Heart rate (HR) variability indicates health condition and mental stress. The development of non-contact HR monitoring techniques with Doppler radar is attracting great attention. However, the performance of heartbeat detection via radar signals easily degrades due to respiration and body motion. In this paper, first, a stochastic gradient approach is applied to reconstruct the high-resolution spectrum of heartbeat by proposing the zero-attracting sign least-mean-square (ZA-SLMS) algorithm. To correct the quantized gradient of cost function and penalize the sparse constraint on updating the spectrum, a more accurate heartbeat spectrum is reconstructed. Then, to better adapt to the noises of different strengths caused by subjects' movements, an adaptive regularization parameter is introduced in the ZA-SLMS algorithm as an improved variant, which can adaptively regulate the proportion between gradient correction and sparse penalty. Moreover, in view of the stability of the location of the spectral peak associated with the HR when the size of time window slightly changes, a time-window-variation (TWV) technique is further incorporated in the improved ZA-SLMS (IZA-SLMS) algorithm for more stable HR estimation. Through the experiments on five subjects, our proposal is demonstrated to bring a significant improvement in accuracy compared with existing detection methods. Specifically, the IZA-SLMS algorithm with TWV achieves the smallest average error of 3.79 beats per minute when subjects type on a laptop.