# Time-Domain Blind ICI Compensation in Coherent Optical FBMC/OQAM System
## 2 CO-FBMC/OQAM System with Laser Phase Noise
After a time synchronization is performed perfectly, a time-frequency translation can be completed by a hardware-efficient structure of PPN for filter bank analysis followed by a fast Fourier transform (FFT). Eventually, the complex-valued received demodulated signal at the (\(n_{0}\),\(m_{0}\))th time-frequency index can be written as,  
\[ \begin{array}{ll}D_{n_{0},m_{0}}&=\sum\limits_{k=0}^{L_{g}-1}r[k]g^{*}_{n_{0 },m_{0}}[k]\\ &=\sum\limits_{k=0}^{L_{g}-1}r[k]g[k-n_{0}\frac{M}{2}]\cdot e^{-j\frac{2\pi(m _{0}+n_{0})}{2}}e^{-j\frac{\pi(m_{0}+n_{0})}{2}}\end{array}  \tag{5}\]  
here, the upper script \"\(*\)\" denotes conjugate operation. By substituting Equation (4) into Equation (5), \(D_{n_{0},m_{0}}\) can be further expressed as [18],  
\[ \begin{array}{ll}D_{n_{0},m_{0}}&=\sum\limits_{k=0}^{L_{g}-1}r[k]g[k-n_{0} \frac{M}{2}]\cdot e^{-j\frac{2\pi}{M}m_{0}k}e^{-j\frac{\pi(m_{0}+n_{0})}{2}}\\ &=\eta_{0,0}a_{n_{0},m_{0}}+\sum\limits_{(p,q)\
eq(0,0)}a_{n_{0}+p,m_{0}+q} \eta_{p,q}+N_{n_{0},m_{0}}\end{array}  \tag{6}\]  
here, \(N_{n_{0},m_{0}}\) denotes the filter processed additive noise, and \(\eta_{p,q}\) is defined as  
\[ \begin{array}{ll}\eta_{p,q}&=\sum\limits_{k}e^{-j\frac{\pi(p+q)}{2}}e^{-j \frac{2\pi}{M}k}g[k-n_{0}\frac{M}{2}]\\ &\cdot g^{*}[k-\frac{M}{2}(n_{0}+p)]\bigg{[}\frac{1}{M}\sum\limits_{s=0}^{M-1 }e^{-j\frac{2\pi}{M}n_{0}}e^{j\varphi_{n_{0}}[s]}\bigg{]}\end{array}  \tag{7}\]  
at \((p,q)=(0,0),\ \eta_{0,0}\) is described by the following expression. \[ \begin{array}{ll}\eta_{0,0}&=\sum\limits_{k}g\Big{[}k-n_{0}\frac{M}{2} \Big{]}\cdot g^{*}\Big{[}k-n_{0}\frac{M}{2}\Big{]}\left[\sum\limits_{M}^{M-1}e^ {j\varphi_{n_{0}}[s]}\right]\\ &=\frac{1}{M}\sum\limits_{s=0}^{M-1}e^{j\varphi_{n_{0}}[s]}\end{array}  \tag{8}\]  
In Equation (6), \(\eta_{0,0}\) is defined as the CPE for the information symbol \(a_{n_{0},m_{0}}\), and it leads to phase rotation of the received constellation diagram. The second item of Equation (6) is defined as the ICI, which causes divergence of the received constellation diagram. The ICI in CO-FBMC/OQAM systems is a convolution between the coefficient \(\eta_{p,q}\) and PAM symbols from different FBMC/OQAM blocks, while the ICI in CO-OFDM systems is a convolution between the ICI coefficients and different sub-carriers in one OFDM symbol. Hence, it is more difficult to compensate the ICI in CO-FBMC/OQAM systems. In many previous studies, assuming that the PN time-domain samples are the same in one FBMC/OQAM symbol period, the complex-valued received signal \(D_{n_{0},m_{0}}\) can be approximately rewritten as [25; 26; 27; 28; 29],  
\[ \begin{split} D_{n_{0},m_{0}}&\approx\eta_{0,0}a_{n_ {0},m_{0}}+\eta_{0,0}\sum\limits_{k=0}^{L_{q}-1}\sum\limits_{(p,q)\
eq(0,0)}a_{ n_{0}+p,m_{0}+q}g_{n_{0}+p,m_{0}+q}[k]g_{n_{0},m_{0}}^{*}[k]+N_{n_{0},m_{0}}\\ &=\eta_{0,0}(a_{n_{0},m_{0}}+ju_{n_{0},m_{0}})+N_{n_{0},m_{0}} \end{split}  \tag{9}\]  
where \(u_{n_{0},m_{0}}\) is called the IMI in CO-FBMC/OQAM systems, and it is the interference term which results from neighboring PAM symbols on the interested (\(n_{0}\),\(m_{0}\))th position. Based on the approximation expression above, various studies have focused on the cancellation of the IMI by using pilot symbols [25; 26; 27]. However, only CPE compensation effect can be improved through the complete removal of the IMI. Because the ICI effect is the convolution in frequency domain, it is useless to perform the ICI compensation by removing the IMI thoroughly. So in the rest of the paper, the blind ICI compensation is proposed for the CO-FBMC/OQAM systems.