# Eyelid Dynamics Characterization with 120 GHz mmW Radar
## 2 Theoretical Background
### Radar Parameters  
The radar used in this study, shown in Figure 2, was developed in our research laboratory. The system uses a System-on-a-Chip (SoC) 120 GHz radar front-end (TRX_120_001) from [16] feeding a plastic lens antenna. As shown in Figure 3, the instantaneous frequency transmitted by the radar is provided by a Direct Digital Synthesizer (DDS) based on a 1 Gs/s 14-bit DDS chip, programmed as a sawtooth linear FM generator. The local oscillator of the TRX_120_001 is phase-locked to the DDS output, generating a reference submultiple frequency of \(1/(64\times 40)\) the transmitted frequency in the 120 GHz band. The target range  
Figure 1: Signal processing algorithm.  
is determined from the frequency peak in the beating tone spectrum. After applying the FFT to the echo signal, and inverting (6), we obtain  
\[R=\frac{c}{2}\tau=\frac{T_{\text{M}}c}{2\Delta f_{\text{b}}}f_{\text{b}} \tag{8}\]  
with \(f_{\text{b}}=\mu\tau\). For a micromotion measurement we evaluate the phase evolution of the range cell of interest containing the target. With \(\phi_{b}=-\omega_{0}\tau=-2 kR\), we obtain  
\[R=\frac{\lambda}{4\pi}\phi_{\text{b}}, \tag{9}\]  
where \(\phi_{\text{b}}\) is the phase of the beating tone of a backscattering point, and \(\lambda=\frac{c}{f_{0}}\) is wavelength corresponding to the transmitted frequency. After detecting the target in a \(k\) frequency domain sample corresponding to the observed body range \(R\) resolution cell, the phase and amplitude of this target are evaluated over the entire measurement of \(l\) chirps (FM sweeps). One single chirp consists of \(k=0,\dots,N_{\text{Smp}}\) samples. The range determination based on the frequency of the beating tone is not ambiguous if the sampling frequency of the ADC fulfils the Nyquist sampling frequency corresponding to the maximum expected range. The range resolution of a radar operating with a bandwidth \(\Delta f\) is given by  
\[\Delta R=\frac{c}{2\Delta f}. \tag{10}\]  
Since the eyelid motion is small compared to the radar range cell, eyelid monitoring is based on phase rather than range resolution. The cyclic characteristic of the phase introduces a maximum unambiguous phase-based change in range as follows:  
\[R_{\text{un}}=\frac{\lambda}{2}. \tag{11}\]  
For this reason, if the range changes exceed \(R_{\text{un}}\), the phase must be unwrapped to obtain the correct displacement.  
Table 1 shows the radar parameters used for this study.  
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameter** & **Calculation** & **Value** \\ \hline Center frequency & \(f_{0}\) & 122.5 GHz \\ Bandwidth & \(\Delta f\) & 1–3 GHz \\ Sweep time & \(T_{\text{M}}\) & 1.5 ms \\ Slope & \(\mu=\frac{\Delta f}{T_{\text{M}}}\) & (1–3) GHz/1.5 ms \\ Sweep repetition period & \(SRP\) & 3 ms \\ \hline \hline \end{tabular}
\end{table}
Table 1: System parameter for simulations and measurements.  
Figure 2: The radar sensor module. The radar front-end is a TRX_120_01 RFE chip mounted onto an FR4 Printed circuit board. (**a**) Circuit board of the radar. (**b**) Prototype of the 120 GHz FMCW radar.