## 2 System Overview
### Target Response Recovery and Demodulation  
Assuming the TX signal from the radar in CW mode is  
\[TX(t)=A_{TX}\cos(2\pi F_{C}t+\theta_{TX}), \tag{8}\]  
it follows that by modulating the phase of the carrier with \(\theta_{m}(t)\), which includes the initial phase constant \(\theta_{TX}\), this equation can be modified to  
\[TX(t)=A_{TX}\cos(2\pi F_{C}t+\theta_{m}(t)). \tag{9}\]  
The target vital sign or gesture information is then mixed with the carrier signal as it reflects off the target, which is then downconverted in the quadrature mixer and measured at the output. The RX signal is downconverted with the unmodulated carrier \(\cos(2\pi F_{C}x(t)+\theta_{0})\), and the initial phase constants can be combined into a DC component. \(\theta_{m}(t)\) is then replaced by (2) in the PSK mode or (7) in the PMCW mode. Since this radar design uses a quadrature mixer, the outputs of the radar to the DAQ are the desired \(I\) and \(Q\) signals. As adapted from [25], the CW, PSK, and PMCW \(I/Q\) baseband responses can be defined as:  
\[B_{CW_{I}}(t)=A_{I}\cos(4\pi x(t)/\lambda_{C}+\Delta\varphi(t))+DC_{I} \tag{10}\]  
\[B_{CW_{Q}}(t)=A_{Q}\cos(4\pi x(t)/\lambda_{C}+\Delta\varphi(t))+DC_{Q} \tag{11}\]  
\[B_{PSK_{I}}(t)=A_{I}\cos(4\pi x(t)/\lambda_{C}+\Delta\varphi(t)+\theta_{PSK}( t))+DC_{I} \tag{12}\]  
\[B_{PSK_{Q}}(t)=A_{Q}\cos(4\pi x(t)/\lambda_{C}+\Delta\varphi(t)+\theta_{PSK}( t))+DC_{Q} \tag{13}\]  
\[B_{PMCW_{I}}(t)=A_{I}\cos(4\pi x(t)/\lambda_{C}+\Delta\varphi(t)+\theta_{PMCW }(t))+DC_{I} \tag{14}\]  
\[B_{PMCW_{Q}}(t)=A_{Q}\cos(4\pi x(t)/\lambda_{C}+\Delta\varphi(t)+\theta_{PMCW }(t))+DC_{Q} \tag{15}\]  
In these equations, \(x(t)\) is the phase shift imparted into the reflected waveform by the target's motion relative to the radar. \(\lambda_{C}\) is the wavelength of the carrier frequency, \(\Delta\varphi(t)\) is the noise in the environment, \(A_{I}/A_{Q}\) are the AC amplitude of the signal out of the mixer, and \(DC_{I}/DC_{Q}\) are the DC offset of the \(I/Q\) channels, respectively. As the modulation schemes used by the FPGA are cyclic and known, the main challenge faced in  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \(a_{i}\) & **n** & **0** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** & **12** & **13** & **14** \\ \hline \(a_{3}\) & 0 & 1 & 1 & 1 & 1 & 0 & 1  
baseband processing is the FPGA clock and the oscilloscope clock being asynchronous. To address this, the modulation scheme \(s_{in}(t)\), as well as the \(I/Q\) channel data, is measured with the oscilloscope sampling well over twice of \(F_{Clk}\). An oversampling factor greater than two is needed to satisfy the Nyquist sampling theorem and recover at least one point per modulation symbol for accurate demodulation. Therefore, to ensure high-resolution sampling of the asynchronous modulation signal and to account for clock speed imprecision, a sampling rate of \(50\times F_{Clk}\) was chosen. With this information, various approaches can be taken to demodulate the PMCW baseband \(I/Q\) signals that can also be extended to the PSK baseband signals. For this radar system, all the signal processing was performed in MATLAB. PMCW radar systems when used for measuring distance have a range resolution of \(c/(2\times F_{Clk})\), as the phase difference between the transmitted and received code will have transitioned one or more encoded symbols over the signal's travel time. For these experiments, \(F_{Samp}\) and \(F_{Clk}\) are chosen to be much lower than the frequency needed to accurately measure phase differences between the transmitted and received signals, so target distance cannot be determined. There will still be a time offset between the recovered \(I/Q\) basebands and \(s_{in}\) due to differences in the signal processing chains, in which \(I/Q\) are AC coupled and amplified, so a small offset is applied to align the logical transition edges of measured signals, which will not be separated by more than \(1/F_{Clk}\).