This paper presents an algorithm for the detection of wakeful state, rapid eye movement sleep (REM) and non-REM sleep based on the analysis of respiratory movements acquired through a bioradar. We used the data from 29 subjects without sleep-related breathing disorders who underwent a polysomnography study at a sleep laboratory. A leave-one-subject-out cross-validation procedure was used for testing the classification performance. Cohen's kappa of 0.56 ± 0.16 and accuracy of 75.13 ± 9.81 % were achieved when compared to polysomnography results. The results of our work contribute to the development of home sleep monitoring systems.