## 4 Detection Performance and Discussion
### Detection Performance  
The improvement in SNCR using the proposed method is now illustrated using the datasets obtained using a human subject at distances of 300 cm to 1100 cm from the antenna in through-wall conditions. The well-known CFAR method is used as a reference. The Cramer-Rao low bound (CRLB) of an unbiased estimate of the variance of the respiration frequency is given by [37]  
\[\text{var}(\hat{f}_{r})\geq\frac{1}{\hat{f}(f_{r})}\approx\frac{3v^{2}}{2\pi^{ 4}A^{2}(\chi T_{s})\mu^{2}SNR}, \tag{31}\]  
where \(\mu=400\) MHz is the effective bandwidth, \(A=8\) mm is the respiration signal amplitude, and \(\chi\) is the number of pulses. This shows that \(\text{var}(\hat{f})\) increases as the SNR decreases, so the detection performance can be improved by increasing the SNR [37]. With no interference at the respiration frequency (which is reasonable), the SNR can be estimated as  
\[SNR=20\log_{10}\left(\frac{\sum\limits_{n=\mu_{r}-1}^{\mu_{r}+1}\abs{\widetilde {\mathbf{O}}[\widetilde{\tau},n]}}{\sum\limits_{n=\gamma_{1}}^{\mu_{r}-2}\abs{ \widetilde{\mathbf{O}}[\widetilde{\tau},n]}+\sum\limits_{n=\mu_{r}+2}^{\gamma _{2}}\abs{\widetilde{\mathbf{O}}[\widetilde{\tau},n]}}\right), \tag{32}\]  
where \(\widetilde{\tau}\) and \(\mu_{r}\) are given by (28) and (29), respectively. The SNR decreases with distance due to signal attenuation [37]. Thus, the improvement in SNR can be determined by considering the detection performance at different distances. The estimated range \(\widetilde{L}\) using four frequency accumulators is shown  
Figure 12: (**a**) Range results after removing the static and non-static clutter and the linear trend, (**b**) after filtering in fast time, (**c**) after filtering in slow time, and (**d**) after respiration signal improvement using the data acquired with the human subject stood behind the wall at 600 cm from the radar facing the antenna. Figure 14: Results after performing the DWFT with a human subject at distances between the antenna and human subject of (**a**) 300 cm, (**b**) 600 cm, (**c**) 900 cm, and (**d**) 1100 cm. Figure 15: Results using the CFAR method with a human subject at distances between the antenna and human subject of (**a**) 300 cm, (**b**) 600 cm, (**c**) 900 cm, and (**d**) 1100 cm. The red squares denote the estimates while the black ellipses denote the true values. These results show that the subject range can be estimated accurately using the proposed method even at a distance of 1100 cm, while the CFAR method is only accurate at a distance of 300 cm. Table 2 presents the SNR values obtained using (32) for the CFAR, MHOC, AM, and proposed methods at distances of 300 cm, 600 cm, 900 cm, and 1100 cm. These results show that the proposed method provides the best SNR at all distances, while the CFAR method provides minimal SNR improvement. Although the MHOC and AM methods have a better SNR improvement than the CFAR method, it will be shown that they provide poor detection results in long-range conditions. The experimental data obtained from the second experiment with the subject facing the antenna at distances of 700 cm and 1000 cm from the antenna in through-wall conditions was used to further validate the accuracy of the proposed method. The detection results are shown in Figure 17 and indicate that the range can be determined more accurately indoors. This is due in part to the presence of wind outdoors. For example, the maximum error \(\aleph\) obtained using (30) is only 6.24 cm when the distance between the antenna and subject is 1000 cm. The corresponding respiration frequency estimate is 0.235 Hz. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & **300 cm** & **600 cm** & **900 cm** & **1100 cm** & **Comparison** \\ \hline Proposed & 7.58 & 4.62 & 1.27 & \(-\)2.75 & Higher \\ CFAR & \(-\)4.54 & \(-\)6.22 & \(-\)17.86 & \(-\)20.64 & Very lower \\ MHOC & \(-\)3.67 & \(-\)4.85 & \(-\)12.88 & \(-\)16.29 & Very lower \\ AM & 2.35 & \(-\)1.84 & \(-\)8.69 & \(-\)11.71 & Lower \\ \hline \hline \end{tabular}
\end{table}