1. Introduction

With the continuous improvement in transmission capacity and spectral efficiency of optic fiber communication systems, the launch power of signals is increasing, and the resulting fiber nonlinearity has become the most important factor limiting the capacity improvement of system [1]. One class of strategy to counteract this problem is to use a series of techniques to mitigate fiber nonlinear effects, including optical phase conjugation technique [2], Volterra-series based equalizers [3], digital back propagation method or neural network based techniques [4–6], etc. As another revolutionary strategy, nonlinear frequency division multiplexing (NFDM) system uniquely takes the fiber nonlinearity as an intrinsic property of system, and modulates the signals onto the nonlinear spectrum utilizing nonlinear Fourier transform (NFT) [7]. It conveys information based on the fact that the nonlinear spectrum evolves linearly along optical fiber, which naturally has advantage of overcoming nonlinear impairments [8].

At present, the researches on NFDM technology have experienced a process from the origin of discrete spectrum (DS) modulation [7], gradually developed to continuous spectrum (CS) modulation by borrowing the idea of orthogonal frequency division multiplexing (OFDM), with achieving a record transmission capacity of 10.83 Tb/s [9] as well as the current full-spectrum modulation which jointly combines DS and CS modulation technologies [10]. In the meantime, a lot of related works have been carried out on fast and accurate NFT and inverse nonlinear Fourier transform (INFT) algorithms [11], improved NFDM modulation and demodulation technology [9], as well as NFDM system impairments and equalization techniques [12,13], etc. This manuscript mainly focuses on DS-NFDM 16/64 amplitude phase shift keying (APSK) system, for it could transmit stable soliton pulses and the b-coefficient distribution of APSK is more consistent with the noise distribution model of DS-NFDM system [14]. Nevertheless, there are a number of critical issues that need to be addressed at the receiver digital signal processing (DSP) side for DS-NFDM system, such as unavoidable amplifier spontaneous emission (ASE) noise and fiber attenuation, laser frequency offset (FO), linewidth (LW)-induced carrier phase noise (CPN), residual channel impairment (RCI) and NFT calculation errors, etc. It is known to all that ASE noise effect would cause the eigenvalue and its nonlinear spectrum to be perturbed from irregular random fluctuation during fiber transmission [15]. Besides, both FO and CPN induce the constellation points to produce different degrees of rotation, eventually leading to erroneous decisions. More importantly, an often-overlooked channel impairment is the RCI, which originates from the eigenvalue shift due to laser FO and ASE noise, would cause the ideal NFDM channel equalization function to be no longer accurate and applicable. Unfortunately, we have not found the related references to construct and analyze the joint model of FO, CPN and RCI for DS-NFDM system.

To address the aforementioned FO and CPN issues in DS-NFDM systems, some solutions have been proposed. Firstly, frequency offset estimation (FOE) schemes could roughly be divided into two categories. The first option is to insert training symbols (TS) in the nonlinear frequency domain (NFD) [16,17], its FOE range could reach half of the baud rate of NFDM system, while the insertion of TS sacrifices a portion of the frequency spectrum. The second one is to utilize the classical 4^th power fast Fourier transform-Like (FFT-Like) FOE scheme [18]. Nevertheless, the theoretical FOE range of this scheme is only as 1/8 of the baud rate, which does not satisfy the FO fluctuation range of the experimental extra cavity lasers in the GHz scale. Secondly, the classical M-th power and blind phase search (BPS) schemes have been utilized for carrier phase recovery (CPR) in DS-NFDM system [19–21]. Whereas the LW of M-th power scheme that can be effectively compensated is very limited and the scheme often exhibits amplitude ambiguity for 16/64APSK [19]. The complexity of BPS scheme is relatively high, and to the best of our knowledge, there are no application reports of BPS in 64APSK NFDM system. Finally, our prior work has demonstrated that the combined effect of FO and ASE noise results in a perturbative shift of eigenvalues [16], and also introduces additional amplitude noise and phase noise for the received DS-NFDM signals. However, this work did not take into account the RCI effect and the corresponding joint equalization of FO, CPN and RCI. Hence, it is necessary to propose a low-complexity joint equalization scheme of FO, CPN and RCI impairments applicable to DS-NFDM 16/64APSK systems.

Aiming at the joint equalization of FO, CPN and RCI impairments, we firstly deduce a joint impairment model for DS-NFDM 16/64APSK systems in this manuscript. This model reveals that the combined effect of FO, CPN and RCI would cause the b-coefficients to exhibit varying degrees of phase rotation, and specially, the RCI impairment also induces additional amplitude noise. In order to ensure the correct reception of DS-NFDM signals, we have proposed a joint equalization scheme based on two-stage cascaded extended Kalman filter (TSC-EKF), which uses 1024 symbols to track FO in the first stage and subsequently performs CPR and RCI equalization in the second stage. More significantly, based on the characteristics of 16/64APSK constellation arrangement, we ingeniously fuse the minimum Euclidean distance and phase difference between the received symbols and the ideal 16/64APSK constellations to calculate the innovation of TSC-EKF. Its effectiveness has firstly been verified by 2 GBaud single-eigenvalue DS-NFDM 16/64APSK back-to-back (BTB) and fiber transmission simulation scenarios. Furthermore, 2 GBaud DS-NFDM 16APSK experiments have been carried out to investigate the performance of TSC-EKF scheme. The results demonstrate that when the 7% hard decision forward error correction (HD-FEC) threshold is reached, the maximum FOE range and LW tolerance that can be jointly equalized by TSC-EKF scheme achieve 9.6 times and 3.3 times as that of the FFT-Like + M-th power scheme, respectively. Moreover, with the complexity being only 63.4% of the NFD + M-th power scheme, the FOE range and LW tolerance of TSC-EKF scheme can be improved by 20% and 233%, respectively.

The rest of this paper is organized as follows: firstly, in Section 2, the joint models of FO, LW, and RCI in DS-NFDM systems are derived and analyzed, and then the principles of the proposed TSC-EKF scheme are described detailedly. In Section 3 and Section 4, the effectiveness of TSC-EKF scheme has been fully verified by 2 GBaud DS-NFDM 16/64APSK simulations and 16APSK experiment systems, respectively. Subsequently, the complexity of TSC-EKF scheme has been analyzed and compared with other schemes in Section 5. Finally, the conclusions are presented in Section 6.

2. Principles of the proposed TSC-EKF scheme

2.1 Impairment model of FO, CPN and RCI for DS-NFDM system

In order to deduce the impairment model of FO, CPN and RCI, Fig. 1 illustrates the schematic diagram of DS-NFDM 16/64APSK systems. Firstly, binary bits are encoded and modulated to 16/64APSK symbols in a manner of $b({{\lambda_{k0}},0} )$ to achieve NFD modulation at the transmitter side, where ${\lambda _{k0}}$ represents the initial eigenvalue of the k-th symbol. Then $b({{\lambda_{k0}},0} )$ is transformed into the nonlinear time domain signal ${q_k}(\tau )\; $ using INFT. After that, ${q_k}(\tau )$ is sent to an IQ modulator to generate the time-domain NFDM signals. During optical fiber transmission, the NFDM signals are amplified by erbium-doped fiber amplifier (EDFA), while this process inevitably introduces ASE noise. Besides that, two lasers located at the transmitter and receiver necessarily suffer from the effects of FO and CPN. Finally, the impaired signal ${Q_k}(\tau )$ is obtained by a coherent receiver and the offline DSP processing is performed. Specifically, the offline DSP consists of normalization, NFT, channel equalization and TSC-EKF processes. The processed DS-NFDM signals after each of above the mentioned processes are denoted as ${Q_k}(t )$, $b({\lambda_{kz}^{\prime},z} )$, and $b^{\prime}({\lambda_{kz}^{\prime},z} )$, respectively, where $\lambda _{kz}^{\prime}$ and z denote the k-th received eigenvalue and transmission distance, respectively.

 

Fig. 1. The schematic diagram of the DS-NFDM system. SSMF: standard single-mode fiber

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According to the above descriptions, the k-th b-coefficient $b({\lambda_{kz}^{\prime},z} )$ after NFT could be expressed as:

Besides that, the received eigenvalue $\lambda _{kz}^{\prime}$ can be expressed as [16]:

If we substitute Eq. (2) into Eq. (1) and simplify, $b({\lambda_{kz}^{\prime},z} )\; $ is written as:

After channel equalization of Rx DSP [8], $b^{\prime}({\lambda_{kz}^{\prime},z} )$ is further obtained using:

Additionally, by separating the corresponding real and imaginary parts of eigenvalues, the initial eigenvalue ${\mathrm{\lambda }_{\textrm{k}0}}$ and the perturbation term $\mathrm{\Delta }{\mathrm{\lambda }_{{\textrm{k}_{\textrm{RCI}}}}}$ can be given as Eq. (5) and Eq. (6), respectively.

Next, by substituting Eq. (5) and Eq. (6) into Eq. (4), $b^{\prime}({\lambda_{kz}^{\prime},z} )$ can be further calculated by:

To simplify Eq. (7), we could decompose the RCI impairment into amplitude noise ${\alpha _{{k_{RCI}}}}$ and phase noise ${\varphi _{{k_{RCI}}}}$, which are separately represented by:

On the above basis, we substitute Eq. (8) into Eq. (7) to obtain the final representation of the b-coefficients as follows:

Finally, the joint action model of FO, CPR and RCI can be represented as

Focusing on the analysis of Eq. (10), it reveals that the joint action model of FO, CPR and RCI lead to different degrees of phase rotation of b-coefficients $b^{\prime}({\lambda_{kz}^{\prime},z} )$, and specifically the RCI impairment results in an additional amplitude noise ${\alpha _{{k_{RCI}}}}$. Therefore, the impairments of FO, CPN and RCI must be jointly equalized in order to ensure the correct receiving of $b^{\prime}({\lambda_{kz}^{\prime},z} )$.

2.2 Principle of the proposed TSC-EKF scheme

On the basis of above joint model of FO, CPN and RCI impairments for DS-NFDM 16/64APSK systems, we have proposed a corresponding solution. It utilizes extended Kalman filters (EKFs) to track FO in the first stage and subsequently performs CPR and RCI equalization in the second stage, so this solution is abbreviated as TSC-EFK. It should be emphasized that although Ref [22]. has proposed a joint linear impairment equalization scheme using a two-stage cascade Kalman structure for coherent optical system, there are many distinct differences in the application scenario, equalization goal and scheme architecture compared with our work. This scheme cannot be simply transplanted to the DS-NFDM 16/64APSK systems. Note that the amplitude noise of RCI can be mitigated by the normalization operation in the DSP flow at the receiver side. According to Eq. (10), the two parameters to be tracked by TSC-EFK are represented by $\Delta f,{\theta _{CPN\& RCI}}$. Figure 2 illustrates the detailed schematic diagram of TSC-EKF. Both stages could be further split into two steps: time update and measurement update. Besides that, in order to emphasize the importance of the adopted innovations, the innovations in both stages are also shown at Fig. 2(c).

 

Fig. 2. The detailed schematic diagrams of the proposed TSC-EKF scheme: (a)Frequency offset compensation;(b) CPR and RCI equalization;(c) Calculation of fused innovation. The input (d) and output (e) of b-coefficient distributions of DS-NFDM 16APSK signals, respectively.

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Stage 1 Frequency offset estimation and compensation

The purpose of this stage is to estimate the FO value and perform FOC using EKF, which is depicted in Fig. 2(a). For the first step of time update in this stage, the proposed scheme needs to initialize the initial state vector ${\hat{x}_0} = [{\Delta {f_0}} ]$, the error covariance matrix ${P_0}$ and the measurement covariance matrix ${R_0}$ of EKF. Generally speaking, the optimum values of ${\hat{x}_0}$, ${P_0}$ and ${R_0}$ are set 0,1e-5 and 1e-3, respectively. Besides that, it should be noted that only initial ${N_{sym}}$ symbols need to be tracked for FOE in Stage 1. If the index n of current symbol is less than $\textrm{}{N_{sym}}$, then ${Q_0} = 1e - 3$, otherwise ${Q_0} = 0$. Secondly, the priori estimation of the state vector ${\hat{x}_{k|k - 1}}$ and the error covariance matrix ${P_{k|k - 1}}$ are calculated according to Ref. [25]:

Then, the measurement update is carried out in step 2. Specifically, the priori estimation of the state vector ${\hat{x}_{k|k - 1}}$ is firstly assumed to be $\Delta {f_k}$, then it is substituted into Eq. (13) to obtain the FO-compensated signal $y_k^{FOC}$.

Subsequently, the measurement vector $h({{{\hat{x}}_{k|k - 1}}} )$ is computed, and the measurement vector is derived from the state vector to obtain Jacobi matrix ${H_k}$. Additionally, the innovation ${e_k}$ is calculated according to the flow of Fig. 2(c). Finally, with the aid of ${e_k}$, the gain matrix ${G_k}$, the posteriori estimation of the state vector ${\hat{x}_{k - 1}}$ and the error covariance matrix ${P_k}$ are sequentially updated as follows:

Stage 2 CPR and RCI equalization

The main task of Stage 2 is to utilize EKF to jointly perform CPR and RCI equalization using the FO-compensated symbols of Stage 1. The detailed flows are shown in Fig. 2(b).

It is worth noting that most of the steps and execution processes in Stage 2 are the same as those of Stage 1. The differences are listed as follows: firstly, ${\hat{x}_0} = [{{\theta_{{0_{CPN\& RCI}}}}} ]$ and ${Q_0}$ need to be set 0 and 1e-3 in step 1, respectively. Secondly, the calculation formula for joint CPR and RCI equalization should be modified to:

Calculations of fused innovation

As we know, the appropriate measurement matrix and innovations are crucial for EKF to perform fast parameter tracking and improve the parameter estimation accuracy in Stage 1 and 2. Taking 16APSK as an example, Fig. 2(c) demonstrates the detailed flow of fused innovation calculation specifically designed for DS-NFDM 16/64APSK systems.

This flow could be roughly divided into two steps: the determination of the optimum constellation and phase innovation calculations. Initially, as illustrated in Fig. 3(b), step 1 calculates the minimum Euclidean distance between the received $({\lambda_{kz}^\mathrm{^{\prime}},z} )$ and the standard constellations of ${\boldsymbol c}_l^{ideal}$, where $l \in \{{1,2,\ldots M} \}$ and M denotes the total number of ideal constellations. Next, the optimum constellation ${c_{min\textrm{}}}\textrm{}$ is determined using the minimum Euclidean distance between $b^{\prime}({\lambda_{kz}^\mathrm{^{\prime}},z} )$ and ${\boldsymbol c}_l^{ideal}$, which could be expressed by:

 

Fig. 3. Taking DS-NFDM 16 APSK as an example, the innovation calculation diagram: (a) the received 16APSK b-coefficients, (b) determination of the optimum constellation, (c) calculation of the phase innovation. ${\theta _k}$ and ${\theta _{min}}$ denote the phase of $b^{\prime}({\lambda_{kz}^{\prime},z} )$ and ${c_{min}}$, respectively.

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Subsequently, as depicted in Fig. 3(c), step 2 calculates the phase difference ${h_{pha.}}({{{\hat{x}}_{k|k - 1}}} )$ between $b^{\prime}({\lambda_{kz}^{\prime},z} )$ and ${c_{min}}$ according to Eq. (19). Finally, the fused phase innovation is obtained using Eq. (20).

3. Simulation setup and results analysis

3.1 Simulation setup

To verify the effectiveness of the TSC-EKF scheme, we have conducted a 2 GBaud single-polarization DS-NFDM 16/64APSK transmission system using VPI Transmission Maker 11.2 and MATLAB software, as illustrated in Fig. 4. Firstly, random binary bit sequences are mapped into 2 GBaud 16/64APSK symbols within the transmitter DSP. Then these symbols are encoded into b-scattering coefficients on an eigenvalue of 0.25j or 0.1 + 0.25j. Subsequently, the time domain pulse signals can be obtained using the INFT technique. The baseband electrical signals are modulated onto optical carrier using an IQ modulator. The center wavelength of transmitter laser is set to 1550 nm. Next, the performances are evaluated in both BTB and optical fiber transmission scenarios. In the BTB scenario, optical signal-to-noise ratio (OSNR) is regulated by changing the ASE noise power. Additionally, an optical bandpass filter (OBPF) with bandwidth of 0.8 nm is employed to filter the out-of-band noise, as depicted in Fig. 4(a). For the optical fiber transmission scenario, Fig. 4(b) illustrates that the fiber loop comprises a standard single-mode fiber (SSMF) with length of 80 km, an EDFA with noise figure (NF) of 6 dB, and an OBPF with bandwidth of 0.8 nm.

 

Fig. 4. Schematic diagrams of 2 GBaud DS-NFDM 16/64APSK simulation system under scenarios of (a) BTB and (b) fiber transmission.

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In order to evaluate the performance of TSC-EKF scheme, different FOs and LWs are configured in the LO laser at coherent receiver. After coherent reception, the receiver DSP (Rx DSP) firstly initiates time synchronization. Subsequently, normalization and NFT operations are carried out, with the NFT operations being performed using the FNFT software package [23]. Next, channel equalization is applied to mitigate the effects of dispersion and nonlinearity in fiber channel. Following that, the TSC-EKF scheme is applied for the joint equalization of FOE, CPE, and RCI. As the corresponding comparisons, schemes of FFT-Like FOE [18] and TS insertion in NFD [17] are used in the FOE flow, and the M-th power scheme [19] is utilized in the CPR flow. Finally, BERs are calculated. It is important to note that each Rx DSP process acquires 60 sets of symbols, with each set containing 32,768 symbols, to calculate the average value of BERs.

3.2 BTB performance

Firstly, in the DS-NFDM 16/64APSK BTB scenario, we have investigated the absolute FOE errors under different FO values and 100 kHz LW. Here the absolute FOE error is defined as $FO{E_{error}} = |{FO{E_{est.}} - FO{E_{true}}} |$, where $FO{E_{est.}}$ and $FO{E_{true}}$ denote the estimated FOE values and actual FO values using three kinds of FOE schemes, respectively. When OSNR is set to be 15 dB, Fig. 5(a) demonstrates that the FOE errors are less than 0.4 MHz for 16APSK system using all three schemes, and the maximum FOE ranges of NFD and FFT-Like power schemes are only ±1000 MHz and 125 MHz, respectively. In contrast, the TSC-EKF scheme achieves a minimum average FOE error of only 0.18 MHz and a maximum FOE range of ±1200 MHz. Furthermore, these above conclusions also hold true when modulation format is upgraded to 64APSK under OSNR of 28 dB. The TSC-EKF scheme also gains a maximum FOE range of ±1200 MHz and a minimum average FOE error of 0.24 MHz.

 

Fig. 5. Absolute FOE error versus different FOs for 2 GBaud (a) DS-NFDM 16APSK system under OSNR of 15 dB, and (b) DS-NFDM 64APSK system under OSNR of 28 dB.

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Secondly, in order to elaborate the convergence performance, Fig. 6 illustrates the tracking curves of FO and CPN using TSC-EKF scheme. Here the OSNR, FO, LW and length of symbol sequence N are set as 15 dB, 200 MHz, 100 kHz and 32768, respectively. According to Fig. 6(a), it can be observed that the proposed scheme achieves convergence of FOE only using around 1024 symbols. It demonstrates that the TSC-EKF scheme has advantages of fast convergence and good real-time performance. Figure 6(b) shows the CPN tracking curves. It clearly depicts that the TSC-EKF scheme is capable of accurately tracking the CPN variations for DS-NFDM 16APSK system.

 

Fig. 6. Tracking curves of FO and CPN using the TSC-EKF scheme in a 2 Gbaud DS-NFDM 16APSK system: (a) FOE tracking curve of 100 MHz FO; (b) CPN tracking curve under 100 kHz LW. The insets in Fig. 6(a) illustrate the enlarged FO tracking curves with symbols ranging from 0 to 1250 and 980 to 1140, respectively.

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3.4 Fiber transmission performance

Thirdly, when LW is set 100 kHz, the FOE performance has also been verified in an optical fiber transmission scenario for 2 GBaud DS-NFDM 16/64APSK systems. Using three schemes, Fig. 7 illustrates the absolute FOE errors at different FOs. From Fig. 7(a), we can learn that after transmission of 1200 km, the average FOE errors for 16APSK system reach 0.24 MHz, 0.325 MHz and 0.04 MHz using the NFD, FFT-Like and TSC-EKF scheme, respectively. It is clearly observed that the TSC-EKF scheme obtains the finest FOE accuracy. Moreover, under condition of FOE error less than 0.6 MHz, the TSC-EKF scheme achieves a wider FOE range of 200 MHz and 1075 MHz than that of the NFD and FFT-Like schemes, respectively. Similar conclusions can be obtained according to Fig. 7(b). It describes that after fiber transmission of 640 km, the average FOE errors for 64APSK system are only less than 0.2 MHz using the TSC-EKF scheme, which is superior to both NFD and FFT-Like schemes. Under the same condition of FOE error less than 3 MHz, our scheme achieves a maximum FOE estimation range of 9.6 and 1.2 times as that of the FFT-Like and NFD schemes, respectively.

 

Fig. 7. FOE error curves versus different FOs for different 2 GBaud DS-NFDM signals, (a) 16APSK, (b) 64APSK system.

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After that, when FO and LW are set 100 MHz and 100 kHz, respectively, Fig. 8 illustrates the BER curves with different transmission distances for 2 GBaud DS-NFDM 16/64APSK systems. Figure 8(a) depicts that when transmission distance varies from 1040 km to 1600 km, the obtained BERs by TSC-EKF scheme are consistently lower than that of the NFD + M-th power and FFT-Like + M-th power schemes for 16APSK system. Especially, the BER of TSC-EKF is an order of magnitude lower than that of the NFD + M-th power scheme after fiber transmission of 1360 km. Additionally, when transmission distance increases from 240 km to 1200 km, Fig. 8(b) clearly demonstrates that our TSC-EKF scheme evidently outperforms two other schemes for 64 APSK system under the same conditions. Concretely, the NFD + M-th power, FFT-Like + M-th power and TSC-EKF schemes are able to reach maximum transmission distances of 604 km, 480 km and 680 km under the 7% HD-FEC threshold, respectively. Furthermore, the insets in Fig. 8 prove that the TSC-EKF scheme can obtain the optimum b-coefficient distributions.

 

Fig. 8. BER curves with varying distance for 2 GBaud DS-NFDM of (a)16 ASPK system and (b) 64 APSK system. After using FFT-Like + M-th power and TSC-EKF schemes, the insets illustrate the b-coefficient distributions for (a) 16APSK system after 1360 km transmission and (b) 64APSK system after 640 km transmission.

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Finally, we have evaluated the joint equalization performance for 2 Gbaud DS-NFDM 16APSK system under different FOs and LWs. After transmission distance of 1200 km, Fig. 9(a)-(c) illustrate the equalization results of NFD + M-th power, FFT-Like + M-th power and TSC-EKF schemes in DS-NFDM 16APSK system with eigenvalue of 0.25j, respectively. It should be noted the blue grid plane represents the 7% HD-FEC threshold plane. Figure 9 (a) depicts that when achieving 7% HD-FEC threshold, the maximum FOE range and compensated LWs of NFD + M-th power scheme are 1000 MHz and 150 kHz, respectively. Meanwhile, Fig. 9 (b) indicates a worse joint equalization performance for the FFT-Like + M-th power scheme, its maximum FOE range and LWs are 125 MHz and 150 kHz, respectively. By comparison, Fig. 9 (c) demonstrates that for each possible combination of FO, LW and RCI, the TSC-EKF scheme could jointly equalize the maximum FO and LW up to 1200 MHz and 500 kHz under condition of 7% HD-FEC threshold, respectively. Similarly, after 960 km transmission, Fig. 9 (d)-(f) show the joint equalization results using the three schemes when the eigenvalue is set 0.1 + 0.25j. It can be observed that when BERs are lower than the 7% HD-FEC threshold, the maximum FOE ranges using the NFD, FFT-Like and TSC-EKF schemes are 1000 MHz, 125 MHz and 1200 MHz, respectively. Additionally, the maximum LW tolerances for the M-th power and TSC-EKF schemes are 150 kHz, 150 kHz and 500 kHz, respectively. Therefore, the TSC-EKF scheme still achieves the best joint equalization performance of FO, CPN and RCI in the presence of real parts of eigenvalues.

 

Fig. 9. When the eigenvalue is set 0.25j after 1200 km transmission, the equalization performance of 2 GBaud DS-NFDM 16APSK system under different FOs and LWs using (a) NFD + M-th scheme, (b) FFT-Like + M-th power scheme and (c) TSC-EKF scheme. When the eigenvalue is set 0.1 + 0.25j after 960 km transmission, the equalization performance using (d) NFD + M-th scheme, (e) FFT-Like + M-th power scheme and (f) TSC-EKF scheme. The insets illustrate the corresponding b-coefficient distributions after using three schemes under the same 100 MHz FO and 500 kHz LW.

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Meanwhile, Fig. 10 gives the equalization comparison results using these three schemes for 2 Gbaud DS-NFDM 64APSK system. It should be mentioned that the transmission distances for eigenvalues of 0.25j and 0.1 + 0.25j are set 640 km and 400 km, respectively. We can draw the similar conclusions as in Fig. 9. Under condition of 7% HD-FEC threshold, regardless of whether the eigenvalues contain real parts or not, the proposed TSC-EKF scheme could jointly equalize the maximum FO and LW up to 1200 MHz and 500 kHz, respectively. Consequently, we can deduce that for the transmission scenarios of DS-NFDM 16/64APSK system, the FOE range of TSC-EKF scheme is 0.2 and 8.6 times higher than that of the NFD and FFT-Like FOE schemes, respectively, and the LW tolerance of TSC-EKF is 2.3 times higher than that of the M-th power scheme.

 

Fig. 10. When the eigenvalue is set 0.25j after 640 km transmission, the equalization performance of 2 GBaud DS-NFDM 64 APSK system under different FOs and LWs using (a) NFD + M-th scheme, (b) FFT-Like + M-th power scheme and (c) TSC-EKF scheme. When the eigenvalue is set 0.1 + 0.25j after 400 km transmission, the equalization performance using (d) NFD + M-th scheme, (e) FFT-Like + M-th power scheme and (f) TSC-EKF scheme. The insets illustrate the corresponding b-coefficient distributions after using three schemes under the same 100 MHz FO and 500 kHz LW.

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4. Experiment results

4.1 Experiment setup

To further validate the equalization performance of TSC-EKF scheme for DS-NFDM systems, we have constructed 2 GBaud single-polarization DS-NFDM 16APSK experimental system. Its detailed setup is shown in Fig. 11. Firstly, at the transmitter side, the DS-NFDM 16APSK waveforms are generated by a 65 GSa/s arbitrary waveform generator (AWG, Keysight M8195A) using Tx DSP. The initial eigenvalue location is set 0.25j. The specific Tx-DSP flows are the same as that in Section 3.1. Meanwhile, two narrow LW lasers (NKT BASIK MIKRO E15, LW < 100 Hz) are utilized as the optical source of transmitter and coherent receiver, whose center wavelengths are both set 1550.12 nm. After the signals are modulated and transmitted, for the BTB scenario, the link OSNRs are adjusted by using an ASE noise source, an EDFA with noise figure of ∼6.5 dB and a variable optical attenuator (VOA). The actual OSNRs are observed using an optical spectrum analyzer (OSA) with resolution of 0.1 nm. In the BTB scenario, OSNRs are varied from 19 dB to 25 dB. Subsequently, the signals pass through an optical filter with bandwidth of 0.8 nm to filter out the ASE noise. For the fiber transmission scenario, the fiber loop consists of a SSMF span of 50.4 km, an EDFA and an OBPF with bandwidth of 0.8 nm. The dispersion, nonlinearity and attenuation coefficients of SSMF are set $17.6\; ps/({nm \cdot km} )$, $1.3\; {w^{ - 1}}k{m^{ - 1}}$ and $0.2\; dB/km$, respectively. Besides that, the total length of fiber transmission is controlled by an optical fiber loop controller. Furthermore, we mainly adjust the FO to verify the joint equalization capability of FO, LW and RCI in BTB and fiber transmission scenarios. Specifically, the center wavelength of LO laser is shifted from 1550.120 nm to 1550.128 nm, with a step of 1 pm (equivalent to ∼125 MHz).

 

Fig. 11. The experimental setup of 2 GBaud single-polarization DS-NFDM 16APSK system. (a) back-to-back scenario, (b) fiber transmission scenario.

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At the receiver side, the NFDM signals are captured and processed by offline Rx DSP using a real-time oscilloscope (Teledyne Lecroy, Labmaster 10Zi), which operates at 80 GSa/s sampling rate and electrical bandwidth of 36 GHz. The Rx-DSP flows are the same as those described in Fig. 4. Finally, the joint equalization performances are evaluated by acquiring 50 sets of experimental data, where each set consists of 32,768 symbols.

4.2 Experimental results

Firstly, when the FO is set 125 MHz, Fig. 12(a) illustrates the BER curves under different OSNRs using three kinds of schemes in the BTB scenario. It depicts that when OSNR varies from 19 dB to 25 dB, the obtained BERs by using the TSC-EKF scheme are consistently lower than that of the NFD + M-th power and FFT-Like + M-th power schemes for 16APSK system. Especially, when the BER reaches the 7% HD-FEC threshold, it is clearly found that the OSNR penalties are reduced by 0.61 dB and 1.42 dB using the TSC-EKF scheme, compared with the NFD + M-th power and FFT-Like + M-th power scheme, respectively.

 

Fig. 12. Experimental results using three kinds of schemes for 2 GBaud DS-NFDM 16APSK system. (a) BER curves with different OSNRs under 125 MHz FO; (b) the normalized FOE errors versus FOs at OSNR of 21 dB; (c) the Q²-factors curves versus FOs after fiber transmission of 504 km. The insets illustrate the corresponding b-coefficient distributions using the TSC-EKF and FFT-Like + M-th power schemes: (a) under conditions of 121.57 MHz FO, 100 Hz LW and 21 dB OSNR, (b) under conditions of 259.4 MHz FO, 100 Hz LW and 21 dB OSNR, (c) under conditions of 282.9 MHz FO, 100 Hz LW and 504 km fiber transmission.

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After that, we have investigated the joint equalization performance of FO, LW and RCI under different FO values. The normalized FOE error is utilized as the measure index. It is defined as $NFO{E_{error}} = \frac{1}{{50}}\sum _{m = 1}^{50}\left|{\frac{{F{O_{true}} - F{O_{est.}}}}{{{R_s}}}} \right|$, where m denotes the index number of experiment data. As shown in Fig. 12(b), when OSNR is fixed at 21 dB, it is obvious that the maximum FOE range of FFT-Like scheme is only about 124 MHz under the normalized FOE error of 0.0089 (corresponding to the 7% HD-FEC threshold). In contrast, the maximum FOE ranges could be [16 MHz, 997.25 MHz] for the TSC-EKF and NFD schemes. However, the average normalized FOE errors for the NFD and TSC-EKF schemes are 0.00582 and 0.00468, respectively. Hence, our proposed TSC-EKF scheme achieves significantly higher FOE accuracy than the NFD scheme.

Finally, we have further verified the joint equalization performances after 504 km fiber transmission of DS-NFDM 16APSK signals. Figure 12(c) depicts the Q²-factor curves under different FOs, which are derived from ${Q^2} = 20{\log _{10}}\left[ {\sqrt 2 erf{c^{ - 1}}({2BER} )} \right]$. It exhibited that the maximum FOE range is only [29.1 MHz, 121.1 MHz] for the FFT-Like + M-th power scheme when Q²-factors achieve higher than the 7% HD-FEC threshold. In comparison, the maximum FOE ranges are both [29.1 MHz, 984.5 MHz] for the TSC-EKF and NFD + M-th power schemes, whereas the average Q²-factor obtained by TSC-EKF is 0.71 dB higher than that of the NFD + M-th power scheme. Notably, when the FO is fixed at 116.5 MHz, our TSC-EKF scheme obtains a Q²-factor gain of about 1.2 dB compared with the FFT-Like + M-th power scheme. Furthermore, the insets in Fig. 12 demonstrate that the TSC-EKF scheme could obtain the best quality for 16APSK signals among these three schemes under the same conditions.

5. Computational complexity

It is known to all that the computational complexity is one of the key indicators to evaluate the availability and performance of joint equalization scheme. It is important to state that these three schemes, TSC-EKF, NFD + M-th power and FFT-Like + M-th power, all execute the impairment equalization after the NFT operation within the DSP flows for DS-NFDM system. Accordingly, when performing the complexity comparisons among these three schemes in Table 1, the complexity $O({K + N{{\log }^2}N} )$ of NFT has been omitted [24].Specifically, the operations of our TSC-EKF scheme can be divided into four categories based on the ideal of optimum implementation, which are real multiplication, real addition, comparison and look-up table (LUT) calculations. Table 1 summarizes the complexity comparison results for NFD + M-th power, FFT-Like + M-th power and TSC-EKF schemes

Table 1. Complexity comparisons of NFD + M-th power, FFT-Like + M-th power and TSC-EKF

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According to Table 1, it can be observed that the overall complexities of the NFD + M-th power and FFT-Like + M-th power schemes could be denoted as $O(N )$ and $O({N\log (N )} )$, respectively. In comparison, our TSC-EKF scheme totally requires $51N + 51K$ real multiplications, $40N + 40K$ real additions, $16N + 16K$ comparisons and $18N + 18K$ LUTs. Here N and K represent the number of symbols used for FOE and CPR by these three schemes, respectively. Consequently, the total complexity of TSC-EKF scheme is on the order of $O(N )$. Taking the joint equalization of 16/64APSK signals as example, if we acquire $K = 32768$ symbols and set $B = 64$, where B represent the number of searched angles, the classical coefficients for the NFD + M-th power, FFT-Lik + M-th power and the TSC-EKF schemes are set $L = 4096\; \; {N_{NFD}} = 128$, ${N_{FFT - Like}} = 4096$ and ${N_{TSC - EKF}} = 1024$, respectively. It can be found that the NFD + M-th power, FFT-Like + M-th power and TSC-EKF scheme need 2,717,952, 960,066 and 1,723,392 real multiplications, respectively. Obviously, our scheme requires 63.4% of the number of real multiplications as that of NFD + M-th power scheme, which is higher than that of FFT-Like + M-th power scheme, while it achieves the optimum joint equalization performance of FO, CPN and RCI.

6. Conclusion

In this paper, firstly we have deduced a joint impairment model of laser FO, LW, and RCI for DS-NFDM 16/64APSK systems. This model reveals that the combined effect of FO, CPN and RCI would cause the scattering b-coefficients to exhibit varying degrees of phase rotation, and the RCI impairment results in additional amplitude noise. Subsequently, a TSC-EKF scheme is proposed to jointly equalized FO, CPN and RCI impairments of DS-NFDM 16/64 APSK systems. It uses 1024 symbols to track FO in the first stage and subsequently performs CPR and RCI equalization in the second stage. In order to calculate the TSC-EKF innovation, we ingeniously fuse the minimum Euclidean distance and phase difference between the received symbols and the ideal 16/64APSK constellations.

The effectiveness has been verified in 2 GBaud DS-NFDM 16/64APSK BTB, fiber transmission simulation and 2 GBaud DS-NFDM 16APSK experiment scenarios. The results prove that when impairments of FO, CPN and RCI are jointly equalized, the maximum FOE range of this scheme reaches [-1200 MHz,1200 MHz] and its LW tolerance achieves a maximum of 500 kHz. Compared with the NFD and FFT-Like schemes, this maximum FOE range of our scheme achieves 20% and 860% improvements, respectively. In addition, the maximum LW tolerance reaches 3.3 times as that of the M-th power scheme. More importantly, its overall computational complexity is reduced 36.6% compared to the NFD scheme and on an order of O(N). The scheme exhibits significant improvements in joint equalization performance and complexity. Accordingly, we believe that the proposed TSC-EKF scheme is a good solution for the joint equalization of FO, CPN and RCI impairment of DS-NFDM 16/64 APSK systems.