## 2 Principle  
### Phase Recovery Theory  
In RoF systems, phase noise caused by laser linewidth generates intricate interference. Conventional phase recovery algorithms encounter difficulties in accurately extracting and compensating for phase fluctuations [24]. Neural networks characterize phase noise in the channel by training on extensive datasets and exhibiting distinct benefits [25; 26]. The QPSK symbol sequence produced by the transmitter constitutes a time-domain signal. The width of a laser's light beam causes the signal's phase to shift over time, resulting in random changes in the phase that can be considered a sequence of offsets \(\{\theta(n)\}\). Mathematically, we represent the received signal as complex exponential modulation:  
\[y(n)=x(n)\cdot e^{i\theta(n)}*h(n)+n(n) \tag{1}\]  
where \(\theta(n)\) represents the phase noise of the \(n\)-th symbol, \(*\) denotes the convolution operation, \(h(n)\) is the channel impulse response, and \(n(n)\) is additive Gaussian white noise. By decomposing the time-domain received signal into its real and imaginary components, we formulate a two-dimensional input tensor:  
\[Input= \begin{bmatrix}a(n)\\ b(n)\end{bmatrix} \in R^{M\times 2} \tag{2}\]  
where \(a(n)\) and \(b(n)\) denote the real and imaginary components, \(M\) signifies the duration of the time frame, and the two channels correspond to the real-part and imaginary-part channels. Figure 1 demonstrates the two-dimensional structure of the input data. Continuous symbol segments are retrieved using a sliding window that encompasses the current symbol and \(\frac{M-1}{2}\) neighboring symbols on either side. Each row of the input tensor shows the real and imaginary parts of a single symbol, and each column relates to the real or imaginary parts of nearby symbols, forming a two-dimensional feature matrix that helps with 2D convolution extraction. The temporal direction refers to the phase continuity of neighboring symbols, whereas the chan  
Figure 1: Two-dimensional structure of the input data. nel direction pertains to the amplitude-phase mapping connection between the real and imaginary components. Under optimal circumstances, the real and imaginary components of the channel fulfil \(a^{2}+b^{2}=1\). In the event of a phase deflection \(\theta\), according to [PERSON]'s formula \(e^{i\theta}=\cos\theta+j\sin\theta\), the signal is expressed as:  
\[(a+jb)(\cos\theta+j\sin\theta)=a\cos\theta-b\sin\theta+j(a\sin\theta+b\cos\theta) \tag{3}\]  
The new real component \(a^{\prime}=a\cos\theta-b\sin\theta\) and the imaginary component \(b^{\prime}=a\sin\theta+b\cos\theta\) are derived. Post-normalization, the total number of squares remains equal to 1; however, the phase has altered. Through extensive data training, the convolution kernel can recognize the \"abnormal\" coupling pattern induced by phase rotation. The convolution kernel weights will enhance the response to this pattern, thereby extracting the attributes of phase rotation. Phase correction is analogous to doing a complex conjugate rotation on the incoming signal:  
\[\hat{y}(n)=y(n)\cdot e^{-j\hat{\theta}(n)}=(a(n)+jb(n))\big{(}\cos\hat{\theta} (n)-j\sin\hat{\theta}(n)\big{)} \tag{4}\]  
Upon expansion, the real component is \(\hat{a}(n)=a(n)\cos\hat{\theta}(n)+b(n)\sin\hat{\theta}(n)\), while the imaginary component is \(\hat{b}(n)=-a(n)\sin\hat{\theta}(n)+b(n)\cos\hat{\theta}(n)\). The 2D-CNN does the inverse transformation of phase deflection by acquiring the weights of the convolution kernel. The mathematical representation of the convolution operation is:  
\[\mathbf{Y}^{(l)}=\text{Re}LU\big{(}\mathbf{X}^{(l)}*\mathbf{K}^{(l)}+\mathbf{ B}^{(l)}\big{)} \tag{5}\]  
Here, \(\mathbf{X}^{(l)}\) denotes the output of the \(l\)-th convolutional layer, \(\mathbf{K}^{(l)}\) signifies the convolution kernel, and \(\mathbf{B}^{(l)}\) indicates the bias. ReLU is an activation function that imparts nonlinear properties, allowing the network to learn intricate phase-mapping correlations.