FMCW radar systems enable remote vital sign and range detection but suffer from proximity clutter issues. When a large clutter exists around the target of interest, capture effect occurs and the reflected waveform from clutter would dominate the phase of the baseband signal, destroying the vital sign detection. This paper presents a novel and clutter-resistant demodulation to extract slow-time signal under strong reflected interference signals of a stationary or non-stationary clutter at close range. By applying amplitude-based demodulation, signals of the stationary clutter can be distinguished and excluded. Therefore, vital sign can be clearly determined even under the condition of strong close reflected interferences. Furthermore, extracting heartbeat clearly from FMCW under interference of clutter is relatively difficult using traditional complex signal demodulation algorithms. This paper proposes a novel EEMD-PCA-Correlation algorithm to solve this problem. From the experiment results, the average error of heartbeat with a clutter within 1 cm is 0.537%.