In this paper, random switch antenna array (RSAA) is proposed to apply to the switch antenna array (SAA) frequency-modulated continuous-wave (FMCW) radar system. Firstly, the signal model and signal processing method of RSAA is analyzed and it shows that RSAA can successfully solve azimuth-velocity coupling problem. Then we suppose a method by using RSAA to reduce the switching frequency and sampling rate based on sparse signal representation with multiple measurement vectors (MMV). It is shown in simulations that the proposed algorithms yield better performance in terms of azimuth-velocity decoupling and can obtain high image accuracy with less observation data.