A new method of signal processing by smart antennas is proposed and justified. It allows to improve the accuracy of angle measurement and to restore the image of the object with superresolution. The method is based on the extrapolation of the signals received by each element of the antenna array, outside the aperture. This allows introducing new virtual elements and thus synthesizing significantly larger antenna array. The method is tested in numerical experiments using a mathematical model and the maximum effective angular resolution is found for different cases and objects. Algorithms based on the method of digital aperture synthesis provide angular superresolution 3-7 times greater than that according to the Rayleigh criterion for a signal/noise ratio of 12-13 dB.