# Time-Domain Blind ICI Compensation in Coherent Optical FBMC/OQAM System
## 3 Principle of the Phase Noise Compensation Method
 \tag{18}\]  
where the symbol \(V_{n,m}^{l}\) satisfies  
\[V_{n,m}^{l}=\sum\limits_{i=0}^{L_{q}-1}r_{n}[i]g[i-n\frac{M}{2}]\tau_{i,l}e^{- j\frac{2n}{M}mi}e^{-j\frac{n(m+n)}{2}} \tag{19}\]  
By combining Equation (17) and Equation (18), the received symbol \(A_{n,m}\) can be expressed as  
\[A_{n,m}\approx\sum\limits_{l=0}^{L-1}C_{n}(l)V_{n,m}^{l}-\xi_{n,m} \tag{20}\]  
While ignoring the noise term \(\xi_{n,m}\), the corresponding estimated vector \(\hat{\mathbf{A}}_{n}\) can be approximated as,  
\[\hat{\mathbf{A}}_{n}=\mathbf{V}_{n}\mathbf{C}_{n} \tag{21}\]  
where \(\hat{\mathbf{A}}_{n}=\left[\hat{A}_{n,0},\hat{A}_{n,1},\ldots,\hat{A}_{n,M-1} \right]^{T}\) and \(\mathbf{V}_{n}=\left[\mathbf{V}_{n}^{0},\mathbf{V}_{n}^{1},\ldots,\mathbf{V}_ {n}^{l-1}\right]\left(m\in[0,M-1]\right)\). \(\hat{A}_{n,m}\) is the estimated value of \(A_{n,m}\), and the \(M\times 1\) vector \(\mathbf{V}_{n}^{l}=\left[V_{n,0}^{l},V_{n,1}^{l},\ldots,V_{n,M-1}^{l}\right]^ {T}\) is a column of the matrix \(\mathbf{V}_{n}\). Taking the real part of Equation (21) in the left and right sides, the equation is changed into  
\[\hat{\mathbf{a}}_{n}=\mathbf{P}_{n}\mathbf{Q}_{n} \tag{22}\]  
where \(\hat{\mathbf{a}}_{n}=\left[\hat{a}_{n,0},\hat{a}_{n,1},\ldots,\hat{a}_{n,M-1}\right]\), \(\mathbf{P}_{n}=\left[\mathrm{Re}(\mathbf{V}_{n})-\mathrm{Im}(\mathbf{V}_{n})\right]\), \(\mathbf{Q}_{n}=\left[\mathrm{Re}(\mathbf{C}_{n}^{T})\ \mathrm{Im}(\mathbf{C}_{n}^{T})\right]^ {T}\). \(\hat{a}_{n,m}\) is the estimated value of the transmitted PAM symbol, and \(\mathrm{Im}(\cdot)\) denotes the imaginary projection operator. As above-mentioned, in order to reduce the decision errors, only partially estimated values of the transmitted signal can be obtained by making a pre-decision on the the constellation points in the low decision error probability regions. When the optimized parameter \(\delta\) is selected by balancing performance and complexity of the proposed algorithm, a \(Z_{n}\times 1\) vector is picked out to perform the pre-decision, where \(Z_{n}\) is the total number of the pre-estimated transmitted signals in the \(n\)th FBMC/OQAM block. The pre-estimated transmitted signal vector can be given by \(\mathbf{S}_{n}\hat{\mathbf{a}}_{n}\), where the permutation matrix \(\mathbf{S}_{n}=\left[\hat{A}_{t_{1}},\hat{A}_{t_{2}},\ldots,\hat{A}_{t_{Z_{n}} }\right]^{T}\) is a \(Z_{n}\times M\) matrix, \(t_{2}\) (\(z=1\), \(2\), \(\ldots\), \(Z_{n}\)) represents the subcarrier index of \(z\)th pre-estimated transmitted signal, and \(A_{t_{z}}\) is a \(M\times 1\) vector \(\underbrace{[0,\ldots,0,1}_{t_{2}-1},\underbrace{0,\ldots,0}_{M-t_{2}}]^{T}\). Subsequently, Equation (22) can be rewritten as  
\[\mathbf{S}_{n}\hat{\mathbf{a}}_{n}=\mathbf{S}_{n}\mathbf{P}_{n}\mathbf{Q}_{n} \tag{23}\]  
Finally, the least square (LS) solution of the unknown DCT coefficient vector can be easily found as follows:  
\[\mathbf{Q}_{n}=\left((\mathbf{S}_{n}\mathbf{P}_{n})^{T}(\mathbf{S}_{n}\mathbf{ P}_{n})\right)^{-1}(\mathbf{S}_{n}\mathbf{P}_{n})^{T}(\mathbf{S}_{n}\hat{\mathbf{a}}_{n}) \tag{24}\]  
Once the DCT coefficient vector \(\mathbf{Q}_{n}\) is obtained, the PN, including the CPE and ICI, can be compensated via Equation (22).