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We present a generalized synthesis procedure for designing low-profile frequency selective surfaces (FSS) with bandpass responses of odd-order (N = 3,5,7,...). The FSSs designed using this technique use a combination of resonant and non-resonant sub-wavelength constituting unit cells with unit cell dimensions and periodicities in the order of 0.15 λ0, where λ0 is the free space wavelength. The main advantage of using this technique, compared to traditional FSS design techniques, is that it allows for the design of low-profile and ultrathin FSSs that can provide sharp frequency selectivity. An Nth order FSS designed using this technique typically has an electrical thickness in the order of ~ (N - 1) λ0/50 which is significantly smaller than the overall thickness of a traditionally designed Nth order FSS (~ (N - 1) λ0/4). The proposed synthesis procedure is validated for two FSS prototypes having third- and fifth-order bandpass responses. Principles of operation, detailed synthesis procedure, and implementation guidelines for this type of FSS are presented and discussed in this communication.