## 2 Theory
### Impact of Periodic Nonlinearity on the FMCW Radar
\begin{array}{ll}\exp\{-j2b\sin(\pi f_{e}\tau)\sin[2\pi f_{e}(t-\tau)]\}\\ &=\sum\limits_{m=-\infty}^{\infty}J_{m}[2b\sin(\pi f_{e}\tau)]\times\exp(j\pi mf _{e}\tau)\times\exp(-j2\pi mf_{e})\end{array} , \tag{8}\]  
Figure 2: The generation of the spectrum of the beat signal. (**a**) The ideal frequencies of the transmitted signal (solid line) and received signal (dashed line); (**b**) The ideal frequency of the beat signal; (**c**) The ideal spectrum of the beat signal; (**d**) The frequencies of the transmitted signal (solid line) and received signal (dashed line) under the influence of periodic nonlinearity; (**e**) The frequency of the beat signal under the influence of periodic nonlinearity; (**f**) The spectrum of the beat signal under the influence of periodic nonlinearity. where \(J_{m}(z)\) is the Bessel function of the first class of order \(m\). After performing a fast Fourier transform (FFT), the spectrum of the beat signal under periodic nonlinear modulation is obtained:  
\[S_{IF1}(\omega)=\sum_{m=-\infty}^{\infty}J_{m}[2b\sin(\pi f_{e}\tau)]\times\exp( j\tau mf_{e}\tau)\times S_{IF0}(\omega-2\tau mf_{e}), \tag{9}\]  
where \(S_{IF0}(\omega)\) is the spectrum of the ideal beat signal. Under the influence of periodic nonlinearity in frequency modulation, the beat signal generation process is shown in the second line of Figure 2. In Figure 2d, the solid and dashed red lines represent the frequencies of the transmitted and received signals over time, respectively. After mixing and low-pass filtering, periodic changes in the frequency of beat signal occur, as shown in Figure 2e. Figure 2f is the spectrum based on Equation (9). Periodic nonlinearity is presented in its spectrum in the form of side lobes. The above derivation of the impact of periodic nonlinearity does not rely on approximations, making it suitable for short and long distances, unlike any previous studies [11]. More deeply, the amplitude of the spectrum is determined by the magnitude of the Bessel function from Equation (9). When \(m=0\), the value of the Bessel function represents the amplitude of the main lobe. In other cases, the value of the Bessel function represents the amplitude of the side lobe. As a matter of fact, the periodic nonlinearity in the system is not too large. We take \(f_{e}=350\) kHz based on the previous work [16]. Therefore, the amplitude of the main lobe and each side lobe can be obtained at different target time delays. \(J_{0}\), \(J_{1}\), \(J_{2}\), and \(J_{3}\) represent the amplitude of the main lobe, the first side lobe, the second side lobe, and the third side lobe, corresponding to \(m=0,1,2,3\), respectively. Their variation with respect to target delay \(\tau\) is shown in Figure 3. Since \(J_{1}\) is larger than \(J_{2}\) and \(J_{3}\), \(J_{1}\) is used as the representative of side lobes for analysis. \(J_{0}\) and \(J_{1}\) intersect at \(\tau=\tau_{1}\). The system can only work properly if the main lobe is larger than the side lobes, i.e., the left part of the intersection where \(\tau<\tau_{1}\). In this region, the main lobe \(J_{0}\) decreases as \(\tau\) increases, while \(J_{1}\) increases as \(\tau\) increases. In other words, ratio \(J_{0}/J_{1}\) is smaller with the target being farther.