A technique for decomposing the Optical Transfer Function (OTF) into a novel set of basis functions has been developed. The decomposition provides insight into the performance of optical systems containing both wavefront error and apodization, as well as the interactions between the various components of the pupil function. Previously, this technique has been applied to systems with circular pupils with both uniform illumination and Gaussian apodization. Here, systems with annular pupils are explored. In cases of annular pupil with simple defocus, analytic expressions for the OTF decomposition coefficients can be calculated. The annular case is not only applicable to optical systems with central obscurations, but the technique can be extended to systems with multiple ring structures. The ring structures can have constant area as is often found in zone plates and diffractive lenses or the rings can have arbitrary areas. Analytic expressions for the OTF decomposition coefficients again can be determined for ring structures with constant and quadratic phase variations. The OTF decomposition provides a general tool to analyze and compare a diverse set of optical systems.