###### Abstract
Mode-matching formulations have been used in [29; 41] to investigate the dispersion characteristics of glide-symmetric periodic structures. Conversely, these methodologies are inadequate for elucidating the physical comprehension of the impact of glide symmetry on the periodic structure. In frequency domain solvers, network parameters (scattering (S)-parameters, ABCD parameters) of the unit cell of the periodic structure are determined using multimodal excitation, and the dispersion diagram of the periodic structure is derived by substituting the obtained parameters into the eigenvalue equation [4; 5; 45]. Despite the complex geometry of the periodic structure, it is feasible to precisely derive the dispersion diagram by analyzing the behavior of the eigenvalues using these methods. Nevertheless, full-wave simulators result in excessively prolonged calculation durations. Consequently, investigating and proposing the bandgap characteristics of glide-symmetric parallel plate waveguides using simple and effective computational techniques can drastically reduce design time. An equivalent circuit model has been recently introduced for the analysis of CPPWs featuring glide-symmetric corrugations [36]. This equivalent circuit model allows for the assumption that in the regions excited by discontinuities, only the dominant mode is propagated inside a periodic structure among an infinite set of higher-order modes [46]. This circuit model is a rapid and effective method that calculates the ABCD parameters of the unit cell and produces dispersion results that are comparable to those of full-wave simulators for a broad spectrum of geometrical parameters [36]. By deriving the circuit model of this structure [36] and obtaining the ABCD or S-parameters from the circuit model, the dispersion diagram of the model can only be obtained by solving the eigenvalue equations specified in [5; 36]. Moreover, the auxiliary functions of generalized scattering matrix (AFGSM) method [5] is an extremely successful alternative technique that identifies passband and stopband regions of periodic structures by analyzing the zero crossings of auxiliary functions. This method, explained in detail in [5], does not need to solve the eigenvalue equation (EE). A significant reduction in computation load can be achieved in this way for controlling bandgaps of periodic structures. It has been previously applied in numerous applications, including rectangular waveguides [5; 14], photonic crystals [47], and helix slow-wave structures [14; 48], to determine the passband/stopband regions of periodic structures and unit cell designs. To the best of our knowledge, the AFGSM method has not been applied to the analysis, design, and investigation of CPPW structures with glide-symmetric corrugations. This study proposes a novel design procedure for one-dimensional glide-symmetric double CPPW structures based on the AFGSM method in the open literature. Firstly, we performed analyses to test the effectiveness of the AFGSM method in determining the passband/stopband regions in the considered structures. We compared the results of each analysis with the eigenvalue equation. The next stage involved selecting design requirements and obtaining unit cell designs through dimensional synthesis using the proposed method. This method can yield numerous unit cell dimensions that align with the same design requirements. We evaluated the filtering performances of final designs using finite periodic implementations to understand whether optimal unit cell dimensions determined by the AFGSM method meet the design requirements. The full-wave electromagnetic simulators (HFSS and CST Microwave Studio) were used for testing the filtering performances of unit cells designed with the AFGSM method. We compared the performance of filters designed using the proposed design procedure with the reported literature.