Systems of a catadioptric form generally have central obscurations that give rise to annular-type entrance pupils. These are found in most cases to be well approximated to the region contained within two ellipses which are generally of different eccentricities. The O.T.F. (optical transfer function) for an annular aperture at a given image point may be expressed as the combination of four integrals, one for each of the possible overlap areas which together describe the form of the overlap region between the two ellipses. Each integral is of a similar nature to the one giving the O.T.F. associated with an elliptical aperture. A computer programme has been written to calculate the O.T.F. using Hopkin's algorithm (1957). Because only one analytical example (aberration-free case) could be obtained for a comparison with this programme, O.T.F. measurements were carried out experimentally on a simple doublet having a central obscuration in order that a more substantial check could be made. The maximum error encountered in the programme was in the neighbourhood of 2 per cent. As a practical example, a study has been made of an objective described by Canzek (1971).