# Time-Domain Blind ICI Compensation in Coherent Optical FBMC/OQAM System
## 2 CO-FBMC/OQAM System with Laser Phase Noise  
In this section, the influence of PN is studied in CO-FBMC/OQAM systems, and the analytical expressions of the PN induced interference are derived. As shown in Figure 1, a single polarization CO-FBMC/OQAM BTB system is considered to investigate the PN induced interference and the performance of the proposed PNC algorithm. The single polarization BTB system is used to concentrate on the impact of laser PN in this paper. At the transmitter, the process of OQAM pre-processing transforms the complex-valued QAM symbols into real-valued symbols, which means that the real-valued symbols are formed with the real and imaginary parts of QAM symbols destaggered by half a QAM symbol period. These real-valued symbols are modulated (i.e., pulse amplitude modulation (PAM)) in such a way that the FBMC system maintains the same data rate as the OFDM system without cyclic-prefix. Then, the PAM symbols are mapped onto a set of orthogonal subcarriers generated by an inverse fast Fourier transform (IFFT). The following operation is a cost-efficient poly-phase network (PPN) for filter bank synthesis implementation. With the digital signal processing (DSP) at the transmitter, its discrete time-domain transmitted signal, \(s[k]\), can be expressed as,  
\[s[k]=\sum_{n=0}^{N_{S}-1}\sum_{m=0}^{M-1}a_{n,m}g_{n,m}[k] \tag{1}\]  
where \(a_{n,m}\) and \(g_{n,m}[k]\) are the PAM symbol and the synthesis filter at the (\(n\),\(m\))th time-frequency index, respectively; \(N_{S}\) is the total number of CO-FBMC/OQAM blocks; and \(M\) is the total number of subcarriers in each CO-FBMC/OQAM block. The (\(n\),\(m\))th synthesis filter is described as,  
\[g_{n,m}[k]=g[k-n\frac{M}{2}]\cdot e^{j\frac{2\pi}{M}mk}e^{j\psi_{n,m}} \tag{2}\]  
\[g[k]=\frac{(4\alpha k/T)\cos[\pi(1+\alpha)k/T]+\sin[\pi(1-\alpha)k/T]}{(\pi k/ T)[1-(4\alpha k/T)^{2}]} \tag{3}\]  
where the square root raised cosine function \(g[k]\) is employed as the prototype filter with length \(L_{g}\)[41], \(\psi_{n,m}=(n+m)/2\) is phase modulation factor, \(\alpha\) denotes the roll-off coefficient, and \(T\) represents the duration of the QAM complex symbols. Figure 1: Block diagram of CO-FBMC/OQAM BTB system. In the BTB CO-FBMC/OQAM system, under the impact of laser PN and additive Gaussian white noise (AWGN), the received signal \(r[k]\) can be written as,  
\[ \begin{array}{ll}r[k]&=e^{i\varphi[k]}s_{g}[k]+w[k]\\ &=e^{i\varphi[k]}\bigg{(}\sum\limits_{n=0}^{N_{S}-1}\sum\limits_{m=0}^{M-1}a_{ n,m}g[k-n\frac{M}{2}]\cdot e^{j\frac{2\pi(m+n)}{2}}e^{j\frac{2\pi(m+n)}{2}} \bigg{)}+w[k]\\ &=e^{i\varphi[k]}\bigg{(}a_{n_{0},m_{0}}g[k-n_{0}\frac{M}{2}]e^{j\frac{2\pi(m +n_{0})}{2}}e^{j\frac{2\pi(m+n_{0})}{2}}+\sum\limits_{(n,m)\
eq(n_{0},m_{0})}a_ {n,m}g[k-n\frac{M}{2}]e^{j\frac{2\pi(m+n)}{2}}e^{j\frac{\pi(m+n)}{2}}\bigg{)}+w[ k]\end{array}  \tag{4}\]  
Here, \(\varphi[k]\) denotes the PN in the time-domain, and the notation \(w[k]\) is the AWGN. The laser PN originates from either the external cavity laser at the transmitter or the local oscillator at the receiver. It is modeled as a random walk Wiener process, and the PN change follows a Gaussian distribution with zero mean and a specific variance \(2\pi\cdot\Delta v\cdot T_{S}\), where \(\Delta v\) is the combined laser linewidth, and \(T_{S}\) is the normalized symbol period. Here, \(T_{S}=(1/B_{r})/M\), and \(B_{r}\) is the baud rate of QAM symbols. The combined laser linewidth is defined as the sum of linewidths of the transmitter and receiver lasers. For the sake of simplicity, in the BTB system, the transmitter laser has the same linewidth as the received laser, and the combined laser linewidth is twice the linewidth of the single laser. At the receiver side, the incoming optical wave signal is first coherently detected in a 90\({}^{\circ}\) optical hybrid. Subsequently, the obtained digital data are sent to the DSP modules to recover the original data.