## 3 The Proposed Algorithm  
### The Rotation Array Structure of Proposed Algorithm  
As shown in Figure 2, \(D\) narrowband far-field sources are observed by a rotation array with two antenna elements, \"1\" and \"2\". Assume the baseline 1-2 is rotating around \(Z\)-axis in XOY plane at a constant velocity by anticlockwise direction. The rotate velocity is \(\
u_{{}_{Z}}\), \(\
u_{{}_{Z}}=\omega_{{}_{Z}}r=2\pi f_{{}_{Z}}r\), where \(\omega_{{}_{Z}}\) is rotational angular frequency; \(f_{{}_{Z}}\) is rotation frequency. The rotation period is \(T\). After time \(\Delta t\) (\(\Delta t<T\)), the elements 1 and 2 are rotating to the position \(1^{\prime}\) and \(2^{\prime}\), respectively. Assume the array radius is \(r\); \(l\) is the incident signal direction; \(l^{\prime}  
In order to derive the proposed algorithm, some assumptions are clarified at first,  
1. While the baseline 1-2 is rotating, the baseline is certainly vertical to \(Z\)-axis with absolute uniform velocity;
2. Select \(2M\) elements at uniformly-time interval to make them form the virtual UCA within \(T/2\) period;
3. The signals remain static during the measurement time. At time \(t_{m}\), the elements 1 and 2 start to sample the received signal data, so \(t_{m}\) can be expressed as  
\[t_{m}=t_{1}+(m-1)\Delta\tau \tag{11}\]  
where \(t_{1}=0\), \(m=1,2,\cdots,M\) ; \(\Delta\tau\) is the time delay which is needed for selecting two neighboring elements, \(0<\Delta\tau<T/2\). As we know, the wave path difference between the reference point \(O\) and the \(m\)th element for the \(i\)th signal is \(\tau_{mi}\). In the rotation array model, the element \(m\) is obtained by rotating element 1 through time \((m-1)\Delta\tau\), so the total wave path difference \(\tau_{1m}\) is  
\[\tau_{1m}=\tau_{mi}+(m-1)_{\Delta}\tau \tag{12}\]  
Then Equation (4) is modified as  
\[ \begin{split} x_{1m}(t_{m})=&\sum_{i=1}^{D}\exp(- \operatorname{j}\omega(\tau_{mi}+(m-1)\Delta\tau))s_{i}(t_{m})+n_{1m}(t_{m})\\ =&\sum_{i=1}^{D}\exp(-\operatorname{j}\omega\tau_{mi })\exp(-\operatorname{j}\omega(m-1)\Delta\tau)s_{i}(t_{m})+n_{1m}(t_{m})\\ =&\sum_{i=1}^{D}a_{1m}(\theta_{i},\varphi_{i})\exp(- \operatorname{j}\omega(m-1)\Delta\tau)s_{i}(t_{m})+n_{1m}(t_{m})\end{split}  \tag{13}\]  
Assuming the signal sources remain static, the element rotation will cause the Doppler frequency shift. As shown in Figure 2, \(v\) is denoted as the velocity component of \(v_{z}\) in the source incident direction \(\mathbf{l}^{\prime}\), we obtain  
\[v=v_{z}\sin(\theta-\omega_{z}t)\cos\varphi \tag{14}\]  
According to the Doppler frequency formula \(f_{\mathrm{d}}=v\mathbf{\cdot}f/\operatorname{c}\), \(f_{\mathrm{d}}\) can be written as  
\[f_{\mathrm{d}}=v_{z}\sin(\theta-\omega_{z}t)\cos\varphi\mathbf{\cdot}f/\operatorname{c}\]  
Figure 2: Rotation array structure diagram. Similarly, denoting \(\mathbf{\varPhi}_{2}=\text{diag}[\phi_{{}_{t_{1}}},\ \ \phi_{{}_{t_{2}}},\ \ \cdots\ \,\phi_{{}_{t_{M}}}]\); \(\phi_{{}_{t_{M}}}\) is the phase difference of element 2 at time \(t_{{}_{m}}\) relative to the initial position; therefore, the array output matrix of element 2 within _T_/2 is modeled as  
\[\mathbf{X}_{2}=\mathbf{\varPhi}_{2}\mathbf{A}_{2}\mathbf{S}+\mathbf{N}_{2} \tag{23}\]  
When the array elements are rotating more than _T_/2 period, we can select 2\(M\) elements to construct a virtual UCA. Then, integrate \(\mathbf{X}_{1}\) with \(\mathbf{X}_{2}\), \(\mathbf{X}\) can be represented by \(\mathbf{X}_{1}\) and \(\mathbf{X}_{2}\)  
\[\mathbf{X}= \begin{bmatrix}\mathbf{X}_{1}\\ \mathbf{X}_{2}\end{bmatrix} = \begin{bmatrix}\mathbf{\varPhi}_{1}\\ \mathbf{\varPhi}_{2}\end{bmatrix}\begin{bmatrix}\mathbf{A}_{1}\\ \mathbf{A}_{2}\end{bmatrix} \mathbf{S}+ \begin{bmatrix}\mathbf{N}_{1}\\ \mathbf{N}_{2}\end{bmatrix} =\mathbf{\varPhi}\mathbf{A}\mathbf{S}+\mathbf{N} \tag{24}\]  
where \(\mathbf{\varPhi}= \begin{bmatrix}\mathbf{\varPhi}_{1}\\ \mathbf{\varPhi}_{2}\end{bmatrix} \), \(\mathbf{A}= \begin{bmatrix}\mathbf{A}_{1}\\ \mathbf{A}_{2}\end{bmatrix} \), \(\mathbf{N}=