The angular resolution of optical system is limited by the ratio of the wavelength to the aperture of the entrance pupil, indicating that the optical system with large aperture has a high spatial resolution. Sparse aperture imaging is one of the effective solutions to the problem that the telescope is bulky, heavy and difficult to manufacture. According to the self-similarity and multi-scale characteristics of fractal configuration, we propose a sparse aperture array and analyze its performance for synthetic aperture imaging system. In the array Golay-3 is used as a structural unit to expand a multi-layered fractal configuration in a self-similar manner. Given the analytical expression of the pupil function which is reduced by dimensionless parameters, we calculate the modulation transfer functions (MTFs), the practical cut-off frequencies and the middle spatial frequency characteristics of the fractal configuration under different fill factors and different outer layer rotational angles. We analyze both the MTF values and the performance parameters of the fractal structure for the cases of N= 3, 9, and 18, respectively. The results show that the decrease of fill factor does not significantly change the MTF curve nor the practical cutoff frequency in a range of fill factor between 0.0952 and 0.2246. The outer layer rotational angle has a periodicity, and the change in the angle has no large influence on the practical cutoff frequency. When the reduced aperture parameter is $ {d_0} = 1$ and the fill factor is 22.46%, the middle spatial frequency of N= 18 array is more stable and the practical cut-off frequency is higher. Using the fractal self-similarity, the aperture of the system can be expanded effectively while maintaining the middle spatial frequency characteristics. The computing results are of scale invariance due to the adoption of the reduced aperture parameter.