## 3 Derivation of the Proposed Method  
### Construction of Virtual Array Signal Model  
Assume that the length of the signal sequence fragment in each time window is \(N_{w}\) samples. We first pick a subsequence of length \(L\) from it with indices generated according to the combinatorics theory, i.e., the indices are a combination obtained by arbitrarily picking \(L\) numbers from the sequence of \(\{1,2,\ldots,N_{w}\}\). Assume the index combination is \(i_{1}=\{i_{1}(l)\}\):\(l=1,2,\ldots,L\}\) and the resulting subsequence is \(s_{1}=\{s_{1}(l):l=1,2,\cdots,L\}\), where  
\[s_{1}(l)=\sum\
olimits_{m=0}^{M}c_{1}^{(m)}e^{j\{[i_{1}(l)-i_{1}(1)](\Delta \varphi_{{\rm CFO}}+\Delta\varphi_{m})+\varphi_{1}^{(m)}\}}+w_{1}(l), \tag{7}\]  
where \(c_{1}^{(m)}\) and \(\varphi_{1}^{(m)}\) denote the amplitude and phase of the \(m\)-th signal component in the first signal sample of the subsequence, respectively, \([i_{1}(l)-i_{1}(1)](\Delta\varphi_{{\rm CFO}}+\Delta\varphi_{m})\) is the phase difference of the \(m\)-th signal component in the \(l\)-th signal sample relative to the same signal component in the first signal sample of the subsequence, and \(w_{1}(l)\) is the white Gaussian noise in the \(l\)-th signal sample. Let \(\Delta i_{l}=i_{1}(l)-i_{1}(1)\), and then (7) can be rewritten as follows:  
\[s_{1}(l)=\sum\
olimits_{m=0}^{M}c_{1}^{(m)}e^{j[\Delta i_{l}(\Delta\varphi_{{ \rm CFO}}+\Delta\varphi_{m})+\varphi_{1}^{(m)}]}+w_{1}(l). \tag{8}\]  
Then, we add (\(P\)\(-\)1) constants to \(i_{1}\) in the way of \(i_{p}=i_{1}+(p-1)\Delta n_{s}\), where \(p=2\), \(3\), \(\ldots\), \(P\) and \(\Delta n_{s}\) is a constant, and we will obtain another (\(P\)\(-\)1) index combinations. With these index combinations, we pick out another (\(P\)\(-\)1) subsequences from the original signal sequence fragment. Since the phase of this signal sequence fragment is linear, as mentioned in Section 2, it is obvious that any signal component contained in a signal sample in the (\(P\)\(-\)1) subsequences can be regarded as a result of the same component in the corresponding signal sample of the first subsequence shifting a phase of \(\Delta\varphi_{p}=(p-1)\Delta n_{s}(\Delta\varphi_{{\rm CFO}}+\Delta\varphi_{ m})\), \(p=2\), \(3\), \(\ldots\), \(P\). Thus, the other (\(P\)\(-\)1) signal subsequences can be denoted as \(s_{p}=\{s_{p}(l):l=1,2,\cdots,L\}\), \(p=2\), \(3\), \(\ldots\), \(P\), where  
\[ \begin{array}{ll}s_{p}(l)&=s_{1}(l)e^{j(p-1)\Delta n_{s}(\Delta\varphi_{{\rm CFO }}+\Delta\varphi_{m})}\\ &=\sum\
olimits_{m=0}^{M}c_{1}^{(m)}e^{j[\Delta i_{l}(\Delta\varphi_{{\rm CFO }}+\Delta\varphi_{m})+\varphi_{1}^{(m)}]}e^{j(p-1)\Delta n_{s}(\Delta\varphi_{ {\rm CFO}}+\Delta\varphi_{m})}+w_{p}(l)\end{array} . \tag{9}\]  
Since \(\Delta\varphi_{0}\) is known, we can compensate for this phase component in the signal and then integrate each subsequence to improve the SNR. The resulting signal of each subsequence can be expressed as follows:\[ \begin{array}{ll}s(p)&=\sum_{l=\ 1}^{L}s_{l}(l)e^{-jA_{l}A_{l}A_{\Phi 0}}\\ &=\ \sum_{l=\ 1}^{L}\sum_{m=\ 0}^{M}c_{1}^{(m)}e^{i[\Delta t_{l}(\Delta\varphi_{CG }+\Delta\varphi_{m}-\Delta\varphi_{0})+\varphi_{1}^{(m)}]}e^{j(p-1)\Delta t_{s}( \Delta\varphi_{CG}+\Delta\varphi_{m})}+\sum_{l=\ 1}^{L}w_{p}(l)e^{-j\Delta t_{l}A_{\Phi 0}}\\ &=\ \sum_{m=\ 0}^{M}e^{i(p-1)\Delta t_{s}(\Delta\varphi_{CG}+\Delta\varphi_{m})}c_{1} ^{(m)}e^{i\varphi_{1}^{(m)}}\sum_{l=\ 1}^{L}e^{j\Delta t_{l}(\Delta\varphi_{CG}+\Delta\varphi_{m}-\Delta\varphi_{0}) }+\sum_{l=\ 1}^{L}w_{p}(l)e^{-j\Delta t_{l}A_{\Phi 0}}\\ \end{array} , \tag{10}\]  
where \(p=1,\ 2,3,\ldots,P\). On this basis, let  
\[ \begin{array}{ll}\mathbf{s}\ =\ (s(1),s(2),\cdots,s(P))^{\mathbf{T}},\end{array}  \tag{11}\]  
\[