## 2 Theory  
### Impact of Periodic Nonlinearity on the FMCW Radar  
The transmitted signal of an ideal FMCW Radar system [17] can be expressed as  
\[s_{t0}(t)=rect(\frac{t}{T_{p}})\exp[j2\pi(f_{c}t+0.5 Kt^{2})], \tag{1}\]  
where \(T_{p}\) denotes the single frequency modulation cycle, \(f_{c}\) denotes the center frequency, \(K\) denotes the frequency sweep rate. To obtain frequency, the derivative operation on the phase of the transmitted signal is performed. The result can be expressed as \(f_{c}+Kt\). The variation of \(t\) ranges from 0 to \(T_{P}\), so frequency \(f\) varies from 0 to \(f_{c}+KT_{p}\), which completes the frequency sweep. Thus, square term 0.5 \(Kt^{2}\) in the phase drives the frequency sweep. Because the bandwidth of the transmitted signal is \(B=K\times T_{p}\), the start and end frequencies are \(f_{l}=f_{c}\), \(f_{h}=f_{c}+B\), respectively. The illustration of the radar wave propagation in free space is shown in Figure 1. The wave is reflected from the target with a time delay of \(\tau\). The rectangular envelope limits the time range of the transmitted signal but does not affect the phase of it. Therefore, the rectangular envelope is generally ignored for the sake of simplicity in formulas. Ignoring the rectangular envelope, the ideal reflected wave can be represented as  
\[s_{r0}(t)=\exp[j2\pi(f_{c}(t-\tau)+0.5K(t-\tau)^{2})]. \tag{2}\]  
The reflected signal is mixed with the transmitted signal and then filtered by a low-pass filter (LPF) to obtain the ideal beat signal,  
\[ \begin{array}{ll}s_{IF0}(t)&=LPF\{s_{t0}(t)\times s_{r0}(t)\}\\ &=LPF\{\exp[j2\pi(f_{c}\tau+K\tau t-0.5K\tau^{2})]\\ &\quad\quad+\exp\{j2\pi[(2f_{c}-K\tau)t+Kt^{2}-f\tau+0.5K\tau^{2}]\}\}\\ &=\exp[j2\pi(f_{c}\tau+K\tau t-0.5K\tau^{2})]\end{array} , \tag{3}\]  
where \(LPF\{\cdot\}\) represents low-pass filtering of signals. The schematic of the above process is shown in the first line of Figure 2. In Figure 2a, the solid and dashed lines represent the frequencies of the transmitted and received signals over time, respectively. The transformation of frequency with time is linear, i.e., linear modulation. After the above mixing and low-pass filtering, a single-frequency beat signal is obtained, as shown in Figure 2b. Fast Fourier transform (FFT) is performed on \(s_{IF0}(t)\) and the result is denoted as \(S_{IF0}(\omega)\). \(S_{IF0}(\omega)\) is the spectrum of \(s_{IF0}(t)\), which is illustrated in Figure 2c. Figure 1: Free space wave propagation. When the periodic nonlinearity in modulation (periodic nonlinear modulation) occurs, a periodic phase is added to the transmitted signal. To describe that, \(\epsilon(t)\) is defined as the phase it brings which is assumed to follow a cosine-like pattern [11],  
\[\epsilon(t)=b\cos(2\pi f_{e}t), \tag{4}\]  
where \(b\) denotes the amplitude of the periodic nonlinear phase, and \(f_{e}\) denotes the frequency of the periodic nonlinear phase. Therefore, the actual transmitted signal [11] can be represented as  
\[s_{t1}(t)=\exp\Big{\{}j[2\pi(f_{e}t+0.5 Kt^{2})+\epsilon(t)]\Big{\}}. \tag{5}\]  
After reflection from the target with time delay \(\tau\), the periodic nonlinearity is carried into the received signal. The actual reflected wave is expressed as  
\[s_{t1}(t)=\exp\Big{\{}j[2\pi(f_{e}(t-\tau)+0.5K(t-\tau)^{2})+\epsilon(t-\tau)] \Big{\}}. \tag{6}\]  
It is mixed with the actual transmitted signal and then filtered by a low-pass filter (LPF) to obtain the actual beat signal,  
\[ \begin{array}{ll}s_{t1}(t)&=LPF\{s_{t1}(t)\times s_{t1}(t)\}\\ &=\exp[j2\pi(f_{e}\tau+K\tau t-0.5K\tau^{2})]\times\exp\{-j2b\sin(\pi f_{e}\tau )\sin[2\pi f_{e}(t-\tau)]\}\\ &=s_{t10}(t)\times\exp\{-j2b\sin(\pi f_{e}\tau)\sin[2\pi f_{e}(t-\tau)]\}\\ \end{array} . \tag{7}\]  
In order to carry out the derivation of the Fourier transform, the additional term in Equation (7) other than \(s_{t10}(t)\) is first expanded using the Bessel function. Specifically, it can be written as  
\[