## 2 Materials and Methods
### Signal Processing Method  
The sensor used in this study is a radar device that is installed at a certain distance from the person whose vital signs are being measured. Its illustration is shown in Figure 3.  
This radar emits millimeter waves to the human body and, simultaneously, captures the waves reflected by the human body. These signals are used in further processing to generate heartbeat rate and breathing rate. The transmitting power of this radar is 12 dBm [4]. This concurrent, contactless measurement of vital signs makes measuring vital signs convenient for the patient and can be performed for long-term monitoring and home-based monitoring. The system block diagram for the non-contact measurement of vital signs is shown in Figures 4 and 5. ADC is an analog to digital converter. The input to the ADC is a signal from the Low Pass Filter (LPF). The LPF is installed at the output of the mixer. The mixer and the LPF are inside of the TI IWR1443 [4]. The functional block diagram of TI IWR1443 is shown in Figure 6.  
Figure 3: Illustration of a concurrent detection. Figure 4: System block diagram up to Range Angle Map. Figure 5: Block diagram of the system after Range Angle map [5]. To make it easier to understand, the signal processing will be explained with the signal originating from one antenna. The mixer mixes the signal called \(r(t)\), originating from the receiving antenna Rx and the transmitted signal \(s(t)\). The output of the LPF is called the beat signal \(b(t)\). The beat signal can be formulated mathematically using Equations (1) and (2) as follows:  
\[b(t)=LPF\;(s(t)r(t))=\frac{\alpha}{2}\cos{(2\pi\frac{Bt_{d}}{T}t+2\pi f_{c}t_{ d}-\pi\frac{B}{T}t_{d}}^{2}). \tag{4}\]  
Let \(t_{d}=2R/c\), \(f_{c}=c/\lambda\), and ignore small values, then \(b(t)\) becomes:  
\[b(t)\approx\frac{\alpha}{2}\cos{\left(2\pi\frac{B}{T}\frac{2R}{c}t+\frac{4\pi R }{\lambda}\right)}=\frac{\alpha}{2}\cos{(2\pi f_{b}t+\phi_{b})}, \tag{5}\]  
where \(f_{b}\) is the frequency of the beat signal; and \(\phi_{b}\) is the phase of the beat signal. The first Fast Fourier Transform (FFT) operation to the ADC output produces the frequency and phase of the beat signal. The range \(R\) can be calculated from these quantities using the following formulas:  
\[f_{b}=\frac{B}{T}\frac{2R}{c}, \tag{6}\]  
\[\phi_{b}=\frac{4\pi R}{\lambda}. \tag{7}\]  
The system uses four receiver antennas and two transmitter antennas. The transmitter antennas operate in alternate mode, so eight receiving antenna signals are received from Rx1 to Rx8. The second FFT is performed to the range data of each range. As there are 8 Rx signals, the 2D-FFT will produce peaks [13] in the same location but with different phases. The measured phase difference \((\delta)\) can be utilized to calculate the angle of arrival \((\theta)\) of the target [14]. It is mathematically written as follows:  
\[\delta=\frac{2\pi d\sin(\theta)}{\lambda}\;\rightarrow\;\theta=\sin^{-1}{ \left(\frac{\lambda\delta}{2\pi d}\right)}, \tag{8}\]  
Figure 6.0 The functional block diagram of TI IWR1443 [4]. where \(d\) is the distance between adjacent Rx antennas. Therefore, this process produces range angle data. Then, it is packed into one map called Range Angle Map. The static data is removed from the data in the Range Angle Map by differential processing because only dynamic data, which comes from chest displacement, is needed for the next process. Other slow-large-scale human random body movements (RBM) [15, 16] can be considered as static data when compared to chest displacement due to breathing and heartbeat rates. Clutters [17] in environments can also be considered as static data, and their influences can be canceled using this differential processing. Subsequently, the smoothing process is carried out to prepare the data in Range Angle Map for further processing to produce vital signs. Then the Phase Unwrapping [18] process is carried out to solve the ambiguity of the phase.