanced filters, sensors, power dividers, couplers, mixers, oscillators, and amplifiers can be designed in many different technologies. As a proof of concept, a 1-D-EBG structure in microstrip technology with a single forbidden band (free of spurious stopband replicas), with attenuation level of 30 dB, fractional bandwidth larger than 100%, and return loss level at the passbands of 20 dB, has been designed and fabricated. The measurements obtained are in very good agreement with the simulations and target specifications, being free of spurious replicas up to the 15th harmonic, showing the robustness and very good performance of the novel design strategy proposed.