# A Novel Scheme for MIMO-SAR Systems Using Rotational Orbital Angular Momentum
## 3 A Novel RDE-Based Scheme  
To solve the problem mentioned above, a novel RDE-based scheme used for the MIMO-SAR system is proposed in this section. The rotational Doppler Effect (RDE) occurs when relative rotation between a vortex EM wave source and an observer exists [20]. For the linear polarized vortex EM waves of the OAM mode \(l\), the rotational Doppler frequency shift \(\Delta f\) is:  
\[\Delta f=\frac{l\Omega}{2\pi} \tag{7}\]  
where \(\Omega\) is the relative rotation speed between the source and the observer. If the phase of each element in UCA antenna varies azimuthally, i.e., the \(n\)th element is fed by \(\varphi_{n-1}=2\pi l(n-1)/N\) after one period of \(\tau\), the rotational OAM beams can be generated, and the equivalent rotation frequency is \(1/N\tau\) r/s. When the UCA antennas in Figure 1 transmit the rotational OAM beams, Equation (5) can be rewritten as:  
\[ \begin{array}{c}s_{p,q}(t_{m},t,l_{p})\approx N\exp\big{\{}il_{p}\pi/2\big{\}}J _{l_{p}}\big{[}ka\sin\theta_{p}(t_{m})\big{]}\omega_{a}[l_{m}-x/v+(p+q-3)L_{a}/2 v]\omega_{r}\big{[}t-2r_{p,q}(t_{m})/c\big{]}\\ \exp\Big{\{}i\pi K_{r}\big{[}t-2r_{p,q}(t_{m})/c\big{]}^{2}\Big{\}}\exp\{i2\pi f _{c}\big{[}t-2r_{p,q}(t_{m})/c\big{]}\big{\}}\exp\{il_{p}\big{[}\Omega_{p}l_{m }+\varphi_{p}(t_{m})\big{]}\big{\}}\end{array}  \tag{8}\]  
where \(\Omega_{p}\) is the rotation speed of the \(p\)th OAM beams. The signal received by the \(q\)th (\(q=1\), 2) antenna is:  
\[s_{q}(t_{m},t)={s_{1}}^{\prime}(t_{m},t)\exp\{il_{1}\Omega_{1}t_{m}\}+{s_{2}}^ {\prime}(t_{m},t)\exp\{il_{2}\Omega_{2}t_{m}\} \tag{9}\]  
where:  
\[ \begin{array}{c}{s_{1}}^{\prime}(t_{m},t)=N\exp\{i{l_{1}\pi/2}\}J_{l_{1}}[ka \sin\theta_{1}(t_{m})]\omega_{a}[t_{m}-x/v+(q-2)L_{a}/2v]\omega_{r}\big{[}t-2r _{1,q}(t_{m})/c\big{]}\\ \exp\Big{\{}i\pi K_{r}\big{[}t-2r_{1,q}(t_{m})/c\big{]}^{2}\Big{\}}\exp\{i2\pi f _{c}\big{[}t-2r_{1,q}(t_{m})/c\big{]}\Big{\}}\exp\{il_{1}\varphi_{1}(t_{m})\},\end{array}  \tag{10}\]  
and:  
\[ \begin{array}{c}{s_{2}}^{\prime}(t_{m},t)=N\exp\{i{l_{2}\pi/2}\}J_{l_{2}}[ka \sin\theta_{2}(t_{m})]\omega_{a}[t_{m}-x/v+(q-1)L_{a}/2v]\omega_{r}\big{[}t-2r _{2,q}(t_{m})/c\big{]}\\ \exp\Big{\{}i\pi K_{r}\big{[}t-2r_{2,q}(t_{m})/c\big{]}^{2}\Big{\}}\exp\{i2\pi f _{c}\big{[}t-2r_{2,q}(t_{m})/c\big{]}\Big{\}}\exp\{il_{2}\varphi_{2}(t_{m})\}.\end{array}  \tag{11}\]  
After azimuth Fourier transforming, Equation (9) in the range-Doppler domain can be expressed by:  
\[S_{q}(f_{a},t)=W\bigg{(}f_{a}-\frac{l_{1}\Omega_{1}}{2\pi}\bigg{)}S_{1}(f_{a},t)+W\bigg{(}f_{a}-\frac{\text{$l_{2}\Omega_{2}$}}{2\pi}\bigg{)}S_{2}(f_{a},t) \tag{12}\]  
where \(f_{a}\) is the azimuth frequency. It can be seen from Equation (12) that, due to the RDE, the echo signals related to the different OAM modes locate at different azimuth frequencies. Assume that \(\Omega_{1}\) is 0 rad/s and \(l_{2}\Omega_{2}/2\pi\) is PRF/2; Equation (12) can be rewritten as:  
\[S_{q}(f_{a},t)=W(f_{a})S_{1}(f_{a},t)+W\bigg{(}f_{a}-\frac{\text{PRF}}{2} \bigg{)}S_{2}(f_{a},t). \tag{13}\]  
To better understand this process, Figure 3 shows the Doppler spectrum of the received signal. The red line denotes the spectrum of \(S_{1}\), which locates at \(f_{a}=0\), while the blue line denotes the spectrum of \(S_{2}\), which locates at \(f_{a}=\text{PRF}/2\). Each spectrum has the same Doppler bandwidth \(\Delta f_{d}\). In the case of aliasing in the Doppler domain, PRF should satisfy:  
\[\text{PRF}>2\Delta f_{d} \tag{14}\]  
Figure 3: Doppler frequency spectrum of the received signal. As shown in Figure 4, for a given receiver, the OAM-based MIMO-SAR data can be processed by the following steps:  
1. Range compression of the raw data;
2. Azimuth Fourier transform of the data;
3. Separating the data of different OAM modes in the range-Doppler domain by bandpass filtering;
4. Imaging processing to obtain the image. The 2-D SAR image can be obtained by the classic SAR imaging algorithm, e.g., the Range-Doppler algorithm or the \(\omega\)K algorithm [12].