## 3 Theory  
### Breathing Signal Detection  
Disordered breathing is characterized by deviations in either amplitude or breathing rate from normal breathing, e.g., shallow or non-existent breathing (apnea) or rapid breathing (hyperpnea) [3]. In order to evaluate these disorders, the breathing frequency and the apnea periods must be detected. The spectrum of the detected signal is obtained in [35]. Both harmonics and cross-products (intermodulation) can be analyzed using the model described in [35]. Some aspects will be revised here for completeness.  
When the transmitted pulse hits the patient, part of it is reflected due to the high reflectivity of the body. The time-of-flight or time-of-arrival (ToA) of this pulse, denoted by \(\tau_{0}\), depends on the nominal radar distance \(d_{0}\). Due to breathing and heart motion, the chest cavity expands and contracts periodically, meaning that the distance travelled, \(d(t)\), varies periodically around the nominal distance \(d_{0}\). The signal received can be represented as the sum of the responses of the channel and the variation due to breathing:  
\[r(t,\tau)=\sum_{i}A_{i}p(\tau-\tau_{i})+Ap(\tau-\tau_{d}(t)) \tag{1}\]  
where \(p(\tau)\) is the normalized received pulse, \(A_{i}\) is the amplitude of each multipath component, \(\tau_{i}\) its delay, and \(A\) is the amplitude of the reflected pulse on the body that depends on the body radar cross section (RCS) and the body orientation.  
The time delay \(\tau_{d}\) (associated with vital signs) is modeled as the sum of the ToA (\(\tau_{0}\)) plus two delays associated with a periodical chest movement due to breathing (with a frequency \(f_{b}\)) and heartbeats (with frequency \(f_{h}\)):  
\[\tau_{d}(t)=2d(t)/c \tag{2}\]  
\[d(t)=d_{0}+m(t)=d_{0}+m_{b}\sin(2\pi f_{b}t)+m_{h}\sin(2\pi f_{b}t)\approx d_ {0}+m_{b}\sin(2\pi f_{b}t) \tag{3}\]  
where \(c\) is the speed of light, and \(m_{b}\) and \(m_{h}\) are the breathing and heart amplitude, respectively. The peak-to-peak chest motion due to respiration in adults ranges from 0.4 to 1.2 cm [41]. Thus, in this study, only the breathing signal is considered because the heart amplitude is considerably lower than the breathing amplitude [42]. In addition, the heartbeat signal can be filtered because the frequency range varies between 1-2 Hz, whereas the frequency range of breathing signal is between 0.1-0.7 Hz.  
The received waveforms \(r\) are measured periodically in slow time \(t=nT_{s}\) (\(n=1\),2,...,\(N\)). \(N\) discrete-time sequences are stored after the received signal is sampled. These values are stored in a matrix \(R\), the elements of which are:  
\[R[m,n]=r(\tau=mT_{f},t=nT_{s}) \tag{4}\]  
where \(T_{f}\)is the sampling period in fast time.  
In a static environment, the resulting clutter represents a DC component in the slow-time direction. In such an environment, the only movement is caused by the person's breathing and heart activity, and the background clutter does not depend on slow time \(t\). The background clutter is removed by subtracting the average of all received waveforms from the original signal. The DC component is blocked by subtracting the average of all samples in fast time. More details about the clutter suppression will be done in the next section. The signal of interest \(y\) is recovered, after the clutter removal process:  
\[y(t,\tau)=Ap(\tau-\tau_{d}(t)) \tag{5}\]  
As \(m_{b}\)\(<<\)\(d_{0}\), the breathing movement can be seen as a perturbation of the delay and it can be classified as a micro-movement (\(m_{b}\) less than range resolution \(\Delta\)R = c/(2B), where B is the pulse bandwidth and \(c\) the light velocity). The signal \(y\) can then be expanded in a Taylor series around \(\tau=\tau_{0}=2d_{0}/c\):  
\[y(\tau,t)=Ap(\tau-\tau_{0