Ultra wideband (UWB) Doppler radar has many biomedical applications, including remote diagnosis of cardiovascular disease, triage and real-time personnel tracking in rescue missions. It uses narrow pulses to probe the human body and detect tiny cardiopulmonary movements by spectral analysis of the backscattered electromagnetic (EM) field. With the help of super-resolution spectral algorithms, UWB radar is capable of increased accuracy for estimating vital signs such as heart and respiration rates in adverse signal-to-noise conditions. A major challenge for biomedical radar systems is detecting the heartbeat of a subject with high accuracy, because of minute thorax motion (less than 0.5 mm) caused by the heartbeat. The problem becomes compounded by EM clutter and noise in the environment. In this paper, we introduce a new algorithm based on the state space method (SSM) for the extraction of cardiac and respiration rates from UWB radar measurements. SSM produces range-dependent system poles that can be classified parametrically with spectral peaks at the cardiac and respiratory frequencies. It is shown that SSM produces accurate estimates of the vital signs without producing harmonics and inter-modulation products that plague signal resolution in widely used FFT spectrograms.