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This paper describes and validates an advanced synthesis design process of Frequency-Selective Surfaces (FSSs) with elliptic band-stop responses. A systematic procedure based on the Generalized Chebyshev Function and the extracted pole technique enables control of the position of the transmission zeros and the attenuation level to obtain an equiripple rejection response. A systematic process is followed to obtain the lumped LC values of the resonator circuits extracted as poles and the impedance inverters. Then, equivalent dipoles and transmission lines are obtained to carry out the electromagnetic design at normal incidence for a linearly polarized field. The impact of the higher-order modes of the periodic structure on the electrical response of the FSS, which can be relevant due to the stringent selected specifications, has been also analyzed. A fourth-order band-stop filter with a 3 GHz bandwidth centered at 30 GHz and its attenuation at 50 dB has been designed considering three different implementations: two filters using a vacuum as a transmission line with different connection lengths and a third one using a dielectric substrate to enable its manufacturing. In order to verify the design procedure using experimental results, the third filter with printed dipoles in the dielectric substrate has been manufactured and measured, thus validating the developed process. [ABSTRACT FROM AUTHOR]