# Time-Domain Blind ICI Compensation in Coherent Optical FBMC/OQAM System
## 3 Principle of the Phase Noise Compensation Method  
In this section, a blind PNC scheme is proposed to effectively compensate both CPE and ICI in CO-FBMC/OQAM systems. Figure 2 shows a block diagram of the proposed blind PNC scheme. In this method, a time-domain PNC model is firstly built by using the DCT approximation method, where each PN sample can be expanded as a linear combination of DCT coefficients and a group of DCT bases. To calculate these DCT coefficients in the time-domain model, it is necessary to pre-estimate the transmitted frequency-domain signals. As a result, the CPE is firstly compensated by using the M-BPS algorithm [28], and then the pre-decision process is performed on a part of the compensated signals to obtain the pre-estimated values of the transmitted signals. After implementing PPN for filter bank analysis and FFT, the M-BPS method is firstly used to compensate the CPE. In the M-BPS method, the signal is firstly rotated by several test phases. Figure 2: Block diagram of the proposed blind phase noise compensation scheme. The values of test phases are defined as \(\phi_{n,y}=(y/Y)\cdot\pi-\pi/2\), where \(y=1,2,\ldots,Y\), and \(Y\) is the total number of test phases. The rotated version of \(D_{n,m}\) can be given by  
\[\hat{D}_{n,m,y}=D_{n,m}\cdot e^{-j\phi_{n,y}} \tag{10}\]  
The estimated value of the CPE is obtained by minimizing the sum of distances between the hard decision of the received samples after test phase compensation and their projections on the real axis,  
\[\hat{\phi}_{n,y}=\operatorname*{argmin}_{\phi_{n,y}}\sum_{m=0}^{M-1}\left| \operatorname{Re}(\hat{D}_{n,m,y})-DD(\operatorname{Re}(\hat{D}_{n,m,y}))\right| \tag{11}\]  
where \(\operatorname{Re}(\cdot)\) denotes the real projection operator, and \(DD(\cdot)\) is the direct pre-decision operator. Subsequently, the conventional unwrap operation is performed to partly solve the problem of phase ambiguity [42]. Then, the received PAM symbol is obtained as follows:  
\[\hat{a}^{\prime}_{n,m}=\operatorname{Re}(\hat{D}_{n,m})=\operatorname{Re}(D_{ n,m}\cdot e^{-j\hat{\phi}_{m}^{*}}) \tag{12}\]  
Next, the estimated values \(\hat{a}_{n,m}\) of the transmitted signal are recovered via the pre-decisions upon the signal \(\hat{a}^{\prime}_{n,m}\). Since the performance of the proposed method strongly depends on the pre-decisions, the negative influence of the decision errors should be considered. Here, an effective approach is taken to reduce the decision errors. As shown in Figure 3, taking the 16-OQAM as an example, the constellation points are classified into 4 classes. When the constellation point \(\hat{D}_{n,m}\) falls in the border regions between two neighbouring classes, the decision errors often occur. Then, three shadow border regions are regarded as the high decision error probability regions, where the parameter \(\delta\) is defined as is the width of a shadow rectangle, and it decides the selected range of the high decision error probability regions. Therefore, if \(\hat{D}_{n,m}\) falls in these high decision error probability regions, these signals are thrown away and not used to perform pre-decisions; while \(\hat{D}_{n,m}\) does not fall in the shadow regions, a hard pre-decision is made normally. However, if excessive signals are removed from the pre-decision process, this can lead to the performance degradation of the proposed method. Thus, the pre-decision parameter \(\delta\) needs to be optimized to achieve good performance. Figure 3: Taking the 16-OQAM as an example, the signals with high decision error probability in three shadow regions (\(\hat{a}^{\prime}_{n,m}\in[-2-\delta/2,-2+\delta/2]\cup[-\delta/2,\delta/2]\cup [2-\delta/2,2+\delta/2]\)) not being used to perform the pre-decision. Secondly, a time-domain PNC model is built to estimate the PN with the aid of DCT.