# Motion Compensation for SAA FMCW Radar Based on Specific Switching Scheme
## 2 Motion-Induced Phase Compensation  
In this section, we describe the signal model of the SAA FMCW radar using a conventional switching scheme and proposed a specific switching scheme. Then, the principle and procedure of the motion compensation method based on the specific switching scheme are given. For the conventional sequential switching scheme from 1 to \(N\), the up-chirp beat signal received by the \(n\)th element is modeled similar to works by the authors of [3,4]:  
\[ \begin{split} u_{n}(t)&=s(t)\exp[-j(n-1)(\varphi_{d }+\varphi_{v})]\\ s(t)&=a\exp[j2\pi(f_{0}t+\varphi_{0})]\end{split}  \tag{3}\]  
where \(s(t)\) is the beat signal received by the 1 st element, \(a\) is signal amplitude, \(f_{0}=2 kr/c+2v/\lambda\) is frequency information, \(\phi_{0}=2r/\lambda\) is phase information, \(k\) is the chirp rate, \(c\) is the light speed constant, and \(r\) is the target's range. Note that the triangular wave consisting of up-chirp and down-chirp is used in this paper. The motion-induced phase \(\varphi_{v}\) leads to erroneous DoA estimation [4], and the incorrect DoA \(\theta_{e}\) is determined by the following formula. \[\sin\theta_{e}=\sin\theta+\frac{2m\pi-\varphi_{v}}{\beta d};m=0,\pm 1,\pm 2,\ldots \tag{4}\]  
To compensate for the motion-induced phase and estimate DoA correctly, this paper proposes a specific switching scheme as shown in Figure 3. The receiving elements are switched in the order of \(1,4,2,3,5,6,7,\ldots,N\), which is different from the conventional sequential switching scheme from 1 to \(N\). Figure 3: Timing diagram of specific switching scheme. Figure 4 illustrates the signal processing flow of the proposed motion compensation method. The receiving elements are switched according to the specific switching scheme to obtain a raw array signal, which is expressed as  
\[s_{1}(t) =s(t); n=1\] \[s_{2}(t) =s(t)\exp[-j(\varphi_{d}+2\varphi_{v})]; n=2\] \[s_{3}(t) =s(t)\exp[-j(2\varphi_{d}+3\varphi_{v})]; n=3 \tag{5}\] \[s_{4}(t) =s(t)\exp[-j(3\varphi_{d}+\varphi_{v})]; n=4\] \[s_{n}(t) =s(t)\exp[-j((n-1)\varphi_{d}+\varphi_{v})]; n=5,6,...,N\]  
A fast Fourier-transform (FFT) leads to the range spectrum  
\[x_{1}(r^{\text{es}}) =b(r^{\text{es}})\exp(-j\varphi_{0}); n=1\] \[x_{2}(r^{\text{es}}) =b(r^{\text{es}})\exp[-j(\varphi_{d}+2\varphi_{v}+\varphi_{0})]; n=2\] Figure 3.0 Figure 3.1 Figure 3.2  
where \(r^{\text{es}}\) is the estimated range of the target; \(b(r^{\text{es}})\) and \(\varphi_{0}\) are the amplitude and phase information, respectively. Targets with close DoAs are differentiated by the range cell in the range spectrum. The phase compensation method in this paper is to perform phase extraction and phase compensation for each range cell independently. Therefore, the proposed method is also be efficient for these targets. Observing the phase information of the first three elements, we found that the phase information of the motion-induced phase \(\varphi_{v}\) and the elements satisfies the following formula. Figure 3.3  
where \(\psi_{1}=\varphi_{0},\psi_{2}=\varphi_{d}+2\varphi_{v}+\varphi_{0},\psi_{3}=2 \varphi_{d}+3\varphi_{v}+\varphi_{0}\) are the phase information of the 1 st, 2 nd, and 3 rd elements, respectively. Assuming the extracted motion-induced phase is \(\widetilde{\varphi}_{v}\), the compensated range spectrum is denoted by  
Figure 3.4 Figure 3.5 Figure 3.6 Figure 3.7 Figure 3.8  
Figure 3.9 Signal processing flowchart for motion compensation. The compensated range spectrum can be directly applied to estimate DoA. Note that the raw array signal is rearranged in the order of 1 to \(N\) for DoA estimation.