## 2 Theory
### Impact of Phase-Locked Loop Spuriousness on the FMCW Radar  
A phase-locked loop (PLL) is required for frequency division to generate signals of different frequencies in the frequency modulation continuous wave (FMCW) radar. Figure 4 shows the schematic diagram of a phase-locked loop. Explanations of the modules can be found at the bottom of the figure. Ideally, reference frequency \(f_{Ref}\) generates the desired  
Figure 3: The variation of \(J_{0}\), \(J_{1}\), \(J_{2}\) and \(J_{3}\) with \(\tau\).  
RF signal \(f_{RF}\) via the PLL. The RF signal is sent through the transmitted signal. However, the spurious phenomenon occurs during the actual process of frequency division. It is called phase-locked loop spuriousness. This phenomenon manifests in the impact on the transmitted signal as follows: Symmetric spurious frequencies appear on both sides of the required frequency.  
The phase of the ideal transmitted signal in Equation (1) is noted as  
\[\varphi(t)=2\pi(f_{c}t+0.5 Kt^{2}). \tag{10}\]  
Assuming that the partition between the spurious frequency and the required frequency is \(f_{s}\), the actual transmitted signal under the influence of phase-locked loop spuriousness can be written as  
\[s_{t2}(t)=A_{0}\exp[j\varphi(t)]+A_{s1}(t)\exp[\varphi(t)+j2\pi f_{s}t]+A_{s2 }(t)\exp[\varphi(t)-j2\pi f_{s}t], \tag{11}\]  
where \(A_{0}\) denotes the amplitude of the required frequency, \(A_{s1}(t)\) and \(A_{s2}(t)\) denote the amplitude of the two spurious frequencies, respectively. Moreover, \(A_{0}>>||A_{s1}(t)||_{\infty}\), \(A_{0}>>||A_{s2}(t)||_{\infty}\).  
After reflection from the target with time delay \(\tau\), the reflected wave is expressed as  
\[\begin{array}{ll}s_{t2}(t)&=A_{0}\exp[j\varphi(t-\tau)]\\ &+A_{s1}(t-\tau)\exp[\varphi(t-\tau)+j2\pi f_{s}(t-\tau)]\\ &+A_{s2}(t-\tau)\exp[\varphi(t-\tau)-j2\pi f_{s}(t-\tau)]\end{array}. \tag{12}\]  
Transmitted signal \(s_{t2}(t)\) is mixed with received signal \(s_{t2}(t)\) and then processed through a low-pass filter. After that, the beat signal can be expressed as  
\[\begin{array}{ll}s_{IF2}(t)&=A_{0}^{2}\exp\{j[2\pi(f_{c}\tau+K\tau t-0.5K \tau^{2})\}\\ &+A_{0}A_{s1}(t-\tau)\exp\{j[2\pi(f_{c}\tau+(K\tau+f_{s})t-0.5K\tau^{2})]\}\\ &+A_{0}A_{s2}(t-\tau)\exp\{j[2\pi(f_{c}\tau+(K\tau-f_{s})t-0.5K\tau^{2})]\} \end{array}. \tag{13}\]  
Using FFT, the spectrum of the beat signal can be obtained:  
\[S_{IF2}(\omega)=A_{0}^{2}S_{IF0}(\omega)+A_{0}A_{s1}(t-\tau)S_{IF0}(\omega+2 \pi f_{s})+A_{0}A_{s2}(t-\tau)S_{IF0}(\omega-2\pi f_{s}). \tag{14}\]  
As a result, the beat signal generates spikes at frequencies \(K\tau\), \(K\tau+f_{s}\) and \(K\tau-f_{s}\), respectively. Due to the contrast in magnitude, \(K\tau\) corresponds to the main lobe and the last two correspond to the side lobes. On the one hand, the amplitude of the main lobe, \(\left|\left|A_{0}^{2}\right|\right|_{\infty}\), is much larger than that of the side lobes, \(\left|\left|A_{0}A_{s1}(t-\tau)\right|\right|_{\infty}\), \(\left|\left|A_{0}A_{s2}(t-\tau)\right|\right|_{\infty}\). On the other hand, the amplitude of the main lobe is independent of target time delay \(\tau\), and side lobes vary with \(\tau\).  
From the above derivation, it is evident that both the periodic nonlinearity during frequency and the phase-locked loop spuriousness during frequency division lead to the  
Figure 4: Schematic diagram of the phase-locked loop (PFD = phase frequency detector, LP = low-pass filter, VCO = voltage-controlled oscillator, \(1/\)N = fractional-N frequency synthesizer).  
generation of side lobes in the beat signal. And the amplitude of the side lobe is related to target time delay \(\tau\). The difference between them is that the amplitude of the side lobe generated by the periodic nonlinearity increases with \(\tau\), i.e., the ratio of the amplitude of the side lobe to the main lobe increases. However, the statistical pattern of the amplitude of the side lobe generated by the phase-locked loop spuriousness is not obvious. The distinction is used to identify the cause of clutter in the system.