## 2 Bio-Radar Prototype Description
#### Signal Processing Algorithm  
The Digital Signal Processing (DSP) Module runs an algorithm designed for the rigorous extraction of vital signals information. Figure 2 represents the Bio-Radar system, more specifically the implemented DSP algorithm implemented in the receptor chain [35]. The signal received by our system is initially processed through a step called downconversion, which converts it to a baseband signal. This results in a complex signal, referred to as \(g(n)\). Following this, the signal undergoes a process known as decimation, which reduces its original sampling rate from 100 kHz to 100 Hz. This reduction produces a new signal, denoted as \(g_{D}(n)\). As illustrated in Figure 3a, the complex baseband signal \(g_{D}(n)\) is represented in the complex plane, typically visualized as an arc. It is crucial to understand that the transmitted signal, which our system sends out, is reflected back not just by the target object, such as the subject's chest wall, but also by other objects in the vicinity. These other objects, assuming they are stationary, contribute to the signal in a specific way. Their reflections are added to the received signal as DC offsets, which are constant components that shift the center of the arc in the complex plane. This means that the overall signal received and visualized includes not just the vital information from the subject's chest wall but also these additional static reflections from the surrounding environment. Understanding and accounting for these additional components is important for accurately interpreting the data from our Bio-Radar system. Assuming these objects remain stationary, their reflections are perceived as DC offsets added to the center of the arc. Afterwards, the DC offsets are estimated and subsequently removed from the decimated signal, resulting in signal \(b(n)\). Figure 3 serves as a visual guide to the DC offset removal step, which is crucial for accurate signal analysis. The process begins with the received radar signal, which often includes an unwanted DC component due to signal propagation effects. Using the Optimized Cost (OC) method, proposed in a previous study [35], it was calculated the median of the signal (Figure 3a) and establish a fictitious boundary to contain the cost function optimization (Figure 3b). The lowest cost point within this boundary indicates the precise DC offset to be removed. Once this offset is subtracted, the signal median should theoretically align with the origin (Figure 3c). Finally, it is demonstrated the signal \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Operating Frequency & 5.8 GHz \\ Transmit Power & 2 dBm \\ Sampling Rate & 100 kHz \\ Frequncy Offset & 10 kHz \\ Antenna Type & \(2\times 2\) patch array \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Technical parameters of the Bio-Radar system. Figure 2: Block diagram of the Bio-Radar system including DSP module, adapted from [35]. post-correction with an example of a rotated signal for illustrative purposes (Figure 3d). This process ensures that the signal is appropriately centered and ready for subsequent stages of analysis, such as feature extraction and pattern recognition. In order to avoid wraps caused by the possibility of the signal crossing the 180\({}^{\circ}\) value, it is necessary to rotate the signal to oscillate around the 0\({}^{\circ}\) angle, resulting in \(br(n)\), as shown in Figure 3d. This process not only avoids wraps but also simplifies the signal subsequent analysis. Lastly, to obtain the received vital signs in the time domain, \(y(n)\), the arctangent demodulation is applied, resulting in the signal of Figure 4.  
Figure 3: Sequential illustration of DC offset removal process in radar signal using the Optimized Cost Method. (**a**) Original radar signal with DC offset shown by ’Received signal’ and the guide for optimization ’Fictitious circumference’ with ’Signal median’ as its center.