To date, ultra-wideband (UWB) radar is one of the leading technologies applied in the field of non-contact vital sign monitoring. A number of studies have focused on processing reflected UWB pulse signals into breathing and heart activities; however, most have emphasised the use of stationary subjects in their data collection processes. Therefore, this paper presents a feasible study conducted to extract the human vital signs of a non-stationary subject during sleep and the optimum position of the UWB radar. The proposed algorithm could measure the respiration rate (RR) and heart rate (HR) recorded regardless of any random body movements during sleep. An analysis of the entire slow time region for the signal was performed to remove random movement signals from the subject by implementing a sinusoidal fitting algorithm to monitor the periodic movement of the chest wall. Next, the value of R-squared was used to find the fit between the algorithm and output signals, and then, the signal was transformed into a frequency domain via Fourier transform (FT). This allowed the determination of the dominant peak from the breathing and heart rates before changing to the rate per minute. An experiment was also done to monitor three different UWB radar positions (i.e. top, side, and bottom of the bed) to identify its optimum location during sleep. All results were then compared with the polysomnography signal. The result found that the top position has the lowest error rate percentage for both RR and HR with only 0.72% and 3.71%, respectively.