# Eyelid Dynamics Characterization with 120 GHz mmW Radar
## 2 Theoretical Background  
A RADAR (RAdio Detection And Ranging system) uses radio waves to detect and measure the distance and velocity of objects [13; 14]. In order to achieve accurate measurements, the radar's beamwidth and distance resolution must be considered. To achieve the required micrometric displacement precision using the FMCW radar, the spectrum of the radar receiver output at the baseband is used to obtain a high-range resolution and the target coarse range estimation. From a sequence of periodic observations, the phase evolution of the target resolution cell provides a very precise submillimetric range, allowing us to estimate eyelid dynamics. The FMCW radar system typically consists of a waveform generator that produces a voltage to control the frequency of the transmitted signal, which is usually a sawtooth or triangle frequency deviation. The transmitted saw tooth signal can be expressed as  
\[s(t)=A\cos(2\pi f_{0}t+\theta(t)+\phi_{0}), \tag{1}\]  
with the amplitude \(A\), the center frequency of the bandpass signal \(f_{0}\), the quadratic phase \(\theta(t)\) producing a linear frequency sweep \(\Delta f\), and \(\phi_{0}\) as the initial phase. For a positive or negative linear FM slope, the phase is  
\[\theta(t)=\pm\pi\mu t^{2}, \tag{2}\]  
with \(\mu=\frac{\Delta f}{T_{M}}\) being the modulation slope within \(-\frac{T_{M}}{2}\leq t\leq\frac{T_{M}}{2}\). For a static target, the echo signal received is  
\[s_{r}(t)=\alpha A\cos[\omega_{0}t-\omega_{0}\tau+\theta(t-\tau)+\phi_{0}], \tag{3}\]  
with a signal-level drop \(\alpha\) and a target delay \(\tau\). Mixing the signal received with the signal transmitted using a single mixer, or, equivalently, the In-Phase output of an I/Q mixer, results in the beating tone \(s_{\rm b}(t)\)with a beating frequency \(f_{\rm b}(t)\), which is the difference between the transmitted signal and echo-instantaneous frequencies  
\[f_{\rm b}(t)=f(t)-f_{\rm r}(t)=\frac{1}{2\pi}[\frac{d}{dt}\theta(t)]-\frac{1}{ 2\pi}[\frac{d}{dt}\theta(t-\tau)]. \tag{4}\]  
The resulting beating tone signal is  
\[s_{\rm b}(t)=\frac{\alpha A^{2}}{2}\cos[\omega_{0}\tau+2\pi\mu\tau t-\pi\mu \tau^{2}], \tag{5}\]  
the delay for a target located at a distance (range) \(R\) is \(\tau=\frac{2R}{c_{0}}\). The terms consists of the following:  
* \(\omega_{0}\tau\): The phase shift due to the target distance
* \(2\pi\mu\tau\): The beating frequency that can be expressed as \[f_{\rm b}=\mu\tau=\frac{2\Delta fR}{T_{\rm M}c}\] (6)
* \(\pi\mu\tau^{2}\): The residual video phase, which can be neglected for short distances, if \(\pi\mu\tau^{2}\ll\omega_{0}\tau\). For short-range applications of FMCW radars, \(\tau\ll\frac{\omega_{0}}{\pi\mu}\), and the residual video phase term can be neglected. For every FM sweep, the beating tone (5) at the radar output is digitized using an analog-to-digital converter (ADC). Since the beating frequency for a single target is proportional to \(R\), for an arbitrary scene containing multiple targets, the mixer output willprovide a spectrum of different tones with frequencies proportional to the different target ranges. Therefore, a Fourier transform can be used to isolate and distinguish different targets with a frequency resolution of \(\frac{1}{T_{\text{M}}}\) and a range resolution of \(\Delta R=\frac{c}{2\Delta f}\). This is carried out with a Discrete Fourier Transform (DFT) of the digitized signal as follows:  
\[S_{\text{b}}(k+1)=\frac{1}{N_{\text{Smp}}}\sum_{n=0}^{N_{\text{Smp}}-1}s_{\text {b}}(n+1)\cdot\exp(-\mathrm{j}\frac{2\pi k\pi}{N_{\text{Smp}}}), \tag{7}\]  
To increase the number of samples at the DFT output, the beating tone signal provided by the radar can be extended with zeros before the DFT (zero-padding). The DFT is usually implemented with a Fast Fourier Transform (FFT) algorithm for numerical efficiency reasons [15]. Figure 1 shows a diagram of the signal processing of the applied algorithm, including the phase processing described in Section 5.