## 3 Derivation of the Proposed Method
### Construction of Virtual Array Signal Model
\begin{array}{ll}\mathbf{x}&=[x(0),x(1),\cdots,x(M)]^{\mathbf{T}}\\ &=\left[\begin{array}{ll}c_{1}^{(0)}e^{i\varphi_{1}^{(0)}}\sum_{l=\ 1}^{L}e^{j\Delta t_{l}A_{\Phi CFO}}\\ c_{1}^{(1)}e^{i\varphi_{1}^{(1)}}\sum_{l=\ 1}^{L}e^{j\Delta t_{l}(\Delta\varphi_{CG}+\Delta\varphi_{1}- \Delta\varphi_{0})}\\ \vdots\\ c_{1}^{(0)}e^{i\varphi_{1}^{(0)}}\sum_{l=\ 1}^{L}e^{j\Delta t_{l}(\Delta\varphi_{CG}+ \Delta\varphi_{M}-\Delta\varphi_{0})}\\ \end{array}\right]\,\end{array}  \tag{12}\]  
\[ \begin{array}{ll}\mathbf{B}&=[\mathbf{b}(0),\mathbf{b}(1),\cdots,\mathbf{b}(M)]^{\mathbf{T} }\\ &=\left[\begin{array}{ll}1\\ e^{j\Delta t_{s}(\Delta\varphi_{CG}+\Delta\varphi_{0})}&e^{j\Delta t_{s}( \Delta\varphi_{CG}+\Delta\varphi_{1})}&\cdots&e^{j\Delta t_{s}(\Delta\varphi_{ CFO}+\Delta\varphi_{M})}\\ \vdots&\vdots&\ddots&\vdots\\ e^{j(P-1)\Delta t_{s}(\Delta\varphi_{CG}+\Delta\varphi_{0})}&e^{j(P-1)\Delta t _{s}(\Delta\varphi_{CG}+\Delta\varphi_{1})}&\cdots&e^{j(P-1)\Delta t_{s}( \Delta\varphi_{CG}+\Delta\varphi_{M})}\\ \end{array}\right]\,\end{array}  \tag{13}\]  
\[ \begin{array}{ll}\mathbf{w}&=[w(1),w(2),\cdots,w(P)]^{\mathbf{T}}\\ &=\left[\begin{array}{ll}\sum_{l=\ 1}^{L}w_{1}(l)e^{-j\Delta t_{l}A_{\Phi 0}}\\ \sum_{l=\ 1}^{L}w_{2}(l)e^{-j\Delta t_{l}A_{\Phi 0}}\\ \vdots\\ \sum_{l=\ 1}^{L}w_{P}(l)e^{-j\Delta t_{l}A_{\Phi 0}}\\ \end{array}\right]\,\end{array}  \tag{14}\]  
then, (10) can be rewritten as follows:  
\[ \begin{array}{ll}\mathbf{s}\ =\ \mathbf{B}\mathbf{x}+\mathbf{w}.\end{array}  \tag{15}\]  
We can see that (15) has the general form of an array signal, which means that we obtain a virtual array using the above derivation. Thus, we call \(\mathbf{s}\) the constructed virtual array snapshot, \(\mathbf{B}\) the virtual array manifold, \(\mathbf{x}\) the virtual signal source vector, and \(\mathbf{w}\) the virtual noise vector. Repeat above steps \(N_{sn}\) times, and we will obtain \(N_{sn}\) virtual array snapshots. Using the constructed snapshots for virtual spatial spectrum estimation, we can obtain \(M\)+1 frequency estimations of the \(M\)+1 virtual signal sources, i.e., \(M\) in-band interferences and one expected signal. By utilizing the influence of the symbol polarity on the MSK signal frequency, we can distinguish the frequency component of the expected signal from those of the in-band interferences from the frequency estimations obtained with the virtual array snapshots constructed from the first two sub-training sequences. According to (13), it can be inferred that when the symbol polarity is positive in the sub-training sequence, the frequencies of the virtual signal sources that can be estimated from virtual spatial spectrum estimation are \(\Delta\hat{f}+\hat{f}_{0}\), \(\Delta\hat{f}+\hat{f}_{1}\),..., \(\Delta\hat{f}+\hat{f}_{M}\), whereas, when the symbol polarity is negative, the corresponding frequency estimations that can be obtained are \(\Delta\hat{f}-\hat{f}_{0}\), \(\Delta\hat{f}+\hat{f}_{1}\),..., \(\Delta\hat{f}+\hat{f}_{M}\). Thus, according to the difference between frequency estimations obtained from the first two sub-training sequences, it is easy to distinguish the \"expected signal frequency + CFO\" from the \"interference frequencies + CFO\".