In this paper, a novel quasi-periodic structure for rectangular waveguide technology is proposed. The constituent unit cells of the structure feature a resonant behavior, providing high attenuation levels in the stopband with a compact (small period) size. By applying a smooth taper-like variation to the height of the periodic structure, very good matching is achieved in the passband while the bandwidth of the stopband is strongly increased. Moreover, by smoothly tapering the width of the structure, a band-pass frequency behavior is obtained. In order to demonstrate the capabilities of the novel quasi-periodic structure proposed, a band-pass structure with good matching, wide rejected band, and high-power handling capability has been designed, fabricated, and measured obtaining very good results.