# A Noise-Tolerant Carrier Phase Recovery Method for Inter-Satellite Coherent Optical Communications
## 2 System Model  
The transmitter topology of the inter-satellite coherent optical communication system is illustrated in Figure 1. An independent and identically distributed (i.i.d) data sequence of length \(K\), represented as \(\mathbf{b}=\{b_{k}|k=0,1,\ldots,K-1\}\), is processed utilizing a field programmable gate array (FPGA). \(b_{k}\) undergoes encoding via the Reed-Solomon (RS) encoder, resulting in the encoded sequence \(\mathbf{c}=\{c_{k}|k=0,1,\ldots,K-1\}\). Subsequently, \(c_{k}\) is mapped onto complex constellation symbols \(\mathbf{a}=\{a_{k}|k=0,1,\ldots,K-1\}\) utilizing the quadrature phase shift keying (QPSK) modulation scheme. \(a_{k}\) is then shaped through a unit-energy transmit filter, denoted as \(h(t)\), before being converted into analog signals by digital-to-analog converters (DACs). The Mach-Zehnder modulator (MZM) [24,25] is  
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employed to modulate the laser carrier with the analog signal. Ultimately, the modulated signal is transmitted into free space through the optical antenna. The transmitted signal is disturbed by Gaussian additive white noise (AWGN) after transmission and the signal received by the optical antenna can be expressed as follows [8]:  
\[r(t)=\sqrt{E_{s}}\sum_{k=0}^{K-1}a_{k}h(t-kT)e^{\left[2\pi f_{c}t+\theta_{c}(t) \right]}+n(t), \tag{1}\]  
where \(E_{s}\) represents the energy per symbol over the interval of \(T\), while \(f_{c}\), \(\theta_{c}(t)\) denote the carrier frequency and the PN of the carrier, respectively. The notation \(n(t)\sim\mathcal{CN}(0,N_{0}/2)\) signifies the complex AWGN with zero mean and variance \(N_{0}/2\), where \(N_{0}\) is the single-sided power spectral density. The PN \(\theta_{c}(t)\) can be modeled as a Wiener process [18; 26], expressed as follows:  
\[\theta_{c}(t)=\int_{-\infty}^{t}G(x)\mathrm{d}x, \tag{2}\]  
where \(G(x)\sim\mathcal{N}(0,\sigma_{g}^{2})\) is a random variable that follows a Gaussian distribution with zero mean and variance \(\sigma_{g}^{2}=2\pi\delta_{v}T\), with \(\delta_{v}\) representing the 3 dB linewidth of the carrier. In Figure 2a, the waveform of \(\theta_{c}(t)\) is depicted, showcasing its continuous and gradually varying nature. Taking the black curve as an example, the process initiates from an initial value of zero, and over time, the PN evolves due to the accumulated random Gaussian increments. This results in a phase trajectory that fluctuates over time, reflecting the cumulative impact of random noise. In Figure 2b, the statistical characteristics of the phase noise increments are presented, where the probability density function (PDF) of the phase increment over small time intervals is plotted. It can be observed that the PDF of the increments closely aligns with a Gaussian distribution. Figure 1: The transmitter topology of inter-satellite coherent optical communication system. The red line and red arrow represent the optical signal. In the receiver depicted in Figure 3, the received signal \(r(t)\) is mixed with the LO \(e^{i[2\pi f_{\text{LO}}+\theta_{\text{LO}}(t)]}\), where \(f_{\text{LO}}\) and \(\theta_{\text{LO}}\) denote the frequency and PN of the LO, respectively, through a \(90^{\circ}\) optical hybrid. The optical outputs from the hybrid are subsequently transformed into analog electrical signals by a balanced detector, resulting in the currents for the in-phase (I) and quadrature (Q) components, as follows [27]:  
\[I(t)=\sqrt{E_{s}}a_{k}\text{cos}[2\pi\Delta ft+\theta_{c}(t)-\theta_{\text{LO} }(t)]+n_{\text{I}}(t), \tag{3}\]  
\[Q(t)=\sqrt{E_{s}}a_{k}\text{sin}[2\pi\Delta ft+\theta_{c}(t)-\theta_{\text{LO} }(t)]+n_{\text{Q}}(t), \tag{4}\]  
where \(\Delta f=f_{c}-f_{\text{LO}}\) represents the residual Doppler FO, which arises from compensation errors due to the high-speed relative motion between satellites. The notations \(n_{\text{I}}(t)\) and \(n_{\text{Q}}(t)\) denote the AWGN affecting the I and Q components, respectively.