# Time-Domain Blind ICI Compensation in Coherent Optical FBMC/OQAM System
## 3 Principle of the Phase Noise Compensation Method
The compensated signal \(\hat{r}_{n}[i]\) can be obtained by multiplying the received signal \(r_{n}[i]\) by the complex conjugate of the estimate of PN,  
\[\hat{r}_{n}[i]=r_{n}[i]e^{-j\phi_{n}[i]} \tag{13}\]  
where \(\varphi_{n}[i]\) denotes the \(i\)th time-domain PN sample in the \(n\)th CO-FBMC/OQAM block. The high frequency components in the PN can be neglected, and the complex conjugate of PN can be expressed as the linear combination of a group of DCT basis and DCT coefficients [37],  
\[\mathbf{\Phi}_{n}\approx\mathbf{\tau}\mathbf{C}_{n} \tag{14}\]  
where \(\mathbf{\Phi}_{n}=\left[e^{-j\phi_{n}[0]},e^{-j\phi_{n}[2]},\ldots,e^{-j\phi_{n}[ KM-1]}\right]^{T}\) and \(\mathbf{C}_{n}=\left[C_{n}(0),C_{n}(1),\ldots,C_{n}(L-1)\right]^{T}\) are \(L\times 1\) unknown DCT coefficient vector. Here, \([\cdot]^{T}\) is the transpose operator, and \(L\) refers to the length of DCT coefficients. The \(L_{g}\times L\) matrix \(\mathbf{\tau}\) of DCT basis is given by  
\[\mathbf{\tau}=\left[ \begin{array}{cccc}\frac{\sqrt{2}}{2}&\cos(\frac{\pi(1+0.5 )}{KM})&\cos(\frac{2\pi(1+0.5)}{KM})&\cdots&\cos(\frac{(L-1)\pi(1+0.5)}{KM})\\ \frac{\sqrt{2}}{2}&\cos(\frac{\pi(2+0.5)}{KM})&\cos(\frac{2\pi(2+0.5)}{KM})& \cdots&\cos(\frac{(L-1)\pi(2+0.5)}{KM})\\ \frac{\sqrt{2}}{2}&\cos(\frac{\pi(3+0.5)}{KM})&\cos(\frac{2\pi(3+0.5)}{KM})& \cdots&\cos(\frac{(L-1)\pi(3+0.5)}{KM})\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ \frac{\sqrt{2}}{2}&\cos(\frac{\pi(KM+0.5)}{KM})&\cos(\frac{2\pi(KM+0.5)}{KM})& \cdots&\cos(\frac{(L-1)\pi(KM+0.5)}{KM})\\ \end{array} \right] \tag{15}\]  
substituting Equation (14) into Equation (13), the compensated temporal signal \(\hat{r}_{n}[i]\) can be rewritten as,  
\[\hat{r}_{n}[i]=\sum_{l=0}^{L-1}r_{n}[i]\tau_{i,l}\mathbf{C}_{n}(l) \tag{16}\]  
hence, the frequency domain compensated symbol after analysis processing can be expressed as,  
\[ \begin{array}{ll}\hat{R}_{n,m}&=\sum\limits_{i=0}^{L_{g}-1}\hat{r}_{n}[i]g[i -n\frac{M}{2}]\cdot e^{-j\frac{2\pi}{M}mi}e^{-j\frac{\pi(m+n)}{2}}\\ &=\sum\limits_{i=0}^{L_{g}-1}e^{-j\phi_{n}[i]}r_{n}[i]g[i-n\frac{M}{2}]\cdot e ^{-j\frac{2\pi}{M}mi}e^{-j\frac{\pi(m+n)}{2}}\\ &=\sum\limits_{i=0}^{L_{g}-1}e^{-j\phi_{n}[i]}\Big{(}e^{j\varphi_{n}[i]}s_{n}[ i]+w_{n}[i]\Big{)}g[i-n\frac{M}{2}]\cdot e^{-j\frac{2\pi}{M}mi}e^{-j\frac{\pi(m+n)}{2}} \\ &\approx\underbrace{\sum\limits_{i=0}^{L_{g}-1}s_{n}[i]g[i-n\frac{M}{2}] \cdot e^{-j\frac{2\pi}{M}mi}e^{-j\frac{\pi(m+n)}{2}}}_{A_{n,m}}+\xi_{n,m}\end{array}  \tag{17}\]  
where \(A_{n,m}\) is the received symbol when the PN is perfectly compensated -- i.e., the transmitted PAM symbol \(a_{n,m}\) can be recovered by taking real part of \(A_{n,m}\) -- and \(\xi_{n,m}\) is the noise term. By replacing \(\hat{r}_{n}[i]\) from Equation (16) into Equation (17), the symbol \(\hat{R}_{n,m}\) is rewritten into \[ \begin{array}{ll}\hat{R}_{n,m}&=\sum\limits_{i=0}^{L_{q}-1}e^{-j\phi_{n}[i]}r_{n} [i]g[i-n\frac{M}{2}]\cdot e^{-j\frac{2n}{M}mi}e^{-j\frac{n(m+n)}{2}}\\ &=\sum\limits_{i=0}^{L_{q}-1}\sum\limits_{l=0}^{L_{q}-1}C_{n}(l)\tau_{i,l}r_{n} [i]g[i-n\frac{M}{2}]\cdot e^{-j\frac{2n}{M}mi}e^{-j\frac{n(m+n)}{2}}\\ &=\sum\limits_{l=0}^{L-1}C_{n}(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}}\\ &=\sum\limits_{l=0}^{L-1}C_{n}(l)V_{n,m}^{l}\end{array}