## 3 Results  
### Analysis of Experimental Outcomes  
The data analysis was conducted using a custom Python 3.8 script, focusing on the mean absolute deviation (MAD) as the primary metric to quantify the discrepancy between the RR measurements obtained from the ESP32 and those recorded by the Vernier belt. The MAD calculation is based on the following formula:  
\[MAD=\frac{\sum_{i=1}^{n}\lvert N_{i}^{gt}-N_{i}^{pd}\rvert}{n}, \tag{15}\]  
where \(n\) is a total number of dataset, \(N_{i}^{gt}\) is RR obtained from the belt at index \(i\) and \(N_{i}^{pd}\) is RR obtained from the ESP32 at index \(i\). The overall result of bpm for 8 participants with calibrated parameters is shown in Figure 13. This visualization demonstrates that the predicted CSI data, after signal processing and RR extraction, closely matches the ground truth obtained from the Vernier belt. However, significant discrepancy can be observed at certain frames. A detailed statistical analysis is presented in Tables 2-5. From 30-s windows, we interpret RR in bpm in the following tables and present the MAD values for three distinct breathing patterns (normal, fast, and slow) under various parameter adjustments. Since, all the 8 participants are adults, we define 12-16 bpm as a normal breathing type during rest as stated by [38]. Therefore, bmp under normal breathing time is slow breathing type, which is useful to understand the RR pattern which may potentially lead to sleep apnea. The bpm higher than 16 is outlined as fast breathing type. \begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline
**Breathing Type** & **Subject** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **AVG** \\ \hline Normal (12–16 bpm) & 2.67 & 1.08 & 1.11 & 1.36 & 4.04 & 3.30 & 2.89 & 3.09 & 2.44 \\ Fast (>16 bpm) & 4.89 & 2.67 & 4.26 & 3.77 & 7.82 & 6.27 & 7.05 & 6.40 & 5.39 \\ Slow (<12 bpm) & 1.21 & 1.97 & 2.10 & 2.00 & 2.06 & 1.78 & 2.78 & 1.92 & 1.98 \\ AVG & 2.92 & 1.91 & 2.49 & 2.38 & 4.64 & 3.78 & 4.24 & 3.80 & 3.27 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Evaluation result when \(\alpha=1\), \(\beta=-50\), \(\gamma=35\), \(\delta=2\), \(\epsilon=20\), \(\tau=5\) and \(\zeta=120\). Figure 13: Overall result of bpm, with the personal calibrated parameter. The experiments tested at multiple parameter combinations show that the overall average MAD from 8 participants across all three breathing types is optimal with \(\alpha=1\), \(\beta=-50\), \(\gamma=15\), \(\delta=2\), \(\epsilon=20\), \(\tau=5\), and \(\zeta=120\), yielding an MAD value of 2.60. This parameter combination also provides the best result for normal breathing with an MAD of 1.38. However, it fails to optimize results for fast and slow breathing rates. For fast breathing, the best MAD is 1.81 with \(\epsilon=30\), keeping other parameters the same. For slow breathing, which is suitable for sleep apnea, the optimal MAD is achieved with \(\gamma=35\) while keeping the other parameters constant. The observation indicates that low \(\epsilon\) is beneficial for slow breathing RR estimation because the Butterworth filter effectively eliminates noise. Similarly, \(\epsilon\) is efficient for fast breathing RR estimation. Observations indicate that the amplitude is much smoother in the smaller room, allowing for the use of a low \(\gamma\) value for the Gaussian filter. Conversely, a higher \(\gamma\) value is needed for a larger room with less smooth amplitude. The parameter \(\epsilon\) emerged as a significant factor influencing the accuracy of our results, acting as a filter for the frequencies detected within the signal. A higher \(\epsilon\) value tends to disregard rapid breathing patterns as noise, preserving only normal and slow breathing rates. Conversely, a lower \(\epsilon\) setting results in the retention of fast breathing rates but also includes more noise, which adversely affects the clarity of normal and slow breathing rate data.