Realizing highly accurate and noncontact heart rate estimation with frequency modulated continuous wave (FMCW) radar is a big challenge under the interference of background noise and respiration harmonics. In this paper, various methods are employed to eliminate the interference, including impulse noise removal, improved complete ensemble empirical mode decomposition with adaptive noise (ICEEMDAN) algorithm, peak-to-valley amplitude difference processing and peak-to-peak time interval processing. A novel heart rate estimation scheme that can efficiently suppress noise, interference and respiration signal for vital sign detection is proposed. After preprocessing the radar raw data, the scheme first removes the impulse noise of the vital signal. Then, the ICEEMDAN algorithm is used for further denoising, and the appropriate component is selected from the decomposition results to reconstruct the heartbeat signal. The heart rate is estimated in time domain and frequency domain, respectively. In the time domain, peak-to-valley amplitude difference and peak-to-peak time interval are used to eliminate noise and interference. In the frequency domain, fast Fourier transform (FFT) and Rife algorithms are applied to improve the estimation accuracy of the heart rate. Finally, the estimated data in the time and frequency domains are fused as the estimated heart rate of the scheme. Extensive experiments reveal that, compared with other methods, the root mean square error (RMSE) and mean absolute percentage error (MAPE) are greatly improved and the estimation accuracy of the heart rate is significantly enhanced by using the proposed scheme.