Convolutional Neural Network-Based Fiber Optic Channel Emulator and Its Application to Fiber-Longitudinal Power Profile Estimation

Abstract

This paper proposes an accuracy enhancement method for fiber-longitudinal power profile estimation (PPE) based on convolutional neural networks (CNN). Two types of CNNs are designed. The first network treats different polarization streams identically and is denoted as CNN. The second network considers the difference between the contributions of different polarization streams to the nonlinear phase shift and is denoted as enhanced CNN (ECNN). The numerical simulation results confirm the effectiveness of the method for a 64 Gbaud/s quadrature phase-shift keying (QPSK) polarization-division-multiplexed (PDM) coherent optical communication system with a fiber length of 320 km. The effects of finite impulse response (FIR) filter length, power into the fiber, and polarization mode dispersion on the PPE accuracy are examined. Finally, the results of the proposed method are monitored in the presence of several simultaneous power attenuation anomalies in the fiber optic link. It is found that the accuracy of the PPE substantially improves after using the proposed method, achieving a relative gain of up to 71%. When the modulation format is changed from QPSK to 16-ary quadrature amplitude modulation (16-QAM), and the fiber length is increased from 360 km to 480 km, the proposed method is still effective. This work provides a feasible solution for implementing fiber-longitudinal PPE, enabling significantly improved estimation accuracy in practical applications.

1. Introduction

Robust optical performance monitoring (OPM) plays a crucial part in the reliable operation of modern optical networks, which are evolving into agile, disaggregated, and dynamic networks [1,2]. In recent years, a new OPM technology, called the fiber-longitudinal power profile estimation (PPE), has attracted a significant amount of attention [3,4]. It is based upon the idea of digital longitudinal monitoring (DLM), where “digital” indicates that the monitoring function is performed using digital signal processing (DSP) at the receiver side, and “longitudinal” refers to the distributed monitoring of key parameters along the fiber optic link. The existing DLM-based OPM can monitor the following parameters: optical power profile [5], chromatic dispersion (CD) [6], amplifier gain tilt [7], optical filter detuning [8], optical-signal-to-noise ratio (OSNR) [9], polarization-dependent loss (PDL) [10], and differential group delay (DGD) [11]. The DLM-based optical PPE, often called in-site PPE, can predict power evolution along an optical link in a distributed manner. The advantage of fiber-longitudinal PPE over optical time–domain reflectometer (OTDR) is that no dedicated equipment is needed to perform manual on-site span-by-span measurement. Therefore, fiber-longitudinal PPE is chosen as the research topic in this paper.

The existing fiber-longitudinal PPE methods can be classified into two main types: correlation-based methods (CMs) and minimum-mean-square-error-based methods (MMSEs) [4]. It has been analytically shown in a previous work [4] that CM-based methods have limited spatial resolution and measurement accuracy, even in noise- and distortion-free environments. In contrast, MMSE-based methods are not restricted by this limitation. There are several variants of MMSE-based methods, such as Volterra-based MMSE [12], linear least squares (LS) [13], and deep-learning networks [14]. In terms of the oversampling rate for signal reconstruction, fiber-longitudinal PPE can be classified into waveform-level and symbol-level methods [15]. In the waveform-level methods, the reference signal is oversampled by a factor of two, whereas the symbol-level methods do not involve any up-sampling.

For the fiber-longitudinal PPE, the emulation of a fiber optic channel has a crucial impact on the estimation accuracy, especially for deep learning-based methods. It is well known that the propagation of an optical signal in an optical fiber follows a nonlinear Schrödinger Equation. The LS-based analytical estimation methods usually establish a fiber channel emulator based on the regular perturbative solution of the nonlinear Schrödinger Equation [13]. On the contrary, the deep learning-based methods usually create an artificial neural network that emulates the fiber optic channel according to the split-step Fourier (SSFM) algorithm for the nonlinear Schrödinger Equation. For example, an optical fiber channel emulator was proposed using a deep learning network (DNN), leveraging asymmetric SSFM to implement fiber-longitudinal PPE [3]. Another study [16] designed a double-effect digital back propagation (DBP) algorithm to simultaneously enable data communication and sensing of nonlinear fiber parameters. However, the correlation between the signal power of neighboring symbols when constructing DNNs was ignored in both of these previous works [3,16]. It is well known that chromatic dispersion causes an optical pulse to spread as it propagates through an optical fiber, which “spills” the optical signal power into neighboring symbols. The nonlinear phase shift at a specific symbol location is correlated with the power of multiple consecutive symbols. Furthermore, optical signals from different polarization streams contribute differently to the nonlinear phase shift [17]. Such a phenomenon may be used to enhance the robustness of fiber channel emulators and improve the PPE estimation accuracy. However, only a few studies exist on this topic.

To cover this research gap and further enrich the fiber-longitudinal PPE implementation methods, this paper proposes a fiber channel emulator based on convolutional neural networks (CNNs). It can account for the power overlap of consecutive symbols and the discrepancy between different polarization streams, which are issues often ignored in published works. The research results demonstrate that the emulator can be successfully employed with fiber-longitudinal PPE and is applicable to 64 Gbaud/s fiber optic transmission systems. The findings of this paper offer a valuable reference for designing more efficient fiber-longitudinal PPE methods. The rest of this paper is organized as follows. Section 2 presents the overall framework of the method and describes the design philosophies of the channel emulator and the signal processing flow at the receiver side. Section 3 describes in detail the numerical simulation methodology for the 64 Gbaud/s fiber optic transmission system and provides the key parameters of the CNNs. The influences of polarization mode dispersion (PMD) and fiber launch power are carefully analyzed, and the PPE accuracy with and without the CNNs is also compared. Section 4 discusses the advantages and limitations of the proposed method. The conclusions of this paper are presented in Section 5.

2. Principles of the Fiber-Longitudinal PPE Based on CNN

In this paper, a typical long-haul M-ary quadrature amplitude modulation (M-QAM) coherent fiber optic transmission system is considered. However, the signal processing is tailored for both the transmitter and receiver sides. The holistic configuration of the fiber-longitudinal PPE based on the proposed CNN is shown in Figure 1. The DSP in the transmitter primarily consists of M-QAM symbol mapping, pulse shaping, and digital pre-emphasis. The DSP for implementing PPE in a coherent receiver is also shown in Figure 1. First, the electrical signal from coherent detectors is matched-filtered, and resampled. Second, the output signal goes through CD compensation, adaptive equalization using the Least-Mean-Square (LMS) algorithm and pilot-assisted carrier phase recovery (CPR), respectively. A perfect compensation of the frequency offset is assumed in this paper. After CPR, the DSP in the coherent optical receiver is divided into upper and lower branches. In the upper branch, the signal passes through the DSP chain for reference waveform reconstruction. The residual phase noise (RPN) within the signal is estimated and compensated in the upper branch, and subsequently, a symbol decision is rendered. In this paper, a blind phase search (BPS) algorithm with 64 test phase angles is employed to estimate the RPN. Next, the reference symbols reloaded with RPN are produced and fed into the CNNs in the waveform reconstruction module. In the lower branch, the signal output from the CPR is reloaded with the CD, divided into batches, and finally fed into the channel emulator, as depicted in Figure 2 and Figure 3 for DBP.

The fiber channel emulator is the most important part of DSP shown in Figure 1. The method used for constructing the channel emulator is vital to improving the accuracy of PPE. The channel emulator is a virtual and digital counterpart of a real fiber that simulates the reverse transmission of a signal. When the generated optical signal is launched into a standard single-mode fiber (SSMF), its propagation satisfies the nonlinear Schrödinger Equation. Due to the random variation and coupling of the state of polarization (SOP) in a long-haul SSMF, the following Markov Equation is considered here instead of the Markov-PMD Equations:

$${\displaystyle \frac{\partial A}{\partial z}}=-{\displaystyle \frac{\alpha }{2}}A+j{\displaystyle \frac{{\beta }_2}{2}}{\displaystyle \frac{{\partial }^{2}A}{\partial t^2}}-j{\displaystyle \frac{8}{9}}\gamma {‖A‖}^{2}A.$$

(1)

Here, $A={\left[\begin{array}{cc}{A}_x& A_y\end{array}\right]}^T$, where $A_x$ represents the optical signal waveform on the x polarization stream, and $A_y$ represents the y polarization stream. Subsequently, ${‖A‖}^2={\left|A_x\right|}^2+{\left|A_y\right|}^2$ represents the optical signal power. $\alpha$ is the fiber attenuation coefficient, ${\beta }_2$ is the CD coefficient, and $\gamma$ is the nonlinear coefficient. Let $A\left(z,t\right)=\sqrt{P_{ch}}{A}^{\prime }\left(z,t\right)e^{-{\displaystyle {\int }_0^z\alpha \left(\xi \right)/2d\xi }}$, which is substituted into Equation (1) to obtain the following:

$${\displaystyle \frac{\partial A^{\prime }}{\partial z}}={\displaystyle \frac{j}{2}}{\beta }_2{\displaystyle \frac{{\partial }^2}{\partial t^2}}{A}^{\prime }-j{\gamma }^{\prime }\left(z\right){‖A^{\prime }‖}^2{A}^{\prime },$$

(2)

$${\gamma }^{\prime }\left(z\right)={\displaystyle \frac{8}{9}}\gamma \left(z\right)P_{ch}\exp \left(-{\displaystyle {\int }_0^z\alpha \left(z^{\prime }\right)d{z}^{\prime }}\right).$$

(3)

The reverse propagation of the signal in the virtual fiber is numerically approximated in the digital domain using the DBP based on the SSFM algorithm, where the virtual fiber is divided into many sections, and its parameters are the inverse of those of the real fiber. Within each section, the linear and nonlinear parts of Equation (2) are executed independently. The linear part corresponds to the dispersion effect and is implemented in the frequency domain as follows:

$$\widehat{D}\left[A^{\prime }\left(z,t\right)\right]=IFFT\left\{\exp (j{\displaystyle \frac{{\beta }_2}{2}}{\omega }^{2}h)·FFT\left\{A^{\prime }\left(z,t\right)\right\}\right\},$$

(4)

where FFT stands for Fourier Transform, $h$ represents the step size of each section, and IFFT stands for Inverse Fourier Transform. The nonlinear part corresponds to the Kerr nonlinear effect, implemented in the time domain as

$$\widehat{N}\left[A^{\prime }\left(z,t\right)\right]=\exp \left(-j{\gamma }^{\prime }h{‖A^{\prime }‖}^2\right)A^{\prime }\left(z,t\right).$$

(5)

The shortcoming of Equation (5) is that it ignores the interaction between dispersion and non-linearity. The term ${‖A^{\prime }‖}^2$ in Equation (5) represents the summed power of the symbols at a given time slot. If dispersion is not considered, the shape of the propagating optical signal does not broaden, and the powers of different symbols do not overlap with each other. Thus, the nonlinear phase shift at a certain time slot is only related to the power of the current symbol. The chromatic dispersion of optical fibers should not be ignored, as it could cause the optical signals to broaden as they propagate and make the powers of different symbols overlap at a given time slot location. Accordingly, the nonlinear phase shift must account for the power of the different symbols overlapping simultaneously in one time slot. Taking this phenomenon into account, Equation (5) is modified as follows:

$$\widehat{N}\left[A^{\prime }\left(z,t\right)\right]=\exp \left(-j{\gamma }^{\prime }h{\displaystyle \sum _{k=-\left(L-1\right)/2}^{\left(L-1\right)/2}{c}_k\left[{\left|A_x\left(t-k{T}_s\right)\right|}^2+{\left|A_y\left(t-k{T}_s\right)\right|}^2\right]}\right)A^{\prime }\left(z,t\right),$$

(6)

where L is an odd integer representing the number of symbols used to account for a nonlinear phase shift and $c_k$ is the real-valued weight coefficient. Furthermore, the intra-polarization and inter-polarization symbols contribute differently to the nonlinear phase shift. Consequently, Equation (6) is modified as

$$\widehat{N}\left[A^{\prime }\left(z,t\right)\right]=\exp \left(-j{\gamma }^{\prime }h{\displaystyle \sum _{k=-\left(L-1\right)/2}^{\left(L-1\right)/2}{c}_k\left[a{\left|A_x\left(t-k{T}_s\right)\right|}^2+b{\left|A_y\left(t-k{T}_s\right)\right|}^2\right]}\right)A^{\prime }\left(z,t\right).$$

(7)

For the x polarization, a and b denote the intra-polarization and inter-polarization parameters, respectively, while for the y polarization, the parameters are reversed.

Since there are similarities between the computational architectures of an SSFM and an artificial neural network [18], this paper converts the channel emulators described by Equations (5)–(7) to artificial neural networks as shown in Figure 2. The inputs of these neural networks are time-domain waveforms sampled at the rate of 2 samples/symbol. Each neural network consists of many layers. Based on Equations (5)–(7), three types of networks with different types of layer structures are designed, as shown in Figure 3. These neural networks are discussed as follows:

(1) The first one is illustrated in Figure 3a, which is an ordinary DNN that contains a linear operator given by Equation (4) and a nonlinear activation function given by Equation (5).

(2) The second one is designed according to Equation (6). It considers the powers of L consecutive symbols when calculating a nonlinear phase shift. The summing operation in Equation (6) used for computing a nonlinear phase shift is equivalent to the convolution of the signal power ${‖A‖}^2$ and a finite impulse response filter (FIR) that has L taps. The tap coefficient of the FIR is denoted as $c_k$. Figure 3b shows the final outcome, i.e., a convolutional layer is inserted between the dispersion and the nonlinear phase-shift sections. It is referred to as a CNN in this paper. Its nonlinear activation function is given by Equation (6).

(3) To further enhance the estimation accuracy of PPE, the third type of neural network accommodates the difference between the contributions of different polarization symbols to a nonlinear phase shift. It is constructed according to Equation (7) and is shown in Figure 3c. The intra- and inter-polarization contributions in this neural network are denoted by a and b, respectively. This paper refers to this third type of neural network as an enhanced CNN (ECNN). Its nonlinear activation function is given by Equation (7).

As seen from Figure 2, after handling dispersion and nonlinearity, the waveforms sampled at 2 samples/symbol are down-sampled to 1 sample/symbol. The last stage of these neural networks, as shown in Figure 1, involves a phase rotation with the reference symbols given by the waveform reconstruction module, which eliminates the residual constellation phase offset. During the network training process, the mean square error (MSE) between the symbols output by these neural networks and the reference symbols was used as the loss function, which is defined as $MSE=E\left\{\left({\left|S_x-{\widehat{S}}_x\right|}^2+{\left|S_y-{\widehat{S}}_y\right|}^2\right)/2\right\}$, where $S_x$ and $S_y$ are the symbols on the two polarization streams of each neural network’s output, and ${\widehat{S}}_x$ and ${\widehat{S}}_y$ are the reference symbols. The effective SNR for symbols is defined as follows:

$$SNR={\displaystyle \frac{1}{B}}{\displaystyle \frac{1}{P}}{\displaystyle \frac{1}{N_s}}{\displaystyle \sum _{k=1}^B{\displaystyle \sum _{j=1}^P{\displaystyle \sum _{i=1}^{N_s}\left(\frac{{\left|{\widehat{S}}_{xi}\right|}^2}{{\left|S_{xi}-{\widehat{S}}_{xi}\right|}^2}+\frac{{\left|{\widehat{S}}_{yi}\right|}^2}{{\left|S_{yi}-{\widehat{S}}_{yi}\right|}^2}\right)/2}}},$$

(8)

where $N_s$ is the number of symbols contained in a waveform sample, $P$ is the number of training epochs, and $B$ is the mini-batch size. The trainable network parameters are denoted by ${\gamma }^{\prime }$, $c_k$, a and b, which are optimized by minimizing the MSE using the Adam optimizer. Once the network is trained, the optimized ${\gamma }^{\prime }$ is determined and deemed as the closest measurement to the true value. Assuming the nonlinear coefficient $\gamma$ is ideal and undisturbed, the optical power at spatial location $z$ is evaluated according to the following: $\widehat{P}\left(z\right)\approx {\gamma }^{\prime }\left(z\right)/\left(8\gamma /9\right)$.

It is important to note that the channel emulators used in the published works [3,16], despite minor differences, are fundamentally identical to the one shown in Figure 3a. Both channel emulators compute the nonlinear phase shift with Equation (5), neglecting interactions between different symbols and polarizations. The difference between them is that the fiber-optic channel emulators of the previous works [3,16] output waveforms with a sampling rate of 2 samples/symbol, while the one described in Figure 3a delivers symbols with a sampling rate of 1 samples/symbol. Due to fiber nonlinearity, optical signals transmitted through long-haul optical fibers are attached with a nonlinear rotation of constellations. For fiber-longitudinal PPE, the nonlinear rotation of the constellations cannot simply be straightened out, as waveform comparison and reconstruction are necessary at the receiver side. However, factors such as PMD, anomalous power attenuation, and other environmental perturbations make it extremely difficult to measure this nonlinear rotation angle. Previous studies [19,20] have observed this issue and explicitly pointed out that MMSE-based PPE requires an additional process to estimate this rotation angle. Especially when the optical power is high, the non-linear rotation of constellations can severely degrade the PPE accuracy. However, both [3,16] fail to explicitly describe the handling method for the nonlinear rotation of constellations. In this paper, the waveforms output from the DNN are down-sampled to 1 sample/symbol, and then the phases of $S_x$ and $S_y$ are aligned with reference symbols, as shown in Figure 2. For the above reasons, and given the necessity of a fair comparison, the fiber channel emulator described by Figure 3a is used as a representative of the published works in this paper. The PPE performance of the proposed method will be compared with it in subsequent investigations.

3. Simulation Setup and Results

A numerical simulation platform equivalent to Figure 1 for polarization multiplexing (PDM) M-QAM coherent optical communication system with a symbol rate of 64 Gbaud/s was constructed to validate the effectiveness of the proposed method. Its modulation format can be specified as quadrature phase-shift keying (QPSK) and 16-ary QAM (16-QAM). All the system and fiber parameters are presented in

Table 1. Key parameters of PDM M-QAM coherent optical communication system.

System Parameters	64 Gbaud/s	Roll-off factor = 0.1	Simulation bandwidth = 512 GHz	QPSK, 16-QAM
Fiber Parameters	α = 0.2 dB/km	γ = 1.3 W−1 km−1	PMD = 0.02~0.2 ps/sqrt(km)	D = 17 ps/nm/km
EDFA Parameter	Noise Figure = 4 dB	Gain = 16 dB	4/6 spans	80 km/span

. The fiber optic link of this system was made up of multiple spans, with the number of fiber spans and the fiber length per span denoted as $N_{span}$ and $L_{span}$, respectively. The fiber under consideration had a nonlinear index of 2.6 × 10⁻²⁰ m²/W and a dispersion parameter of 17 ps/nm/km. The PMD coefficient varied between 0.01 and 0.1 ps/sqrt(km). When the forward transmission of an optical signal was simulated over a real long-haul fiber, asymmetric SSFM was used to handle the dispersion and Kerr nonlinear effects, with a maximum nonlinear step of 20 km and a maximum nonlinear phase shift of 20 rad per step. Moreover, a coarse-step approximation method (also referred to as a waveplate model) was used to simulate the polarization-related effects [21]. The entire fiber optic link was modeled as a cascade of waveplates. The step size for each waveplate was set to 1 km. The average DGD accumulated along a single waveplate was equal to the PMD coefficient multiplied by the square of the waveplate length. The total line width of the transmitter laser and the local oscillator was fixed at 10 kHz. The noise figure of the Erbium-Doped Fiber Amplifier (EDFA) was set as 4 dB. The total CD accumulated on the fiber optic link was perfectly compensated with an electric domain DSP incorporated in the receiver. A frequency domain filter was used. The random revolutions of SOP and PMD were equalized using a data-aided LMS algorithm with 32 taps [22]. The estimation and compensation of PN were divided into two stages within the fiber-longitudinal PPE proposed in this paper. Coarse phase tracking based on the pilot symbols was conducted in the first stage, which was intended to cope with the large phase shifts that may occur. To implement this, one pilot symbol was periodically inserted after 511 payload symbols. The second stage was RPN estimation to ensure accurate waveform comparison. To achieve this goal, a BPS algorithm with 64 test phase angles was employed. During simulations, the analog sampling rate was 8 samples/symbol, while the digital sampling rate was 2 samples/symbol.

The neural networks shown in Figure 2 and Figure 3 were implemented using TensorFlow, and the learning rate of the three neural networks was fixed at 0.0001. The impairment-contaminated waveforms and the reference symbols output by the waveform reconstruction module were simultaneously fed into the three neural networks for training and parameter optimization. The Adam optimizer was used to minimize the loss function during 100 iterations. In each iteration, 10 waveforms were randomly selected out of a total of 1000 waveform samples and input into the three neural networks for training. Each waveform sample contained 4096 random symbols per polarization. Each iteration ran for 200 epochs. The final longitudinal power profile output by the three neural networks was obtained by averaging the results over 100 iterations.

First, the modulation format was set up as QPSK. The performance of the three different neural networks shown in Figure 3 was examined in terms of PPE accuracy. In this example, the total distance of the fiber optic link was 320 km, including four spans of 80 km each. When the SSFM algorithm was mapped to a neural network, each fiber span was partitioned into 40 steps. It was assumed that there were no anomalous attenuation spots on the overall fiber optic link. The average optical power into the fiber was temporarily fixed at 3 dBm. The filter length for the two CNNs shown in Figure 3b,c was chosen as 5 temporarily. The theoretical power profile was compared with the estimate produced by the proposed method at each spatial location.

All results are presented in Figure 4a, where the DNN and CNN results are indicated by solid lines with triangular and square symbols, respectively. The ECNN results are indicated by a solid line with circular symbols. As seen from Figure 4a, the PPE accuracy of the DNN deteriorates considerably at the beginning and end of each fiber span. After using the proposed ECNN, the PPE accuracy at these locations is noticeably improved. To more clearly compare the performance of the three neural networks, the absolute error of PPE was computed at each spatial location as $\widehat{P}\left(z_i\right)-P\left(z_i\right)$, where $P\left(z_i\right)$ is the theoretical power profile at position $z_i$, and $\widehat{P}\left(z_i\right)$ is the power profile estimated by the proposed method. The obtained results are shown in Figure 4b. The DNN gives the worst PPE accuracy, followed by the CNN. The ECNN performs the best, giving the highest PPE accuracy.

Second, the impact of the FIR filter length on the PPE performance of the ECNN was investigated. The fiber span number, length per span, and power into the fiber remained unchanged from the previous section. The FIR filter length for CNNs was increased from 3 to 11 at intervals of 2. The PPE accuracy was quantified using the mean squared deviation (MSD) metric calculated as shown below:

$$MSD={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{\left|\widehat{P}\left(z_i\right)-P\left(z_i\right)\right|}^2},$$

(9)

where $N$ is the number of spatial positions $z_i$. For each filter length, the power profile was calculated ten times for this fiber link based on the proposed ECNN. Subsequently, the MSD for each power profile was derived according to Equation (9). All the obtained results are shown in Figure 5, where the box plots are superimposed upon a scatter plot to provide the statistical distribution of these ten measurements. Each point in the figure represents one MSD of a power profile. The average of these ten measurements is also marked in bold red beside the box plot. It can be observed that the ECNN performs the best in terms of PPE accuracy when the filter length is equal to 5, i.e., the optimal length of the FIR filter is equal to 5. Furthermore, the average MSD of PPE produced by the ECNN is the minimum at only 0.13575. The PPE performance of the ECNN degrades when the FIR filter length is less than or greater than 5. Therefore, the FIR filter length of CNNs was selected as 5 in the following experiments unless stated otherwise.

Third, the impact of different optical power values into the fiber and PMD coefficients on the accuracy of PPE was investigated. For this purpose, the optical power into the fiber was increased from −2 dBm to 8 dBm, and the PMD coefficient of SSMF was chosen as 0.02 ps/sqrt(km). The fiber span number and length per span were kept the same as those in the earlier experiment. For each power value, the power profile of the fiber link was estimated using the DNN, CNN, and ECNN shown in Figure 3. The corresponding MSDs are shown in Figure 6a. It can be observed that the PPE accuracy of DNN gradually degrades as the power in the fiber increases. After the incorporation of the convolutional layer in the neural network, the accuracy of PPE shows an obvious improvement. The PPE performance exhibited by the CNN and ECNN is significantly less dependent on the power transmitted into the fiber. The PPE accuracy of the latter is superior to that of the former. When the power increases from −2 dBm to 8 dBm, the average relative improvement in the MSD of the ECNN is up to 71% compared to the DNN.

To investigate the influence of the PMD coefficient, the power into the fiber was fixed at 3 dBm, while the PMD coefficient was increased from 0.02 to 0.20 ps/sqrt(km) at intervals of 0.03 ps/sqrt(km). For each PMD coefficient, the MSDs of the PPE output from the three neural networks are shown in Figure 6b. Comparing Figure 6a,b, it can be noted that the PMD coefficient has a considerably smaller impact on the PPE accuracy than the power into the fiber. The MSDs of PPE show only minor variations with respect to the PMD coefficients. At a given PMD coefficient, the ECNN has the highest PPE accuracy, followed by the CNN and the DNN.

Lastly, the PPE capability of the proposed method was verified for abnormal fiber links. The fiber span number and the length of fiber per span remained unchanged. The power into the fiber was fixed as 3 dBm, and the PMD coefficient was chosen as 0.02 ps/sqrt(km). It was assumed that the EDFA operated in constant output power mode. Two abnormal scenarios were considered, and the PPE performance of the proposed method was verified in these scenarios. The first scenario involved the appearance of a power attenuation anomaly of 5 dB right in the middle of the third fiber span. In the second scenario, two power attenuation anomalies of 3 dB and 5 dB simultaneously occurred in the middle of the second and third fiber spans, respectively. The PPE results of the proposed ECNN corresponding to the two scenarios are shown in Figure 7a,b, respectively. The actual locations where power attenuation anomalies occur are marked with cyan dots in these figures. It can be observed that, at these abnormal locations, the ECNN outputs and the theoretical results significantly deviate, indicating that the proposed method can successfully detect the anomalous power attenuation in both scenarios.

To meet the growing communication bandwidth requirements, optical fiber transmission systems need enhanced spectral efficiency and advanced modulation formats. To further illustrate the generalization ability of the proposed method, this study modified the modulation format and fiber length and re-evaluated the fiber-longitudinal PPE performance of CNNs in this new environment. The modulation format of the coherent optical communication system shown in Figure 1 was changed from QPSK to 16-QAM, and the number of fiber spans was increased from 4 to 6. The fiber length per span remained unchanged at 80 km. Other fiber parameters, network training methodology, and DSP setup were kept identical to QPSK. Under this new environment, the waveform samples were regenerated to simulate the fiber-longitudinal PPE process. The obtained results are shown in Figure 8, Figure 9 and Figure 10.

The PPE results for three different neural networks with 16-QAM are shown in Figure 8a. In this scenario, no power attenuation anomalies occurred in the fiber link, the power into the fiber was 3 dBm, and the filter length of the CNNs was 5. The absolute error of the PPE extracted from Figure 8a is shown in Figure 8b. Similar to the QPSK results, ECNN has the best PPE accuracy with 16-QAM, followed by CNN and DNN. The optimization results of the FIR filter length for ECNN are updated and shown in Figure 9. It can be seen from the figure that when the modulation format is promoted from QPSK to 16-QAM, the optimized filter length of the ECNN is still equal to 5. The emergence of the same optimal filter length is logically justifiable. It is well known that the extent of optical pulse broadening during fiber propagation is determined by the dispersion coefficient and baud rate. The greater the dispersion coefficient and baud rate, the wider the pulse spread, and consequently, the greater the number of symbols with power overlap. As indicated by Equations (6) and (7), the FIR filter length of the CNNs is bounded by the number of symbols with power overlap. Since the dispersion coefficient and baud rate are identical when changing the modulation format, the optimal FIR filter length obtained in the two cases remains unchanged. The PPE results of the abnormal fiber link for 16-QAM are shown in Figure 10. A power attenuation anomaly of 5 dB occurs at the midpoint of the 3rd fiber span. At this abnormal location, the ECNN output deviates from the theoretical results. Similar to QPSK, the proposed method still successfully detects the power attenuation anomaly.

4. Discussion

Although the above results fully demonstrate the PPE performance of the proposed method, several issues remain to be addressed. The most distinguishing feature of the proposed method is the design and construction of CNN-based fiber channel emulators for the fiber-longitudinal PPE. In recent decades, many fiber-optic channel emulators have been proposed, which have been extensively applied to DBP for nonlinear compensation (NLC) and, more recently, to fiber-longitudinal PPE. While fiber-optic channel emulators for these two applications share similarities, their emphasis differs. For DBP, the focus is to minimize the computational complexity by maximizing the step size while keeping the NLC performance to meet the requirements [23]. In PPE, the fiber-optic channel emulator must enable the distributed monitoring of key parameters. The attainable spatial resolution is then of utmost concern. Several existing fiber-optic channel emulators and their application to DBP and PPE are shown in

Table 2. Several existing fiber-optic channel emulators.

Channel Emulator	Power Overlaps Between Symbols	Discrepancy Between the Polarization Streams	Already Applied to NLC?	Already Applied to PPE?
DNN	X	X	Yes, [23]	Yes, [3,16]
CNN	√	X	Yes, [24,25]	No
ECNN	√	√	Yes, [17]	No
RNN	√	√	Yes, [26]	No
LSTM	√	√	Yes, [27]	No

X: not considered; √: considered.

. Oleg Sidelnikov [24] proposed to apply CNN on DBP for low-complexity NLC. However, it did not account for the discrepancy between different polarization streams [25]. The DBP proposed by Danish Rafique [17] considered the discrepancy in different polarization branches but did not use machine learning to optimize ${\gamma }^{\prime }$, $c_k$, a and b. In other words, it is not a machine learning-based fiber channel emulator. Similar to CNNs, recurrent neural networks (RNNs) and long short-term memory (LSTM) networks are more robust fiber channel emulators that can accurately characterize signal propagation. RNNs and LSTMs have been frequently deployed for DBP [26,27]. However, to the best of the authors’ knowledge, their application in PPE has not been reported in the literature. Their effects on the PPE performance remain to be further investigated. As seen from Table 2, the current machine learning-based channel emulators applied to fiber-longitudinal PPE employ DNN, which ignores the power overlap of consecutive symbols and the discrepancy between the different polarization streams. This paper proposes a CNN-based fiber optic channel emulator that closes this gap and leads to improved accuracy of PPE.

Although the CNN-based fiber channel emulators improve the PPE performance compared with the DNN-based one, they incur increased computational complexity. Hence, the computational complexity introduced by the two CNNs proposed in this paper was also investigated. The real multiplications (RM) required at each sample were used to estimate computational complexity, while the addition operations were not considered. As shown in Figure 2, the two CNNs proposed in this paper require only one more convolutional layer at each step than the DNN. A one-dimensional real-valued convolutional layer with L taps requires L RMs. Computing the power of each polarization stream requires one RM. Thus, the computational complexity per sample is increased by $N_{span}\cdot StPS\cdot \left(2L-2\right)$ RMs when using the CNN in comparison to the DNN. $N_{span}$ is the number of fiber spans, while $StPS$ is the number of steps contained in each span. As shown in Equation (7), one extra RM is required for each polarization stream in ECNN. Consequently, the computational complexity per sample is increased by $N_{span}\cdot StPS\cdot \left(4L-2\right)$ RMs when using the ECNN in comparison to the DNN. These computational complexity increments certainly take more computing resources. However, the costs are justified by the ability to extract more valuable information (i.e., distributed measurements of the optical power along the spatial domain). In future studies, the validity of the proposed method for more modulation formats and fiber links will be investigated. Particularly, the emphasis will be on examining the spatial resolution of the fiber-longitudinal PPE. More advanced network structures will be designed and investigated to enhance the sensitivity of fiber-longitudinal PPE to power attenuation anomalies and to improve the spatial resolution.

5. Conclusions

In summary, this paper proposed a CNN-based method to improve the accuracy of fiber-longitudinal PPE. The working principle, optimization approach, and construction of CNN were first discussed in detail, and the signal processing of the corresponding fiber-longitudinal PPE was elaborated. The efficiency of the proposed method was verified by the numerical simulation of a 64 Gbaud/s M-QAM coherent fiber optic transmission system. Its PPE performances were tested over different modulation formats and fiber optic links. It was verified that the power into the fiber and PMD had a minor influence on the proposed method. The accuracy of PPE was substantially improved by using the proposed method, compared to not using it. The proposed method could correctly estimate the anomalous locations of several power attenuation anomalies on the fiber link. The results of this paper could contribute to the development of DLM-based optical performance monitoring technology. In subsequent studies, the validity of the proposed method will be tested for other modulation formats and fiber links, and network architectures more favorable to spatial resolution will be developed.