Frequency offset estimation for nonlinear frequency division multiplexing with discrete spectrum modulation

Zibo Zheng; Xulun Zhang; Ruihua Yu; Lixia Xi; Xiaoguang Zhang

doi:10.1364/OE.27.028223

1. Introduction

Fiber nonlinearity has become the main obstacle for high-speed long-haul optical fiber communication systems. In the past decade, several methods for equalizing or mitigating fiber nonlinearity have been reported [1]. Among them, nonlinear frequency division multiplexing (NFDM), which was first proposed by Hasegawa and Nyu in [2] and then developed by both Yousefi, Kschischang [3] and Le, Prilepsky in [4], is a promising candidate for solving the problem of fiber nonlinearity. For an NFDM system, the transmitting information is encoded onto the nonlinear spectra by the nonlinear Fourier transform (NFT). These nonlinear spectra propagate independently with nonlinear Schrödinger equation, which means there is no crosstalk among the nonlinear spectral components [3]. Hence, the NFDM system possesses inherent robustness to ﬁber nonlinearity, and it has drawn a considerable amount of attention [2–18]. Some impressive demonstrations that exploited the high order quadrature amplitude modulation (QAM) format modulated on discrete spectrum [5], on continuous spectrum [6], or even on both [7,8] have been reported. The polarization division multiplexing (PDM) technique was also introduced into NFDM systems to increase their transmission capacity [9–13], using either discrete, continuous spectrum and both parts. Despite the aforementioned rapid progress, a series of key challenges still need to be solved for a practical NFDM system. Studies on NFDM systems have mainly focused on theoretical analyses or the proof-of-concepts experiments. Some linear impairments such as frequency offsets, rotations of the state of polarization, and phase noise have seldom been analyzed in the literatures [14–16]. Considering practical implementations, equalization techniques for these impairments in an NFDM system should be investigated and discussed.

Carrier frequency offset (CFO), or simply frequency offset (FO), which is a linear impairment originating from the mismatch of the transmitter and local oscillation lasers, causes a shift of nonlinear spectra [3,14] and leads to detection errors. In NFDM systems, FO was usually mitigated by a time domain method in the linear region prior to NFT processes [4,5]. However, the FO effect can be alternatively compensated for within the nonlinear frequency domain (NFD) after the NFT process, with the similar idea for the phase noise equalization in the NFD [15].

In this paper, a method applied in the nonlinear frequency domain for linear frequency offset estimation is proposed. The proposed method is designed based on the FO effects in the NFD and is suitable for FO estimation and compensation in the NFDM system based on discrete spectrum. It is realized by a data-aided angle search algorithm, which can return the optimum estimation of the FO induced rotation angle of the received scattering data; and hence we can calculate the estimated FO value.

The manuscript is organized as follows. The basic NFT theory and its fundamental properties are firstly reviewed. Then FO impairments are analyzed based on the properties of the NFT and NFDM system. Afterwards, the novel FO estimation methods applied in the nonlinear frequency domain (NFD method) is introduced for the discrete spectrum NFDM system. As the comparison counterpart, the conventional FO estimation method in the linear frequency domain (LFD method) is also introduced. The performance comparison is carried out in back-to-back and fiber transmission scenarios. The back-to-back results reveal that the NFD method can achieve an FO estimation error around 0.1 MHz using 64 training symbols while the estimation error provided by the LFD method is around 10 MHz. The results in the fiber transmission scenario reveal that the NFD method dramatically reduces the TS overhead, saves 75% of TS overhead for 16QAM modulation format and saves at least 87.5% overhead for the QPSK format in the multiple eigenvalue transmission situations. Besides, the estimation accuracy and the implementation stability of the NFD method in the cases of different phase noise levels and link noise are also illustrated. These results reveal that the proposed method is a suitable approach for the discrete spectrum NFDM system using high order modulation formats.

2. NFT Principle and fundamental properties

2.1 Brief review of the NFT theory

The nonlinear Fourier transform (NFT), which is more often referred to as the inverse scattering transform (IST) in mathematics and physics, is a method for solving nonlinear partial differential equations (NPDEs) [18], such as the nonlinear Schrödinger equation (NLSE). It is well known that evolutions of optical signals in a fiber considering the Kerr effect are governed by NLSE [19]. Under the noise-free and lossless assumptions, by performing the unit normalization (see [3], Part I), the normalized NLSE can be given as

(1)$$j{\partial _z}q + {\partial _{tt}}q + 2\kappa {|q |^2}q = 0$$

Here, q(t, z), t, and z denote the complex envelope of the signal, the retarded time, and the transmission distance, respectively. Note that all of these variables are normalized by their corresponding parameters. Variable$\kappa = \pm 1$ determines the focusing or defocusing case of the NLSE, where the former ($\kappa = + 1$) corresponds to anomalous dispersion in the fiber and the latter ($\kappa = - 1$) corresponds to normal dispersion [20].

The main function of the NFT is to calculate the scattering data (also referred to as nonlinear Fourier coefficients) [3]. They vary simply along the nonlinear optic-fiber while the signal in the time domain experiences very complex evolution. The nonlinear Fourier transform is performed by solving the Zakharov–Shabat problem with initial conditions [3,18,20]:

(2)$$\frac{\partial }{{\partial t}}{\textbf{v} = }\left[ {\begin{array}{cc} { - j\lambda }&{q({t,z} )}\\ { - \kappa {q^ \ast }({t,z} )}&{j\lambda } \end{array}} \right]{\textbf{v}},\;\;\;\mathop {\lim }\limits_{t \to - \infty } {\textbf{v}}({t,\lambda } )= \left[ \begin{array}{l} 1\\ 0 \end{array} \right]{e^{ - j\lambda t}}\;.$$

Then, the scattering data $a(\lambda ),\;b(\lambda )$can be calculated as

(3)$$a(\lambda )= \mathop {\lim }\limits_{t \to \infty } {e^{j\lambda t}}{v_1}({t,\lambda } ),\quad b(\lambda )= \mathop {\lim }\limits_{t \to \infty } {e^{ - j\lambda t}}{v_2}({t,\lambda } ).$$

where v = [v₁, v₂]^T is the canonical solution and $\lambda \in {\mathbb C}$is the given spectrum value. Furthermore, the discrete spectrum and continuous spectrum are defined as

(4)$$\begin{array}{l} {Q_d}(\lambda )= {{b({{\lambda_j}} )} \mathord{\left/ {\vphantom {{b({{\lambda_j}} )} {{a_\lambda }({{\lambda_j}} )}}} \right.} {{a_\lambda }({{\lambda_j}} )}}\quad {\lambda _j} \in {{\mathbb C}^ + },\\ {Q_c}(\lambda )= {{b(\lambda )} \mathord{\left/ {\vphantom {{b(\lambda )} {a(\lambda )}}} \right.} {a(\lambda )}}\quad \quad \lambda \in {\mathbb R}. \end{array}$$

${\lambda _j}$ are the discrete eigenvalues that satisfy $a({{\lambda_j}} )= 0$and ${a_\lambda }({{\lambda_j}} )= { {{{da(\lambda )} \mathord{\left/ {\vphantom {{da(\lambda )} {d\lambda }}} \right.} {d\lambda }}} |_{\lambda = {\lambda _j}}}$. It should be clarified that only in the focusing case, i.e., $\kappa = + 1$ in Eq. (1) and Eq. (2), do the eigenvalues and their corresponding discrete spectrum exist. The space in which the NFT operation is performed is denoted as the nonlinear frequency domain. After propagating distance z governed by the NLSE, the variation in the corresponding scattering data and spectrum value satisfy the following rules:

(5)$$\begin{array}{l} {\lambda _j}(z )= {\lambda _j}(0 );\quad a({\lambda ,z} )= a({\lambda ,0} ),\quad b({\lambda ,z} )= {e^{ - 4j{\lambda ^2}z}}b({\lambda ,0} ),\\ \textrm{ accordingly, }{Q_d}({{\lambda_j},z} )= {e^{ - 4j\lambda _j^2z}}{Q_d}({{\lambda_j},0} ),\quad {Q_c}({\lambda ,z} )= {e^{ - 4j{\lambda ^2}z}}{Q_c}({\lambda ,0} ). \end{array}$$

Therefore, the transfer function of NLSE in the nonlinear frequency domain can be considered as an all-pass like filter $H(\lambda )= \exp ({ - 4j{\lambda^2}z} )$.

Moreover, a very crucial assumption is made here that the initial condition in Eq. (2) takes the vanishing boundary form of q(t), i.e., $q(t )\to 0$ for $t \to \pm \infty $. There were also some theoretical studies on the scattering problem of a periodic signal [21–23], which is equivalent to the periodic boundary form [24–26]. However, in this study, we will just focus on the vanishing boundary signal and its eigenvalue transmission.

2.2 Fundamental properties of the NFT

Prior to the discussions of the proposed frequency offset estimation method, two helpful fundamental properties of NFT are reviewed so that the rest of this paper can be easily understood. When we discuss the relationship between the time domain signal and the scattering data $q(t )\leftrightarrow \{{a(\lambda ),b(\lambda )} \}$, we have the two following properties:

(6a)$${e^{ - 2j\omega t}}q(t )\leftrightarrow \{{a({\lambda - \omega } ),b({\lambda - \omega } )} \}.$$

(6b)$${e^{j\phi }}q(t )\leftrightarrow \{{a(\lambda ),{e^{ - j\phi }}b(\lambda )} \}.$$

where $\omega ,\;\phi $ represent the frequency shift and the constant phase shift, respectively. Equation (6a) can be regarded as the frequency offset effect and Eq. (6b) exhibits the phase shift feature. Although the NFT is basically a nonlinear transform, it can be easily concluded from Eq. (6) that the change properties of frequency or phase are very similar to those associated with linear Fourier transforms. The proof of Eq. (6) can be carried out by replacing q(t) with $\exp ( - 2j\omega t)\,q(t )$ or $\exp (j\phi )\,q(t )$ in the Zakharov–Shabat problem, and solving the partial differential equation Eq. (2) with variable substitution. The detailed proof can be found in [3, Appendix B]. The reader can also consult Eq. (12) and Eqs. (26)–(27) in [11] for a more generalized conclusion.

3. Frequency offset and estimation method

In this section, the effects caused by the frequency offset in a nonlinear frequency division multiplexing system were analyzed and illustrated. Based on these effects, an angle search method for the frequency offset estimation in the nonlinear frequency domain was proposed. For comparison, the conventional FO estimation method in linear frequency domain was also introduced.

3.1 Frequency offset effects in the NFDM system

Thus far, most of the NFDM signals in the time domain are transmitted in “burst mode” (except for the periodic boundary case) to satisfy the vanishing boundary requirement [3], which means that there is a pre-defined time window for each burst (and usually the width of the window is fixed). Therefore, the transmitted signal $U(\tau )$ can be written as

(7)$$U(\tau )= \sum\limits_k {{u_k}({\tau - k{T_p}} )} $$

where k indicates the index of the time window, u_k represents the burst waveform in the k-th time slot $[{0,{T_p}} ]$, and T_p is the fixed width of the time window. For any $\tau \notin [{0,{T_p}} ]$, u_k is set to zero. Note that the time parameters $\tau ,{T_p}$ in Eq. (7) are not the normalized time quantities which have the time unit. The information signals are modulated on the discrete part of the nonlinear spectrum. The scattering data of ${u_k}(\tau )$ are defined as $\{{{a_k}(\lambda ),{b_k}(\lambda )} \}$. After propagating a distance z in the fiber, the received signal is detected by a coherent receiver with a certain frequency offset $\Delta \Omega $(with the unit of Hz). Ignoring the additive and phase noise in the link and assuming the bursts in different time slots are interference free, we can process the burst in each time window at the receiver side independently. Therefore, the signal in the processing window is given as

(8)$${u^{\prime}_k}({\tau - k{T_p}} ){e^{j\Delta \Omega \tau }}\buildrel {\textrm{by normalization}} \over \longrightarrow {q^{\prime}_k}({t - {{k{T_p}} \mathord{\left/ {\vphantom {{k{T_p}} {{T_0}}}} \right.} {{T_0}}}} ){e^{j\Delta \Omega {T_0}t}}.$$

here ${u^{\prime}_k}$ is regarded as the output of the noiseless fiber span (also ignore the loss) with the input u_k, and normalization is done by $t = {\tau \mathord{\left/ {\vphantom {\tau {{T_0}}}} \right.} {{T_0}}},\;q = {u \mathord{\left/ {\vphantom {u {\sqrt P }}} \right.} {\sqrt P }}$, where $P = {2 \mathord{\left/ {\vphantom {2 {({\gamma L} )}}} \right.} {({\gamma L} )}},\;L = {{2T_0^2} \mathord{\left/ {\vphantom {{2T_0^2} {|{{\beta_2}} |}}} \right.} {|{{\beta_2}} |}}$, $\gamma $ is the nonlinear coefficient, and ${\beta _2}$ is the group velocity delay parameter of the fiber. T₀ is a free parameter. For the case k = 0, the fundamental property in Eq. (6a) can be directly applied. For the case k > 0, the variable substitution is taken as $\tilde{t} = t - k{{{T_p}} \mathord{\left/ {\vphantom {{{T_p}} {{T_0}}}} \right.} {{T_0}}}$, and the expression becomes ${q^{\prime}_k}({\tilde{t}} ){e^{j\Delta \Omega {T_0}\tilde{t}}}{e^{jk\Delta \Omega {T_p}}}$. Then, the properties of Eqs. (6a)–(6b) can be applied. In general, the scattering data $\{{{{a^{\prime}}_k}(\lambda ),{{b^{\prime}}_k}(\lambda )} \}$ in the k-th window at the receiver is

(9)$$\{{{{a^{\prime}}_k}(\lambda ),{{b^{\prime}}_k}(\lambda )} \}= \{{{a_k}({\lambda + {{\Delta \Omega {T_0}} \mathord{\left/ {\vphantom {{\Delta \Omega {T_0}} 2}} \right.} 2}} ),\;{b_k}({\lambda + {{\Delta \Omega {T_0}} \mathord{\left/ {\vphantom {{\Delta \Omega {T_0}} 2}} \right.} 2}} ){e^{ - jk\Delta \Omega {T_p}}}{e^{ - 4j{\lambda^2}z}}} \}.$$

Let $\omega = - {{\Delta \Omega {T_0}} \mathord{\left/ {\vphantom {{\Delta \Omega {T_0}} 2}} \right.} 2}$ and $\theta = \Delta \Omega {T_p}$; then

(10)$$\{{{{a^{\prime}}_k}(\lambda ),{{b^{\prime}}_k}(\lambda )} \}= \{{{a_k}({\lambda - \omega } ),\;{b_k}({\lambda - \omega } ){e^{ - jk\theta }}{e^{ - 4j{\lambda^2}z}}} \}.$$

Equations (9)–(10) indicate that the nonlinear spectrum will vary by a frequency offset $\Delta \Omega $ with two effects: a shift of the nonlinear frequency and a phase rotation. The shift is determined by the natural property of the NFT, while the phase rotation is mainly raised by the burst-cascaded structure of the NFDM signal. Figure 1 illustrates these two effects induced by FO on the nonlinear frequency $\lambda $ and scattering data. Note that $\lambda $ can be either discrete complex numbers or real numbers.

Fig. 1. FO effects: (a) shift of nonlinear frequency; (b) phase rotation on scattering data$b(\lambda )$. Left column: without FO; right column: with FO.

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It should be noted that the expressions in Eqs. (8)–(9) represent an idealized approximation. If additive noise or phase noise is considered, there will be additional location deviations of discrete eigenvalues [14]; moreover, any additive noise will influence the detection accuracies of the eigenvalues and the scattering data when the NFT algorithm is performed. It might even result in failed detection.

3.2 FO estimation in the nonlinear frequency domain

To realize an effective impairment equalization, the most important thing is to find out how the impairment works or which influence makes on the system. For example, FO makes frequency shift in the linear frequency domain, which is the base we estimate the FO by estimating this frequency shift. According to the analyzed results in Section 3.1, we found that FO would lead to a frequency shift of the nonlinear spectra and a phase rotation of the scattering data in the NFDM system. It implies that FO could be estimated either by finding the eigenvalue shift or the rotation angle in the nonlinear frequency domain (NFD). Therefore, a corresponding algorithm for FO estimation in NFD is proposed in this section.

For an NFDM system utilizing the discrete spectrum, a crucial step for an NFT algorithm at the receiver is to locate the NFT eigenvalue positions of the received burst signals, in order to provide more accurate values of the scattering data. In fact, the shift of the nonlinear frequency can be automatically handled by performing eigenvalue location. According to Eq. (9), we naturally think of estimating the frequency offset $\Delta \Omega $by finding the optimum normalized phase angle $\Delta \Omega {T_p}$. Thus, the proposed algorithm, namely, angle search algorithm, mainly aims at handling phase rotation.

The idea of an angle search algorithm for FO estimation is presented as follows. On the j-th eigenvalue, we insert m-length training symbols in front of the signal data. The transmitted k-th training symbol on the j-th eigenvalue is denoted as $s_{jk}^{TS}$, and the corresponding received one is denoted as $c_{jk}^{TS}$. According to Eq. (10), the received distorted training symbol can be expressed as: $c_{jk}^{TS} = s_{jk}^{TS}{e^{ - jk{\theta _0}}}$. With angle search, we will find the best estimation $\hat{\theta }$ that minimizes the difference or error between the original training symbol and the distorted one. The details of the algorithm are illustrated in Algorithm 1.

Algorithm 1.

Angle search method: denote the length of the training symbol on the j-th eigenvalue as m, the k-th transmitted training symbol on the j-th eigenvalue as $s_{jk}^{TS}$, and the received symbol as $c_{jk}^{TS}$ ; denote the divided angle number as N. Divide $2\pi $ into uniform N areas on constellation map:

\Theta = \{{ - \pi + {\pi \mathord{\left/ {\vphantom {\pi N}} \right.} N}, - \pi + {{3\pi } \mathord{\left/ {\vphantom {{3\pi } N}} \right.} N}, \cdots ,\pi - {\pi \mathord{\left/ {\vphantom {\pi N}} \right.} N}} \}

For each ${\theta _i} \in \Theta $, calculate the decision error e_i and find the optimal angle $\hat{\theta }$.

\begin{aligned}&{Decision\ error}:\quad {e_i} \buildrel \Delta \over = \sum\limits_{j = 1} {\sum\limits_{k = 1}^m {\left[ {{{s_{jk}^{TS}} \mathord{\left/ {\vphantom {{s_{jk}^{TS}} {\left\| {s_{jk}^{TS}} \right\|}}} \right. } {\left\| {s_{jk}^{TS}} \right\|}} - {{c_{jk}^{TS}\exp \left( {jk{\theta _i}} \right)} \mathord{\left/ {\vphantom {{c_{jk}^{TS}\exp \left( {jk{\theta _i}} \right)} {\left\| {c_{jk}^{TS}} \right\|}}} \right. } {\left\| {c_{jk}^{TS}} \right\|}}} \right]} } \\ & {Optimal\ angle}:\quad \hat{\theta } = \arg \mathop {\min }\limits_{{\theta _i}} \left| {{e_i}} \right|\\ & {Estimated\ frequency\ offset}:\quad \Delta {f_{est}} = {{\hat{\theta }} \mathord{\left/ {\vphantom {{\hat{\theta }} {({2\pi {T_p}} )}}} \right.} {({2\pi {T_p}} )}}\end{aligned}

The frequency offset is estimated through the optimal searching angle $\hat{\theta }$. The proposed method provides an alternative FO estimation approach after the NFT process, i.e., estimation of the FO in a nonlinear frequency domain. Therefore, in the following sections of this paper, the proposed method will be referred to as the NFD method.

It should be noted that the proposed angle search algorithm has some similarities with the popular blind phase search (BPS) method. BPS is a powerful tool for carrier phase noise recovery in traditional coherent optical communication system with high order modulation formats. Whether the regular rotation phase of the scattering data induced by FO or stochastic phase noise of constellations induced by laser linewidth, they both impact the symbol in the form of $\exp (j\varphi )$. Therefore, whether FO estimation or phase noise estimation, they both will lead to a so-called ‘phase/angle searching’ algorithm, whose main idea is to discretize the phase space and make a minimum error decision over tested windows. The difference between the BPS and the proposed angle search algorithm mainly depend on the problem source and application scenario. BPS is usually used for phase noise estimation with received block data, while the proposed angle search algorithm in this paper is used for FO estimation in NFD aided by training symbols, and has less complexity compared with BPS.

Actually, there could be more than one scheme to estimate FO in NFD, but the performance needs to be checked. Before developing the proposed FO estimation method in this paper, we had even tried the Mth -power algorithm [27] for FO estimation in NFDM system. However, the results show that the Mth-power algorithm is not very suitable for NFDM scenario, at least in the system with discrete spectrum modulation. The main reason is limitation estimation range ± R_s/2 m, where R_s is the symbol rate and m is the number of constellation states [28–29]. For example, a QPSK modulated 1G baud rate NFDM system, the absolute FO value covered by Mth-power is only around 125 MHz, which is not enough in practical utilization.

It should also be noted that according to Eqs. (9) and (10), the FO $\Delta \Omega $ can be calculated either by the way of finding the eigenvalue shift $\omega = - {{\Delta \Omega {T_0}} \mathord{\left/ {\vphantom {{\Delta \Omega {T_0}} 2}} \right.} 2}$ by performing the eigenvalue locations at the receiver (without using training symbols), or by finding the phase rotation of the training symbols using aforementioned angle search method. It superficially indicates the former way should be preferred owing to its simple implementation and no need of TS. However, some effects make it unstable, and we should adopt the latter one − the proposed angle search method aided by TS because it provides more stable and accurate estimations. A further discussion and the simulation verification results will be presented in section 5.4.

3.3 FO estimation in the linear frequency domain

As the comparison counterpart of the proposed nonlinear frequency domain estimation method, a linear frequency domain estimation method for the frequency offset is introduced in this section. This method is data-aided and estimates the frequency offset with the help of the pilot frequency. The pilot frequency is generated by a periodic pattern of training symbols and time domain waveforms. It is extracted at the receiver side and used for frequency offset estimation. It is a common and convenient method for removing FO impairment in an NFDM system and is usually applied before the direct NFT algorithm at the receiver DSP [5,6,13]. The detailed application of this method is described below.

A periodic training symbol sequence is given at the transmitter side and encoded into the scattering data in the nonlinear frequency domain. Owing to the periodicity of the symbol pattern, the waveform generated by an inverse NFT also shows the same periodicity, which indicates the linear Fourier spectrum of the “training waveform” will have a pilot component at the baseband. Using an appropriate symbol pattern, the amplitude of the pilot will be exceedingly higher than the other frequency components. Figure 2 illustrates the diagram of the periodic symbol pattern, time domain waveform, and its Fourier spectrum.

Fig. 2. Example of the pattern of the training symbols. The time waveform was generated by an inverse NFT and its Fourier spectrum. The pilot frequency is selected as the frequency component at the baseband. T_p is the width of the time window.

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At the receiver DSP side, the training waveform is separated from the received signal, and its Fourier spectrum is obtained by a fast Fourier transform (FFT). The pilot frequency will be extracted by searching the frequency component with the highest component; then, the frequency offset $\Delta {f_{est}}$ is estimated as the shift in the values of the pilots’ positions from its original ones, as shown in right side graph of Fig. 2. In the following sections of this paper, this method will be referred as the LFD method as the FO estimation is done in the linear frequency domain.

4. NFDM system with discrete spectrum modulation

The performance of the proposed NFD estimation method is numerically verified in a single-polarization (SP) NFDM system with one or multiple eigenvalues, i.e., only with discrete spectrum. The performance comparison with the LFD method will be fully demonstrated. Here, it is necessary to clarify that polarization multiplexing (dual-polarization DP) does not influence the operation of either the proposed NFD estimation method or the conventional LFD estimation method. Therefore, the discussion around the SP environment is sufficient, and the discussions can be extended to the DP case.

4.1 Single-/Multi-eigenvalues transmission platform

The structure of the simulation platform is depicted in Fig. 3. The different channels, i.e., fiber transmission (Fig. 3(a)) and back-to-back (BTB) scenarios (Fig. 3(b)), are also shown.

Fig. 3. Simulation platform with different channels: (a) fiber-by-fiber transmission and (b) back-to-back. PRBS: pseudo-random binary sequence; LFD: linear frequency domain; and NFD: nonlinear frequency domain. Only one of these methods will be used at a time.

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In the transmitter DSP, data symbols are mapped from the pseudo-random binary sequence (PRBS). The designed training symbols are added as a TS header on the data stream. Then, the entire symbol stream was encoded in a b-coefficient manner on the single or multiple eigenvalues. The Darboux transform [7] (i.e., inverse NFT) is applied for time domain waveform generation. Because of the limited sampling rate of electronic devices, the sampled points of each pulse are confined. The data frame structures are depicted in Fig. 4. The transmitter laser is configured with a central wavelength of 1550 nm and a fixed linewidth (LW). If a back-to-back scenario is applied, the optical signal will experience only additive white Gaussian noise (AWGN), and the optical signal-noise ratio (OSNR) will be around 30 dB (at 0.1 nm reference bandwidth). Meanwhile, for the fiber transmission scenario, the signal will propagate through cascaded fiber spans. Each span consists of an 80 km non-zero dispersion shift fiber (NZ-DSF) (with attenuation $\alpha $ = 0.2 dB/km, group velocity delay parameter ${\beta _\textrm{2}}$ = −5.75 ps²/nm, and nonlinear coefficient $\gamma $=1.3 W^-1km^-1) and an erbium-doped optical fiber amplifier (EDFA) deployed at the end of each span with a 16 dB gain and 5 dB noise figure. Although the NZ-DSF is not a typical fiber choice in optical communication, it can provide a good performance for a soliton signal transmission and is often used in NFDM transmission [5,12,30]. The simulation process of fiber propagation is implemented by the step-split Fourier method with a step length of 1 km. A 10-nm bandwidth optical band-pass filter was placed ahead of the coherent receiver to remove out-band noise. For the local oscillator, different FOs with respect to 1550 nm and different levels of LW are assumed for coherent detections and performance tests.

Fig. 4. Diagram of the data frame, time domain bursts, and DSP structures for FO estimations using LFD and NFD methods and followed by the FO compensations. The same received signal (training bursts) is used for both LFD and NFD estimation methods.

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In the receiver DSP, an ideal synchronization is processed to extract a pulse in each time window. A direct NFT is implemented by the FNFT software package in [31], which provides functions such as eigenvalue searching and scattering data calculating. Afterwards, the inverse transfer function of the nonlinear frequency domain $H = \exp ({4j{\lambda^2}z} )$ is multiplied with the scattering data. The FO recovery is carried out either by the LFD method (dotted yellow module prior to NFT in Fig. 3) or by the proposed NFD method (dotted yellow module after NFT) to obtain results for comparison. Finally, the phase noise compensation and the Q-factor calculation (from the error vector magnitude, [32]) are processed successively.

For this NFDM simulation platform system, different modulation formats are utilized for optical fiber transmission and for the FO compensation performance test. Quadrature phase shift keying (QPSK) and 16 quadrature amplitude modulation (QAM) symbols are modulated onto the b-coefficients of the scattering data at the transmitter. For the multiple eigenvalue case, the modulation format and modulation carrier (b-modulation) for each eigenvalue are the same. As the generated pulse signal can only be limited in a finite time range in practical use, the time window should cover enough of the pulse energy for the correct NFT operation. We set the (normalized) time window as the [ − t_max, t_max], and t_max = T_p/(2T₀). The value of t_max is determined by the criterion that the window covers more than 99.99% of the energy of the pulse following adjustment by the free normalization parameter T₀. The parameters satisfying this energy requirement for all the different situations are listed in Table 1. The normalized time parameter t_max and occupied linear bandwidth (W) represent a − 20 dB relative bandwidth, which is selected as the maximum value from the results of every possible waveform. The free normalization parameter T₀ of the NFT is decided by the duration of the time window T_p and normalized time range t_max. The sampling number D indicates the number of points utilized to describe a pulse in each time window. The launch power values in different situations are also listed in Table 1.

Table 1. Parameters of different modulation schemes

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4.2 FO compensation by the LFD and NFD methods

As mentioned in sections 3.2 and 3.3, the FO estimation is implemented by both the LFD and NFD methods for the comparison. With the estimated frequency offset $\Delta {f_{est}}$, the FO impairment can be compensated for in the linear frequency domain. For the LFD method, the compensation process is applied in the time domain:

(11)$${S_c}(\tau )= S(\tau )\exp ({ - j2\pi \varDelta {f_{est}}\tau } ).$$

$S(\tau )$is the received time domain signal waveform. The term $\exp ({ - j2\pi \varDelta {f_{est}}\tau } )$ is multiplied by the whole received signal in the time domain to complete the compensation.

For the NFD method, the estimated frequency offset can be given by Algorithm 1 Section 3.3. The estimated angle $\hat{\theta }$ will be sufficient for compensation. According to Eq. (10), the rotation phase is multiplied by all data symbols carried in the k-th time window:

(12)$$b_k^c({{\lambda_i}} )= {b_k}({{\lambda_i}} )\exp ({jk\hat{\theta }} )$$

Subscript i indicates the number of eigenvalues. Figure 4 illustrates the diagram of the data frame, time domain signal, and detailed DSP structures of both estimation methods. In the data frame, the same received signal (training bursts) is used for both LFD and NFD estimation methods.

5. Numerical results and analysis

The results of FO estimation and compensation using both the conventional LFD and the proposed NFD method are illustrated in this section. The results in back-to-back scenario provide the benchmark of performance, while the fiber transmission results show the performance in a practical link. The estimated error used as the metric is defined as estimated error = |estimated FO –true FO|, where the estimated FO is computed by the LFD and NFD methods introduced in section 3. The angle division number in Algorithm 1 is set as 4096 for testing purposes. The phase noise compensation is completed by the Viterbi-Viterbi phase equalizer (VVPE) for QPSK format and blind phase searching (BPS) for 16QAM format. The Q-factor is evaluated only from the de-modulated data symbol.

5.1 Back-to-back results

In the back-to-back scenario, the linewidth is omitted in the transmitter or local laser to prevent the phase noise from affecting the estimation methods; however, a phase noise compensation module is deployed to remove residual FOs. Figure 5 depicts variation in the estimated error with different FO values and the length of the training symbols. Figures 5(a)–5(b) show that the estimated errors of the NFD method are stable with the estimated FO error round 0.1 MHz for both single and multiple eigenvalue transmission cases, while the estimated errors of the LFD method vary between 1 to 10 MHz. Figure 5(a) also indicates that the NFD method is much more accurate than the LFD method for estimating the frequency offset in the range [ − BaudRate/2, BaudRate/2]. Although the estimation range of the NFD method is smaller than the one covered by the LFD method, it is satisfied in practical applications. These results are obtained using 64 training symbols and averaged using 300 independent tests. The fluctuation pattern of the estimation error is caused by the limited linear frequency resolution. The frequency resolution in the LFD method is decided by the time duration of the training bursts for FFT operation, while the resolution in the NFD method is caused by the limited divided angle number. Therefore, for a certain special situation, such as FO = 0, the error estimated using the LFD method becomes extremely low, but it does not make any sense in practical application.

Fig. 5. Average estimation error versus true frequency offset with (a) single eigenvalue transmission {0.3j}; (b) multiple eigenvalues transmission {0.3j,0.6j}. magenta circles: LFD method; blue rectangle: NFD method; (c) estimated FO error changes with the length of the training symbol. Lines and marks in blue: single eigenvalue transmission {0.3j}. Lines and marks in magenta: multiple eigenvalues transmission {0.3j,0.6j}; (d) Q-factor in multiple eigenvalues transmission. Lines and marks in blue: QPSK format. Lines and marks in magenta: 16QAM format. Circle: results obtained by the NFD method; diamond: results obtained by the LFD method.

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Figures 5(c) and 5(d) illustrate the estimation errors with different TS lengths and corresponding Q-factors for QSPK and 16QAM data symbols in the multiple eigenvalue case. The utilized TS length is {16, 32, 64, 128, 512, 1024, 2048}, and a 300 MHz FO is assumed here for the performance evaluation. The Q-factor is calculated from around 2¹⁵ data symbols. Figure 5(c) shows that the estimated error of the NFD method is lower than 0.1 MHz (single eigenvalue) or 1 MHz (multiple eigenvalues) with 16 TS, while with the same TS length, the LFD method cannot work at all. The LFD method requires more than 128 TS to reduce the error to 1 MHz and needs 2048 TS to approach the same error level as the NFD method. The estimated error trends in our proposed single and multiple eigenvalue NFDM systems are very similar; therefore, we only present the corresponding Q-factor in the multiple eigenvalue case in Fig. 5(d). In terms of Q-factor, a 25 dB for 16 QAM and more than 30 dB Q-factor for

QPSK can be achieved by the NFD method. However, the LFD method still requires 128 or more TS to reach that Q-factor level for both QPSK and 16QAM formats. Combining the results in Figs. 5(c) and 5(d), we found that the higher estimated FO error made by the LFD method is suppressed by the phase noise compensator to achieve the same Q-factor using more than 128 TS.

5.2 Fiber transmission results

The performance results for the fiber transmission scenario are presented in this section. First, the phase noise’ impact on the LFD and NFD frequency offset estimation methods are investigated. Figures 6(a)–6(b) illustrate the estimation performances of the two methods with phase noise introduced by different linewidths of the lasers. The linewidth (LW) here refers to the phase noise level for both the transmitter and local lasers, i.e., LW = 50 kHz means the linewidths of lasers at TX and RX are both 50 kHz. The signal is transmitted along 1600 km (single eigenvalue) or 960 km (multiple eigenvalues) fibers. The NFD and LFD methods work with 128 TS and are tested by 300 independent evaluations to obtain the average estimation error. The results reveal that the NFD method still provides a stable estimation error smaller than 1 MHz, while the LFD method presents an error between 1 MHz to 10 MHz. The LFD method is not effected by phase noise because the absolute value of the linear spectrum is inherently immune to phase noise. Meanwhile, the induced phase noise will introduce a small penalty in the NFD estimation performance: a 200 kHz linewidth leads to an increase in error of around 0.3 MHz to 0.7 MHz in single eigenvalue transmission, and 0.3 MHz to 0.9 MHz in multiple eigenvalue transmission. However, this penalty is acceptable, and the NFD method is still superior to the LFD method up to the 200 kHz linewidth.

Fig. 6. Average estimation error versus the true frequency offset with a 128 TS length and different line width levels in the fiber transmission scenario: (a) single eigenvalue transmission {0.3j}; (b) multiple eigenvalues transmission {0.3j,0.6j}. Estimated FO error and Q-factor changes with the length of the training symbol. A 300 MHz FO is set: (c) single eigenvalue transmission {0.3j} after 1600 km propagation (20 spans). LW = 50 kHz; and (d) multiple eigenvalue transmission {0.3j,0.6j} after a 960 km propagation (12 spans) LW = 50 kHz. Blue dots: errors obtained with NFD method; blue diamond: errors obtained with the LFD method; rectangle: Q-factor obtained with the NFD method; triangle: Q-factor obtained with the LFD method. Marks and lines in blue: QPSK format; marks and lines in magenta: 16QAM format.

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Figure 6(c) illustrates the estimated error versus the length of the training symbol in the fiber transmission scenarios. The line width is set to 50 kHz. The transmission distance is also 1600 km for the single eigenvalue case and 960 km for the multiple eigenvalue case. Overall, the NFD method had fewer estimation errors than the LFD method. Furthermore, a decreasing trend is shown with increasing TS length for both the LFD and the NFD methods. For single eigenvalue transmission, the error of the LFD method is lower than 1 MHz at 512 symbols and around 0.2 MHz at 2048 symbols, while the error of the NFD method is lower than 1 MHz at 64 symbols and around 0.2 MHz at 512 symbols. Thus, the NFD method provides a similar estimation error with a lower TS. A similar conclusion can be drawn for the multiple eigenvalue case. Figure 6(d) illustrates the corresponding Q-factor in the multiple eigenvalue transmission scenario. For the multiple eigenvalue case, the Q-factor for QPSK by the NFD method remains high (around 20 dB), while the results obtained by the LFD method converge to that level at 128 TS. Compared with QPSK, 16QAM needs more TS to maintain a higher Q-factor level for both methods. To achieve a high Q-factor, the NFD method needs 128 TS for 16QAM; however, the LFD method needs approximately 512 TS, indicating that the NFD method can save 75% TS overhead with the 16QAM modulation system compared with LFD method. For the QPSK format, as the Q-factor reaches the high level at TS = 16, we believe at least 87.5% of the overhead can be reduced. A larger TS length is required with an increase in the order of the modulation format, which implies the NFD method is more meaningful for high order modulation formats.

We also obtained the performance results as a function of the transmission distance, as illustrated in Fig. 7. The FO, laser linewidth, and TS length are set to 300 MHz, 50 kHz, and 128, respectively. We only illustrate the results in multiple eigenvalue transmission systems, as the same results can be obtained for the single eigenvalue case. The largest distance for multiple eigenvalue transmission is 2560 km (32 spans). Figure 7(a) shows that the estimated errors of the two estimation methods remain unchanged along with the span number. However, the LFD method may still work in this case. This situation is meaningless because the information symbols will still be unable to recover owing to failure of the NFT even if the FO is estimated correctly. Figure 7(b) illustrates the Q-factor variation with different transmission distances. The Q-factors for QPSK format behave almost the same during transmission when using both the LFD and NFD methods, which is consist with the results in Fig. 6(d). However, the 16QAM signal cannot be recovered correctly by the LFD method with 128 TS owing to the insufficient accuracy of FO estimation, while the NFD method has a function for equalizing FO impairment at all test distances. The results indicate that the proposed NFD method can provide stable estimation performance and is feasible for practical long-haul transmission.

Fig. 7. Estimated FO error and Q-factor changes with the length of the training symbol in multiple eigenvalue transmission systems. FO = 300 MHz, TS length = 128, and LW = 50kHz: (a) Estimation error results. Blue dots: error by the NFD method; blue diamond: error by the LFD method; (b) Q-factor results. rectangle: NFD estimation & compensation; triangle: LFD estimation & compensation, marks and lines in blue: QPSK format, and marks and lines in magenta: 16QAM format.

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5.3 Complexity analysis

The complexity of Algorithm 1 is about O(mN_angle), where m is the length of the training symbol and N_angle is the divided angle number. This complexity evaluation was used to evaluate the complexity of the popular used constant modulus algorithm (CMA), which is proportional to input parameters [33]. Therefore, we think complexity of Algorithm 1 is acceptable for DSP implement. However, in fact, NFD method may include extra forward NFT for training symbol. Correspondingly, FFT operation should also be counted as part of LFD method. The overall review of complexity of these two methods are given in the following Table 2. For the NFD method, the main procedures include forward NFT and angle search, while for LFD, FFT and sorting for pilot are unavoidable actions. The forward NFT can achieve a complexity of O(N₁(K + log₂ N₁)) according to [34], where K is number of eigenvalues, N₁ is the sampled point number (from training burst) for NFD method. Accordingly, FFT has the complexity of O(N₂ log₂ N₂), where N₂ is the number for LFD method. However, it should be noticed that the complexity from [34] do not cover the operation of eigenvalues location process, where operation number relies on many other aspects (denote X in the Table 2). In conclusion, the total complexity of NFD method is a little higher, which was verified by the running time consumed when we made the simulations using our computer.

Table 2. Complexity comparison of NFD and LFD method

View Table | View all tables in this article

5.4 Further discussion and perspectives

Besides, there are some issues need to be clarified and explained. The first one is about performance with q-modulation, i.e., using Q_d in Eq. (4) to carry data symbols. All the numerical results above are based on b-coefficient modulation because it has been shown that b-modulation performs better than q-modulation in data transmission [35]. However, q-modulation is still an option. According to our test results, the estimation error provided by q-modulation is similar to the error provided by b-modulation, but with a lower Q-factor owing to the larger noise level accumulated on the Q_d coefficients.

Second, as mentioned in the last paragraph in Section 3.2, the feasibility of estimating the FO by using shift of the eigenvalue should be clarified. In general, this shift can provide FO information, and can be obtained by finding the eigenvalue locations through performing NFT, but it is not accurate enough in practical implementations. The FO estimation accuracy depends strongly on the accuracy of finding location of the eigenvalues. Any distortion of the transmitted waveform will make the eigenvalues drift from their original positions, or even induces some pseudo-eigenvalues, as shown in Fig. 8(a). The factors that making the distortion of the original waveform are imperfections of the transmitter, fiber channel impairments (ASE noise etc.), and the sampling number D at the receiver. However, aided by the training symbols, the angle searching method obtain more accurate estimation, as shown in Fig. 8(b). We can see in Fig. 8(b) that by using finding eigenvalue shift, the FO estimation error is beyond several MHz or more than 10 MHz, and more accuracy needs larger sampling number (more hardware resources). On the other hand, by using angle search method, the estimation error reduces to around 100 kHz. Therefore, we adopt here the angle search method instead of eigenvalue shift location.

Fig. 8. (a) Example of eigenvalue distribution at the receiver; (b) the errors estimated by eigenvalue shift and angle search vary with different sampling numbers D. (c) Estimation performance example in single eigenvalue system with 6G baud rate.

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Finally, we would like to make a comment on: if the angle search algorithm adopted in this paper is also applicable to the NFDM system with higher capacity. We find the algorithm is irrelevant to the baud rate or modulation format when we check the detail code showing in Algorithm 1. Therefore, it can be used for any discrete spectrum modulated system with available baud rate. Figure 8(c) illustrates an example of estimation performance with 6G baud rate, which is very similar to the results in Fig. 5(a)–5(b).

6. Conclusion

Like in linear coherent optical communication systems, frequency offsets are still a challenging issue for NFT-based systems. In this paper, the principles that describe the effects of a frequency offset in an NFDM system are first fully reviewed. Then, a more effective solution for frequency offset (FO) estimation for an NFT-based eigenvalue transmission system is presented. The proposed method is realized by a data-aided angle search algorithm in the nonlinear frequency domain (NFD), while the conventional method, which estimates an FO by a shift of the pilot in the linear frequency domain (LFD), is also introduced. The numerical results verify the better stability and FO estimation accuracy of the NFD method compared with the conventional LFD method, in both back-to-back and fiber transmission scenarios. The NFD method requires shorter training symbols to achieve good Q-factor performance for both QPSK and 16QAM signals. The results also show that the NFD method is robust to linewidth and fiber link noise. Therefore, we believe that the proposed NFD method can be a better option for FO compensation in a practical eigenvalue transmission system with a high order QAM or PSK format.

Funding

National Natural Science Foundation of China (61527820, 61571057, 61575082, 61875247); Beijing Excellent Ph.D. Thesis Guidance Foundation (CX2018113).

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Frequency offset estimation for nonlinear frequency division multiplexing with discrete spectrum modulation

Abstract

1. Introduction

2. NFT Principle and fundamental properties

2.1 Brief review of the NFT theory

2.2 Fundamental properties of the NFT

3. Frequency offset and estimation method

3.1 Frequency offset effects in the NFDM system

3.2 FO estimation in the nonlinear frequency domain

Algorithm 1.

3.3 FO estimation in the linear frequency domain

4. NFDM system with discrete spectrum modulation

4.1 Single-/Multi-eigenvalues transmission platform

4.2 FO compensation by the LFD and NFD methods

5. Numerical results and analysis

5.1 Back-to-back results

5.2 Fiber transmission results

5.3 Complexity analysis

5.4 Further discussion and perspectives

6. Conclusion

Funding

References

Cited By

Figures (8)

Tables (2)

Equations (15)

Optics Express

Eigen-values	Modulation Format	t_max	Baud Rate	Normalization T₀ [ps]	Bandwidth W [GHz]	Sampling Number D	Launch Power [dBm]
0.3j	QPSK	9	2G	28	12	64	1.5
0.3j	16QAM	10.5	2G	24	12	64	2.1
0.3j,0.6j	QPSK	10	1G	50	22	128	0.8
0.3j,0.6j	16QAM	11.5	1G	43	25	128	1.4

	Forward NFT	Angle search	Eig. location
NFD method	O(N₁(K + log₂ N₁))	O(mN_angle)	X
	FFT	Sorting for pilot
LFD method	O(N₂ log₂ N₂)	$O (N_{2}^{2})$