## 2 Method
### Asymptotic Performance of the Maximum Likelihood Estimation  
The periodic signal model (2) includes \(P_{s}+1\) unknown parameters, where \(P_{s}\) is unknown itself. Therefore, it constitutes an irregular estimation problem, whose maximum likelihood estimator is biased, and the classical Cramer-Rao bound does not exist [21]. To accurately describe the performance of the estimator, we established a regularized model similar to that of Equation (2) with a fixed number of unknown parameters:  
\[x[k]=s[k\mathrm{mod}P_{s}] \tag{4}\]  
where \(P_{s}\) is the unknown signal period, and the array \(\tilde{\mathrm{s}}=\left\lceil\tilde{s}[0],\tilde{s}[1],\ldots,\tilde{s}[\hat{ P}_{s}-1],0,..0\right\rceil\) including \(P_{s}\) samples is the waveform in one period, with \(\hat{P}_{s}<P_{s}\) being known. Although \(P_{s}\) is unknown, we assumed that the number of non-zero waveform samples within each period was known and the rest of the signal was 0. The number of observed periods \(\tilde{N}\) was also considered a known constant integer, while it was ignored that an incomplete period might exist, and the total number of samples was \(K=\tilde{N}P_{s}\). There are \(\hat{P}_{s}+1\) unknown parameters in total in Equation (4), and the corresponding Cramer-Rao bound of Equation (4) in white Gaussian noise can be derived.  
Figure 1: Flowchart of heartbeat rate estimation. The raw radar signal with strong respiratory components is high-pass filtered, and only the high-order heartbeat harmonics are preserved for the estimation. The heart rate can be obtained by maximum likelihood estimation, even if the SNR signal is low.  
which can be reorganized with  
\[\sigma^{2}\Bigg{\{}\sum_{p=0}^{P_{s}-1}\bigg{(}\frac{\partial\tilde{s}(t)}{ \partial t}|_{t=p\Delta}\bigg{)}^{2}\Bigg{\}}^{-1}=\sigma^{2}\Bigg{(}\frac{N_{0 }/2}{\varepsilon{F}^{2}}\Bigg{)}. \tag{10}\]  
where \(N_{0}\) is the noise power spectrum density, \(\varepsilon\) is the energy of \(\tilde{\mathrm{s}}\), and \(F^{2}\) is a measurement of the bandwidth of \(\tilde{\mathrm{s}}\), defined as  
\[F^{2}=\frac{\int_{-\infty}^{\infty}{(2\pi f)^{2}\big{|}\tilde{S}(f)\big{|}^{2} df}}{\int_{-\infty}^{\infty}{\big{|}\tilde{S}(f)\big{|}^{2}df}}, \tag{11}\]  
where \(\tilde{S}(f)\) is the Fourier transform of \(\tilde{\mathrm{s}}\)[23]. Equation (10) is exactly the Cramer-Rao bound of the signal delay estimation with the known waveform \(\tilde{\mathrm{s}}\)[24].  
The estimated variance of the maximum likelihood estimator of the model (4) should be asymptotic to the Cramer-Rao bound in Equation (9) [25], and the asymptotic variance of the estimator in Equation (2) can be approximated as  
\[\sigma^{2}\frac{12}{N^{3}-N}\Bigg{\{}\sum_{p=0}^{P_{s}-1}\bigg{(}\frac{ \partial s(t)}{\partial t}|_{t=p\Delta}\bigg{)}^{2}\Bigg{\}}^{-1}=\sigma^{2} \bigg{(}\frac{12}{N^{3}-N}\bigg{)}\frac{N_{0}/2}{\varepsilon\frac{\int_{- \infty}^{\infty}{(2\pi f)^{2}\big{|}\tilde{S}(f)\big{|}^{2}df}}{\int_{-\infty }^{\infty}{\big{|}\tilde{S}(f)\big{|}^{2}df}}} \tag{12}\]  
where \(N=\lfloor K/P_{s}\rfloor\), and \(S(f)\) is the Fourier transform of \(\mathrm{s}\). This can be easily extended to the case of complex signal.