We design a sparse aperture of five sub-apertures surrounded by an annulus. We assume two models of linear and quadratic distributions for the sub-apertures. We calculated the lateral point spread functions (PSFs) in all cases and compared them with the case of uniform circular sub-apertures. We look forward to improving the PSF by adding an annulus surrounding the sparse aperture imaging systems (SAIS). We computed the point array autocorrelations (PAAs) corresponding to the two models, the absence of the surrounding annular aperture and the presence of the annulus. The first secondary peaks of the PSF at different radii of the sub-apertures, with a range from 16 to 64 pixels, are discussed. We computed the contrast of the speckle images using modulated sparse apertures in the presence of the annulus, and we compared it with the case of the annulus' absence.