## 2 System Model
### Model System Setup  
In our model, we implement the periodic insertion of pilot signals using BPSK modulation with a rotation of \(\pi/4\) rad. This rotation helps to differentiate pilot signals from the square modulation used in data transmission. For the transmitter, we used square modulation schemes set through QPSK, 16 QAM, 64 QAM, and 256 QAM constellations. An example of the model transmitting pilots can be seen in Figure 1 for a 16 QAM modulation.  
The pilot symbols were periodically inserted into the transmitted signal, following every 32 randomly generated data symbols. In the following discussion, we demonstrate that this interval can be adjusted. These pilot symbols serve as reference points, enabling the interpolator to accurately track and recover the phase, even in the presence of phase noise. Figure 2 illustrates an example of the transmitted symbols and pilots: the red markers indicate the pilot symbols, while the blue markers represent the data symbols generated by the modulator prior to transmission through the channel.  
Consider the \(n\)th constellation symbol, \(x[n]\), which is transmitted over a complex additive white Gaussian noise (AWGN) channel, where it is subject to additive noise \(\eta[n]\), which models the amplified spontaneous emission (ASE) noise generated from inline amplifiers. Additionally, phase noise, resulting from the frequency and phase offset between the transmitter and local oscillator lasers, is modeled as a multiplicative factor \(e^{i[n]}\). Consequently, the received symbol \(y[n]\) is expressed as:\[y[n]=x[n]\ e^{i\theta[n]}+\eta[n], \tag{1}\]  
where \(\eta[n]\) is a Gaussian noise characterized by a zero mean AWGN with variance \(\sigma^{2}[n]\), generated by optical amplification [32]. In optical coherent systems, phase noise is primarily influenced by the linewidths of local oscillators and manifests itself as a random-walk process. This behavior is also affected by the SNR in the receiver [33]. Assuming an initial condition \(\theta(0)=0\) at time \(t=0\), the phase evolves over time and deviates from 0 due to the random-walk process. At any subsequent time \(t=\Delta t\), the phase difference measured during the symbol period \(T_{S}\) follows a Gaussian distribution, with its variance directly proportional to the line width \(\Delta\
u\)[34], as follows:  
\[E[(\theta(t=0)-\theta(t=\Delta t))^{2}]=2\pi\Delta\
u\Delta t. \tag{2}\]  
The phase noise at time \(t\) can be modeled as a Wiener process [1], as follows:  
\[\theta(t)=\sum_{n=-\infty}^{t}w[n]\, \tag{3}\]  
where \(w[n]\) is a random \(\mathcal{N}(0,2\pi\Delta\
u T_{S})\) Gaussian variable with zero mean and variance given by [32]:  
\[\sigma_{w}^{2}=2\pi\Delta\
u T_{S}. \tag{4}\]  
In our model, \(T_{S}\) denotes the symbol period, while \(\Delta\
u\) represents the combined linewidth of the transmitter laser and the local oscillator (LO). The phase difference between two adjacent symbols follows a zero-mean Gaussian distribution with a variance given in Equation (4).  
Figure 1: Red points denote the BPSK modulation for pilots, which are rotated by \(\pi/4\) rad and inserted within a 16 QAM square modulation.  
Figure 2: Red samples depict the pilots, while blue samples represent the data symbols.  
Following the analytical and geometrical arguments in [27; 35], the phase noise evolution is then described with good approximation as:  
\[\varphi[n]=\varphi[n-1]+w[n]. \tag{5}\]