# Time-Domain Blind ICI Compensation in Coherent Optical FBMC/OQAM System
## 4 Modified Algorithm Based on Overlapped Symbol Structure  
Different from OFDM symbols, neighboring temporal FBMC/OQAM blocks are overlapped with each other. Figure 4 shows the time-domain FBMC/OQAM transmitted blocks with overlapped structure. The \(i\)th sample of \(n\)th received FBMC/OQAM block can be given by  
\[r_{n}[i]=r[i+(n-1)\frac{M}{2}],\ i\in[1,KM] \tag{25}\]  
Then, the last _KM-M/2_ samples of \(n\)th FBMC/OQAM block are the same as the first _KM-M/2_ samples of \((n+1)\)th FBMC/OQAM block--i.e., \(r_{n}[j+M/2]=r_{n+1}[j]\), \(j\in[1,KM-M/2]\). As a consequence, the PN sample also satisfies that \(\varphi_{n}[j+M/2]=\varphi_{n+1}[j]\), where \(\varphi_{n+1}[j]\) is the \(j\)th sample of the \((n+1)\)th PN vector \(\varphi_{n+1}\). Let \(\mathbf{\tau}_{pre}\) and \(\mathbf{\tau}_{last}\) be the first and the last _KM-M/2_ rows of DCT basis \(\mathbf{\tau}\), repectively, and we can obtain that \(\mathbf{\tau}_{last}\mathbf{C}_{n}=\mathbf{\tau}_{pre}\mathbf{C}_{n+1}\). Using LS estimation, the DCT coefficient vector \(\mathbf{C}_{n}\) for \(n\)th CO-FBMC/OQAM block can be obtained as follows,  
\[\mathbf{C}_{n}=\left(\mathbf{\tau}_{last}^{T}\mathbf{\tau}_{last}\right)^{-1}\mathbf{\tau}_{ last}^{T}\mathbf{\tau}_{pre}\mathbf{C}_{n+1} \tag{26}\]  
If the first DCT coefficient vector \(\mathbf{C}_{1}\) is known, the other DCT coefficient vectors can be calculated according to the recursive expression in Equation (26). It is obviously unreasonable in such a way to calculate the other DCT coefficient vectors, because the updating between the PN vectors of \(\mathbf{\varphi}_{n}\) and \(\mathbf{\varphi}_{n+1}\) is not completely neglected. Moreover, two truncated matrices of \(\mathbf{\tau}_{pre}\) and \(\mathbf{\tau}_{last}\) have very small singular values owing to the circularity of the DCT basis matrix \(\mathbf{\tau}\). This leads to the significant degradation of the accuracy in the LS estimation. Therefore, an approximate expression between the DCT coefficient vectors of \(\mathbf{C}_{n}\) and \(\mathbf{C}_{n+1}\) is derived to achieve high estimate accuracy. Although the DCT basis matrix is not an orthogonal matrix in the proposed algorithm, the following condition is still satisfied:  
\[\mathbf{\tau}^{T}\mathbf{\tau}=b\mathbf{I} \tag{27}\]  
where \(b\) is a constant, and \(\mathbf{I}\) is a \(L\times L\) identity matrix. When the difference between the first and the last _M/2_ samples of \(\mathbf{\varphi}_{n}\) and \(\mathbf{\varphi}_{n+1}\) are ignored, \(\mathbf{\varphi}_{n+1}\) can be approximated as the _M/2_ circular shift of \(\mathbf{\varphi}_{n}\). Hence, the terms satisfy that  
\[\left\{ \begin{array}{l}\mathbf{\tau}\mathbf{C}_{n}=\mathbf{\varphi}_{n}\approx\mathbf{\mu}_ {-1}\mathbf{\varphi}_{n+1}=\mathbf{\mu}_{-1}\mathbf{\tau}\mathbf{C}_{n+1}\\ \mathbf{\tau}\mathbf{C}_{n+1}=\mathbf{\varphi}_{n+1}\approx\mathbf{\mu}_{+1}\mathbf{\varphi}_{n}= \mathbf{\mu}_{+1}\mathbf{\tau}\mathbf{C}_{n}\end{array} \right. \tag{28}\]  
Figure 4: Time-domain FBMC/OQAM transmitted blocks with overlapped structure. where \(\mu_{-1}\) and \(\mu_{+1}\) are two \(L_{g}\times L_{g}\) circular shift matrices,  
\[\left\{\begin{array}{l}\mu_{-1}=\left[ \begin{array}{cc}0_{\frac{M}{2}\times(L _{g}-\frac{M}{2})}&\mathbf{I}_{\frac{M}{2}}\\ \mathbf{I}_{L_{g}-\frac{M}{2}}&0_{(L_{g}-\frac{M}{2})\times\frac{M}{2}}\\ 0_{(L_{g}-\frac{M}{2})\times\frac{M}{2}}&\mathbf{I}_{L_{g}-\frac{M}{2}}\\ \mathbf{I}_{\frac{M}{2}}&0_{\frac{M}{2}\times(L_{g}-\frac{M}{2})}\end{array}  \right]\right. \tag{29}\]  
here, \(\mathbf{0}_{m\times n}\) denotes a \(m\times n\) zero matrix, and \(\mathbf{I}_{n}\) is a \(n\times n\) identity matrix. In combination with Equation (27), Equation (28) can be further expressed as  
\[\left\{ \begin{array}{l}\mathbf{C}_{n}=\mathbf{\gamma}_{-1}\mathbf{C}_{n+1}\\ \mathbf{C}_{n+1}=\mathbf{\gamma}_{+1}\mathbf{C}_{n}\end{array} \right. \tag{30}\]  
\[\left\{