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Nonlinear frequency domain PMD modeling and equalization for nonlinear frequency division multiplexing transmission

Xulun Zhang, Lixia Xi, Jiacheng Wei, Shucheng Du, Wenbo Zhang, Jianping Li, and Xiaoguang Zhang

Open Access Open Access

Abstract

Polarization mode dispersion (PMD) is one of the fundamental properties of a standard single-mode fiber. It affects the propagating signals and degrades the performance of high-speed optical fiber communication systems. PMD also gives an effect on the nonlinear spectra or scattering data in nonlinear frequency division multiplexing (NFDM) systems. However, PMD is usually described in the linear frequency domain, and there are few investigations about the influence of PMD in the nonlinear frequency domain (NFD). An NFD-PMD model is needed to understand the impact of PMD in the NFD. In this work, using a linear approximation method, we first propose an NFD-PMD model and verify its effectiveness. With the guide of the NFD-PMD model, a blind NFD-PMD equalization scheme is designed. The simulation results indicate that the proposed NFD-PMD equalization scheme has better performance than the training sequence method based on linear frequency domain equalization.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Kerr nonlinearity has become a main obstacle limiting the capacity of long-haul optical fiber communications [1]. Due to the natural advantage of overcoming the limitations of Kerr nonlinearity, the nonlinear frequency division multiplexing (NFDM) has gotten a lot of attention in recent years [2–6]. It uses the nonlinear Fourier transform (NFT) [7,8] to convert the time domain signal into the nonlinear spectra which evolve linearly and independently along the nonlinear fiber channel. In the NFDM system, information is loaded onto the nonlinear spectra, which means they are free from the crosstalk among them. Thus, NFDM is regarded as a promising technique to break through the limit of fiber nonlinearity [2–6,9,10].

The nonlinear spectra consist of discrete spectrum and continuous spectrum, which can be modulated separately or jointly [11–25]. For the discrete spectrum modulated system, the spectral efficiency (SE) is quite low. Up to now, the used maximum eigenvalues number of the discrete spectrum is 12, and the SE is only about 0.25bit/s/Hz [12]. Naturally, more researchers have paid attention to the continuous spectrum and full spectrum (continuous spectrum plus discrete spectrum) modulated systems for pursuing higher SE [13–18,21–25]. However, due to the limit of accuracy of NFT and inverse NFT (INFT) algorithm, as well as the nonlinear cross-talks between the continuous spectrum and discrete spectrum, the benefit of full-spectrum modulation is hindered [9,16,18,24,25]. Therefore, we only focus on the continuous spectrum modulated NFDM system, which is analogous to the orthogonal frequency division multiplexing (OFDM) system in the linear frequency domain. The NFDM system has been extended from single-polarization to polarization division multiplexing transmission [17–25]. However, polarization impairments seriously affect the propagating signals [26,27]. In this work, we study the effect of polarization mode dispersion (PMD) on the continuous spectrum modulated NFDM system.

PMD, originating from the random birefringence, is one of the fundamental properties of standard single-mode fiber (SSMF) [26–28]. For the NFDM transmission system, there are rare studies on PMD modeling and equalizations. Many proof-of-concept simulations directly ignore the impact of PMD [2–6, 22,25]. In the experiments, the polarization state of the signal is usually manually adjusted by using a polarization controller at the receiver side [19,20,24]. Due to the time-varying property of PMD [29,30], the above solution is not feasible in the practical transmission scenario. There are a few reports about PMD equalization using the linear system signals as training sequences, such as root-raised cosine pulses and OFDM signal [21,23]. However, this scheme complicates the NFDM system and wastes spectral efficiency.

The PMD of SSMF is essentially a linear frequency domain (LFD) impairment. However, in the NFDM transmission system, the data is loaded into the nonlinear frequency domain (NFD). To better understand the impact of PMD and design the equalization scheme, we first construct an NFD-PMD model in this work. Based on the proposed model, a blind PMD equalization scheme is proposed and its effectiveness is verified.

The rest of this paper is organized as follows. Section 2 reviews the channel models of SSMF channel links, the basic properties about NFT, and the multi-section cascade LFD-PMD model. In section 3, according to the basic properties of NFT and channel model, an NFD-PMD model is derived. The effectiveness of the NFD-PMD model is verified in section 4. In section 5, an NFD-PMD model-based PMD equalization scheme is proposed and verified. Finally, the conclusion is summarized in section 6.

2. Channel model

2.1 Channel model without PMD

If the PMD is not considered, the evolution of dual-polarization (DP) signal along the SSMF whose random birefringence varies rapidly with the distance, is governed by the Manakov equation [31,32]

$$i\frac{{\partial {\textbf Q}}}{{\partial z}} - \frac{{{\beta _2}}}{2}\frac{{{\partial ^2}{\textbf Q}}}{{\partial {t^2}}} + \frac{8}{9}\gamma {||{\textbf Q} ||^2}{\textbf Q} = {\textbf 0},$$
where ${\textbf Q} = {[{Q_x}({t,z} ),{Q_y}({t,z} )]^T}$ is the Jones vector containing the two polarization components, ${||{\textbf Q} ||^2} = {{\textbf Q}^\dagger }{\textbf Q}$, and $\dagger $ denotes Hermitian conjugation, $t{({\textrm{ps}})}$ and $z{({\textrm{km}})}$ are retarded time and distance, ${\beta _2}({{{{\textrm{ps}}^2}} / {\textrm{km}}})$ represents group velocity dispersion coefficient, and $\gamma ({{\textrm{W}}^{ - 1}}{{\textrm{km}}^{ - 1}})$ denotes the nonlinearity parameter. The Manakov equation can be normalized as
$$i\frac{{\partial {\textbf q}}}{{\partial l}} + \frac{{{\partial ^2}{\textbf q}}}{{\partial {\tau ^2}}} + 2{||{\textbf q} ||^2}{\textbf q} = {\textbf 0},$$
by introducing the normalization parameters in Eq. (3)
$${\kern 1pt} {\textbf q} = {{\textbf Q} / {{A_S}}},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \tau = {t / {{T_S}}},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} l = {z / {{Z_S}}},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {Z_S} = {{2{T_S}^2} / {|{{\beta_2}} |}},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {A_S} = \sqrt {{2 / {(\frac{8}{9}\gamma {Z_S})}}} ,$$
where ${T_S}$ is set to 1 ns. Using the NFT, the time domain signals are converted into the nonlinear spectra [8,17,25]. The continuous spectrum is defined as
$$\hat{{\textbf q}}(\lambda ) = {{{\textbf b}(\lambda )} / {a(\lambda )}},$$
where $\lambda $ is the nonlinear frequency defined on the real axis, ${\textbf b}(\lambda ) = {[{{b_1}(\lambda ),{b_2}(\lambda )} ]^T}$ and $a(\lambda )$ are called the scattering data. The continuous spectrum $\hat{{\textbf q}}(\lambda )$ and scattering data $a(\lambda )$ satisfy the relations:
$${\kern 1pt} {\kern 1pt} |{a(\lambda )} |= \frac{1}{{\sqrt {1 + {{|{{{\hat{\textrm q}}_1}(\lambda )} |}^2} + {{|{{{\hat{\textrm q}}_2}(\lambda )} |}^2}} }},\textrm{Arg}(a(\lambda )) = {\cal H}({\log |{a(\lambda )} |} ),$$
where $\textrm{Arg}(a(\lambda ))$ denotes the phase of $a(\lambda )$ and ${\cal H}$ is the Hilbert transform. The evolution of scattering data and continuous spectrum along the ideal fiber channel satisfy the rules:
$$a(\lambda ,l) = a(\lambda ,0){\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\textbf b}(\lambda ,l) = {\textbf b}(\lambda ,0){e^{4j{\lambda ^2}l}},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \hat{{\textbf q}}(\lambda ,l) = \hat{{\textbf q}}(\lambda ,0){e^{4j{\lambda ^2}l}}.$$

2.2 Channel model with PMD

PMD is one of the fundamental properties of SSMF. The evolution of DP signal along the SSMF considering PMD is governed by the Manakov-PMD equation [31–33]

$$i\frac{{\partial {\textbf Q}}}{{\partial z}} - \frac{{{\beta _2}}}{2}\frac{{{\partial ^2}{\textbf Q}}}{{\partial {t^2}}} + \frac{8}{9}\gamma {||{\textbf Q} ||^2}{\textbf Q} ={-} i{{{\boldsymbol{\mathrm{\beta}}}}_1}(z)\frac{{\partial {\textbf Q}}}{{\partial t}},$$
where ${{{\boldsymbol{\mathrm{\beta}}}}_1}$ represents PMD and it is a unitary $2 \times 2$ matrix as shown in Eq. (8)
$${{{\boldsymbol{\mathrm{\beta}}}}_1}(z) = \frac{{\Delta {\beta _1}(z )}}{2}{{{\boldsymbol{\mathrm{\sigma}}}}_{\textbf 1}}{\textbf R}(z),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{{\boldsymbol{\mathrm{\sigma}}}}_{\textbf 1}} = \left( {\begin{array}{cc} 1&0\\ 0&{ - 1} \end{array}} \right),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\textbf R}(z) = \left[ {\begin{array}{cc} {\cos {\kappa_z}}&{ - \sin {\kappa_z}{e^{j{\eta_z}}}}\\ {\sin {\kappa_z}{e^{ - j{\eta_z}}}}&{\cos {\kappa_z}} \end{array}} \right],$$
where $\Delta {\beta _1}({{ps} / {km}})$ is the difference of inverse group velocity (DIGV) between the two polarizations, ${\textbf R}(z)$ denotes the rotation matrix of the principal states of polarization (PSP) [33], and ${\kappa _z},{\eta _z} \in [{ - \pi ,\pi } ]$ are uniformly distributed random variables.

If the Kerr nonlinearity is not considered, the fiber channel can be divided into multiple sections to model PMD. Each section has random PSP orientation and DIGV parameter. Equation (9) describes the total PMD of fiber channel in linear frequency domain (LFD) as

$${\textbf U}(f) = \mathop \prod \limits_{m = 1}^M {{\textbf U}_m},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\textbf U}_m} = \left[ {\begin{array}{{cc}} {{e^{j2\pi f\frac{{\Delta {\beta_{1,m}}{\textrm{z}_c}}}{2}}}}&0\\ 0&{{e^{ - j2\pi f\frac{{\Delta {\beta_{1,m}}{\textrm{z}_c}}}{2}}}} \end{array}} \right]\left[ {\begin{array}{cc} {\cos {\kappa_m}}&{ - \sin ({{\kappa_m}} ){e^{j{\eta_m}}}}\\ {\sin {\kappa_m}{e^{ - j{\eta_m}}}}&{\cos {\kappa_m}} \end{array}} \right],$$
where f is linear frequency, ${z_c}$ is the section length, and $\Delta {\beta _{1,m}}{z_c}$ is the differential group delay (DGD) in the mth section. The DIGV parameter $\Delta {\beta _{1,m}}$ satisfies Gaussian distribution with the mean ${\mu _P} = {D_{PMD}}\sqrt {{{\textrm{3}\pi } / 8}{z_c}} $ and standard deviation $0.2{\mu _P}$, where ${D_{PMD}}$ is the PMD coefficient [27,28,33–36]. For the fiber channel of length L, the accumulated DGD satisfies Maxwell distribution and the mean value is ${D_{PMD}}\sqrt L $ [27,28].

For generality, the Manakov-PMD Eq. (7) can be normalized as

$$i\frac{{\partial {\textbf q}}}{{\partial l}} + i{\widetilde {{\boldsymbol{\mathrm{\beta}}}}_1}(l)\frac{{\partial {\textbf q}}}{{\partial \tau }} + \frac{{{\partial ^2}{\textbf q}}}{{\partial {\tau ^2}}} + 2{||{\textbf q} ||^2}{\textbf q} = {\textbf 0},$$
by introducing two additional normalization parameters
$${\widetilde {{\kern 1pt} {{\boldsymbol{\mathrm{\beta}}}}}_1} = \frac{{\Delta {{\tilde{\beta }}_1}}}{2}{{{\boldsymbol{\mathrm{\sigma}}}}_{\textbf 1}}{\textbf R}(z),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \Delta {\tilde{\beta }_1} = \Delta {\beta _1}\frac{{2{T_S}}}{{|{{\beta_2}} |}}.$$

Then, the corresponding normalized LFD-PMD model can be represented as

$$\tilde{{\textbf U}}({\tilde{f}} )= \mathop \prod \limits_{m = 1}^M {\tilde{{\textbf U}}_m},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\tilde{{\textbf U}}_m} = \left[ {\begin{array}{{cc}} {{e^{j2\pi \tilde{f}\frac{{\Delta {{\tilde{\beta }}_{1,m}}{l_c}}}{2}}}}&0\\ 0&{{e^{ - j2\pi \tilde{f}\frac{{\Delta {{\tilde{\beta }}_{1,m}}{l_c}}}{2}}}} \end{array}} \right]\left[ {\begin{array}{{cc}} {\cos {\kappa_m}}&{ - \sin {\kappa_m}{e^{j{\eta_m}}}}\\ {\sin {\kappa_m}{e^{ - j{\eta_m}}}}&{\cos {\kappa_m}} \end{array}} \right],$$
where $\tilde{f} = f{T_S}$ is the normalized linear frequency, ${l_c} = {{{z_c}} / {{Z_S}}}$ is the normalized section length, $\Delta {\tilde{\beta }_{1,m}} = \Delta {\beta _{1,m}} \cdot {{2{T_S}} / {|{{\beta_2}} |}}$ represents the normalized DIGV parameter and $\Delta {\tilde{\beta }_{1,m}}{l_c}$ is the normalized DGD in the mth section.

Affected by PMD, the evolution rule of continuous spectrum or scattering data is not as simple as Eq. (6). To understand the impact of PMD on the continuous spectrum and b-scattering data, in the following section 3, an NFD-PMD model is derived.

3. NFD-PMD model

To derive an NFD-PMD model, two linear approximation properties of NFT and channel model are used.

Property 1: If the time domain signal ${||{{\textbf q}(\tau )} ||_1} \ll 1$, the NFT of ${\textbf q}(\tau )$ tends to the Fourier transform (FT) [2,25] as

$${\textbf q}(\tau ) \leftrightarrow \hat{{\textbf q}}(\lambda ) \to - {{\cal F}^{\ast }}\{{{\textbf q}(\tau )} \}( - {\lambda / \pi }) ={-} \int_{ - \infty }^\infty {{{\textbf q}^\ast }(\tau ){e ^{ - j2\lambda \tau }}d\tau } ,$$
where ${\cal F}$ is the FT as ${\cal F}\{{{\textbf q}(\tau )} \}({\tilde{f}} )= \int_{ - \infty }^\infty {{\textbf q}(\tau ){e^{ - j2\pi \tilde{f}\tau }}d\tau }$ and the asterisk denotes the conjugate.

Property 2: If the time domain signal ${||{{\textbf q}({\tau ,0} )} ||_1} \ll 1$, after the evolution along the SSMF channel with and without PMD (Eqs. (10) and (2)), the signal at a distance $l$ are ${{\textbf q}_{PMD}}(\tau ,l)$ and ${\textbf q}(\tau ,l)$, respectively. Then

$${{\textbf q}_{PMD}}(\tau ,l) \to {{\cal F}^{ - 1}}\{{\tilde{{\textbf U}}({\tilde{f}} ){\cal F}\{{{\textbf q}(\tau ,l)} \}({\tilde{f}} )} \}(\tau ),$$
where ${{\cal F}^{ - 1}}$ is the inverse Fourier transform as ${{\cal F}^{\textrm{ - }1}}\{{{\textbf E}({\tilde{f}} )} \}(\tau ) = \int_{ - \infty }^\infty {{\textbf E}({\tilde{f}} ){e ^{j2\pi \tau \tilde{f}}}d\tilde{f}}$.

Based on the two properties above, we can obtain the following corollary 1. The proof can be found in the appendix.

Corollary 1. If the time domain signal ${||{{\textbf q}({\tau ,0} )} ||_1} \ll 1$, its nonlinear spectrum and b-scattering data are $\hat{{\textbf q}}(\lambda ,0)$ and ${\textbf b}(\lambda ,0)$. After the evolution along the SSMF channel with and without PMD, the signal at a distance $l$ are ${{\textbf q}_{PMD}}(\tau ,l)$, ${\textbf q}(\tau ,l)$, and their corresponding nonlinear spectrum and b-scattering date are ${\hat{{\textbf q}}_{PMD}}(\lambda ,l), {{\textbf b}_{PMD}}(\lambda ,l)$ and $\hat{{\textbf q}}(\lambda ,l),{\textbf b}(\lambda ,l)$, respectively. Then

$${\hat{{\textbf q}}_{PMD}}(\lambda ,l) \to \tilde{{\textbf U}}{( - \lambda /\pi )^\ast }\hat{{\textbf q}}(\lambda ,l) = \tilde{{\textbf U}}{( - \lambda /\pi )^\ast }\hat{{\textbf q}}(\lambda ,0){e^{4j{\lambda ^2}l}},$$
$${{\textbf b}_{PMD}}(\lambda ,l) \to \tilde{{\textbf U}}{( - \lambda /\pi )^\ast }{\textbf b}(\lambda ,l) = \tilde{{\textbf U}}{( - \lambda /\pi )^\ast }{\textbf b}(\lambda ,0){e^{4j{\lambda ^2}l}}.$$

For the NFDM system, the modulation on the b-scattering data behaves better performance than modulation on the continuous spectrum. According to the corollary above, the transmission of the b-scattering data along the fiber channel with PMD can be expressed as

$${{\textbf b}_{PMD}}(\lambda ,l) = ({{\textbf T}(\lambda ) + {\boldsymbol{\mathrm{\varepsilon}}}({\lambda ,l} )} ){\textbf b}(\lambda ,0){e^{4j{\lambda ^2}l}},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\textbf T}(\lambda ) = {\tilde{{\textbf U}}^\ast }({{ - \lambda } / \pi }) = \mathop \prod \limits_{m = 1}^M {{\textbf T}_m}(\lambda ),{\kern 1pt} {\kern 1pt} {\kern 1pt}$$
$${{\textbf T}_m} = \left[ {\begin{array}{{cc}} {{e^{j\lambda \Delta {{\tilde{\beta }}_{1,m}}{l_c}}}}&0\\ 0&{{e^{ - j\lambda \Delta {{\tilde{\beta }}_{1,m}}{z_c}}}} \end{array}} \right]\left[ {\begin{array}{{cc}} {\cos {\kappa_m}}&{ - \sin {\kappa_m}{e^{ - j{\eta_m}}}}\\ {\sin {\kappa_m}{e^{j{\eta_m}}}}&{\cos {\kappa_m}} \end{array}} \right],$$
where ${\textbf T}$ and ${{\boldsymbol{\mathrm{\varepsilon}}}}$ are called the linear and nonlinear terms of the NFD-PMD model. When ${||{{\textbf q}({t,0} )} ||_1} \ll 1$, the nonlinear term ${{\boldsymbol{\mathrm{\varepsilon}}}}$ can be ignored. In the following, the model is verified in section 4.

4. Model verification

To verify the effectiveness of the NFD-PMD model, a b-scattering data modulated dual-polarization NFDM system is designed as shown in Fig. 1. The modulation scheme is expressed as

$${u_{i,n}}(\lambda ) = A\sum\limits_{k ={-} K/2}^{K/2 - 1} {c_{i,n,k}^t\frac{{\sin (\lambda {T_0}/{T_S} + k\pi )}}{{\lambda {T_0}/{T_S} + k\pi }}} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} i \in \{{1,2} \},$$
$${b_{i,n}}(\lambda ) = \sqrt {\frac{{1 - {e^{ - {{|{{u_{1,n}}} |}^2} - {{|{{u_{2,n}}} |}^2}}}}}{{{{|{{u_{1,n}}} |}^2} + {{|{{u_{2,n}}} |}^2}}}} {u_{i,n}}(\lambda ){e^{ - 2jn\lambda {{{T_1}} / {{T_S}}}}},$$
where n is the NFDM signal block index, $A$ is the parameter controlling the power, $K$ denotes the number of nonlinear subcarriers, $c_{i,n,k}^t$ represents the transmitted symbol, ${T_0}$ is the effective block time, and ${T_1}$ is total block time including guard interval (GI). The pre-dispersion compensation (PDC) operation is performed to reduce the GI. Afterwards, the time domain pulse is computed by using the inverse Ablowitz-ladik method (INFT) [17]. We use the slip step Fourier transform method to emulate the SSMF channel with PMD, where the step length is set to 1 km [17]. The fiber channel parameters are listed in Table 1. Four kinds of SSMFs with different PMD coefficients are considered to cover a wide range of PMD. Considering the random property of PMD, 100 times fiber transmission simulations are performed. At the receiver side, after normalization, the scattering data is calculated using the Ablowitz-ladik method (NFT) [17]. Then, the interplay of Kerr nonlinearity and chromatic dispersion is inversed by the phase shift operation. To demonstrate the NFD-PMD model’s effectiveness, it is supposed that the PMD of fiber channel is known. The linear PMD part of the NFD-PMD model is used to inverse the impact of PMD. After that, the ${\textbf u}(\lambda )$ is recovered from ${\textbf b}(\lambda )$ as
$${u_{i,n}}(\lambda ) = \sqrt {\frac{{ - \log (1 - {{|{{b_{1,n}}(\lambda )} |}^2} - {{|{{b_{2,n}}(\lambda )} |}^2})}}{{{{|{{b_{1,n}}(\lambda )} |}^2} + {{|{{b_{2,n}}(\lambda )} |}^2}}}} {b_{i,n}}(\lambda ){e^{2jn\lambda {{{T_1}} / {{T_S}}}}},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} i \in \{{1,2} \},$$

Finally, the transmitted information can be obtained after OFDM decoding.

 figure: Fig. 1.

Fig. 1. NFDM system setup, where the “PMD Inversion”, and “PMD Equalization” plus “Phase Noise Estimation” modules are used in NFD-PMD model verification and model-based PMD equalization scheme verification, respectively.

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Tables Icon

Table 1. The channel link parameters

View Table

In the given B=50 GHz bandwidth, there are K=176 modulated subcarriers. The symbols are taken from a 16QAM constellation. The effective block time can be estimated as ${T_0}\textrm{ = }{K / B} = 3.52\textrm{ }ns$. As described in [37], the NFDM system suffers from severe processing noise, which mainly consists of numerical algorithm error of NFT/INFT and the impact of tails truncation. There may be an interaction between processing noise and PMD. To ensure the accuracy of model verification, an oversampling factor of R0=10 is used to mitigate the impact of numerical algorithm error and an ultra-long GI of 24.64 ns is adopted to avoid the impact of tails truncation. In this case, the data rate is about 50 Gbit/s. After 1600 km lossless and noiseless fiber channel transmission, the performance curves with ideal NFD-PMD inversion (NFD-PMDI) based on the NFD-PMD model linear term are obtained, as shown in Fig. 2, where the link PMD is assumed to be known. The results indicate that the NFD-PMD model is indeed effective. For the small PMD and burst power scenarios, the nonlinear term of the NFD-PMD model can be ignored. When the signal power and PMD increase, the impact of the nonlinear term of NFD-PMD becomes severely.

 figure: Fig. 2.

Fig. 2. Q-factor(dB) as a function of burst power for 50Gbits NFDM system (ultra-long GI) after 1600 km lossless and noiseless fiber transmission, where the channel PMD is assumed to be known and inversed in the NFD.

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Next, considering the realistic transmission scenario, the GI is set to $5.63ns$(0.52×channel memory time) and the data rate is about 153.8Gbit/s. Raman amplifier is used to compensate fiber loss and the amplifier spontaneous emission (ASE) noise is added after each step of slip step Fourier transform [5]. We also simulate the performance of the OFDM system for comparison, as shown in Fig. 3. In the case of without PMD, the optimal power of the NFDM system is about 1 dB higher than that of the OFDM, and NFDM system has a performance advantage of about 0.8 dB over the OFDM system. This means that in this case, the nonlinearity impairment appears and NFT exhibits its advantage. It can be seen that in the nonlinear region with the power greater than -5.1dBm (the optimal power of the OFDM system), when the PMD coefficient is less than 1ps/km1/2, there is no significant gap between the performance curves with and without PMD. It indicates that the linear term of the NFD-PMD model can compensate PMD well. The effect of the nonlinear term of NFD-PMD model can be ignorable in the presence of processing noise and ASE noise, that is, the effects of processing noise and ASE noise are much greater than that of the nonlinear term of the NFD-PMD model. The results demonstrate that, although the NFD-PMD modeling is based on the low power approximation condition, the proposed NFD-PMD model is not only suitable for low power (the linear region with power less than -5.1dBm) scenario, but also suitable for moderate power scenario (the nonlinear region with power greater than -5.1dBm). In addition, according to ITU-T G.652 Recommendations [38], the maximum PMD coefficient is 0.2ps/km1/2 and modern fibers have smaller PMD coefficient (0.04 ∼0.1ps/km1/2) [36]. The linear term of the NFD-PMD model is effective for the modern optical fiber channel. This means that the PMD equalization scheme of the modern fiber link can be designed based on the linear term of the NFD-PMD model.

 figure: Fig. 3.

Fig. 3. Q-factor(dB) as a function of burst power for a 153.8Gbit/s NFDM/OFDM system after 1600 km fiber transmission, where the channel PMD is assumed to be known and inversed in the NFD.

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5. NFD-PMD model-based PMD equalization scheme

5.1 Proposed PMD equalization scheme

After the inverse of channel response induced by chromatic dispersion and Kerr nonlinearity, the b-scattering data affected by PMD, can be expressed as

$${\textbf b}_{PMD}^I(\lambda ) = {{\textbf b}_{PMD}}(\lambda ,l){e^{ - j4{\lambda ^2}l}} = ({{\textbf T}(\lambda ) + {{\boldsymbol{\mathrm{\varepsilon}}}}({\lambda ,l} )} ){\textbf b}(\lambda ,0).$$

As discussed in section 4, the nonlinear term ${{\boldsymbol{\mathrm{\varepsilon}}}}$ of the NFD-PMD model can be ignored. In addition, since ${\textbf T}(\lambda )$ is a unitary matrix, the ${\textbf u}_{PMD}^I(\lambda )$, derived from ${\textbf b}_{PMD}^I(\lambda )$ according to the Eq. (21), can be described as

$${\textbf u}_{PMD}^I(\lambda ) = {\textbf T}(\lambda ){\textbf u}(\lambda ).$$

After decoding from ${\textbf u}_{PMD}^I(\lambda )$, Eq. (24) describes the demodulated symbol as

$${\textbf c}_{n,k}^I = {{\textbf T}_{n,k}}{\textbf c}_{n,k}^t\textrm{ = }\left[ {\begin{array}{{cc}} {\cos {\kappa_{n,k}}{e^{j{\xi_{n,k}}}}}&{\sin {\kappa_{n,k}}{e^{j{\eta_{n,k}}}}}\\ { - \sin {\kappa_{n,k}}{e^{ - j{\eta_{n,k}}}}}&{\cos {\kappa_{n,k}}{e^{ - j{\xi_{n,k}}}}} \end{array}} \right]{\textbf c}_{n,k}^t,$$
where ${{\textbf T}_{n,k}}$ is a unitary matrix operating on the kth subcarrier of the nth signal block. If we consider the impact of phase noise [23], the demodulated symbol becomes
$${\textbf c}_{n,k}^p = {e^{ - j{\delta _n}}}{\textbf c}_{n,k}^I = {e^{ - j{\delta _n}}}\left[ {\begin{array}{{cc}} {\cos {\kappa_{n,k}}{e^{j{\xi_{n,k}}}}}&{\sin {\kappa_{n,k}}{e^{j{\eta_{n,k}}}}}\\ { - \sin {\kappa_{n,k}}{e^{ - j{\eta_{n,k}}}}}&{\cos {\kappa_{n,k}}{e^{ - j{\xi_{n,k}}}}} \end{array}} \right]{\textbf c}_{n,k}^t,$$
where ${e^{j{\delta _n}}}$ is the phase noise in nth NFDM signal block.

Based on the impairment model above, we design a blind PMD equalization scheme with a Kalman filter. The Kalman filter is a strong adaptive equalizer with a recursive nature [39], which has been introduced in optical communications to equalize polarization effects [40–43]. It has three important key issues.

Issue (1): choose an adequate state vector. The impairment matrix induced by PMD can be represented by the three parameters ${\kappa _s},{\xi _s},{\eta _s}$. Thus, the state vector is selected as

$$\hat{{\textbf x}} = {({{\kappa_s},{\xi_s},{\eta_s}} )^T}.$$

Issue (2): perform a proper equalization operation. The equalization of the symbols affected by PMD can be expressed as

$${\textbf c}_{n,k}^e = \left[ {\begin{array}{{cc}} {\cos {\kappa_s}{e^{j{\xi_s}}}}&{ - \sin {\kappa_s}{e^{j{\eta_s}}}}\\ {\sin {\kappa_s}{e^{ - j{\eta_s}}}}&{\cos {\kappa_s}{e^{ - j{\xi_s}}}} \end{array}} \right]{\textbf c}_{n,k}^P.$$

Issue (3): select appropriate measurement and innovation parameters. For the perfect PMD equalization, the symbols constellation has an obvious feature. For example, the 16QAM modulation format constellation should converge to three circles. In addition, because the ${\textbf T}(\lambda )$ changes slowly in the adjacent nonlinear frequency point, ${K_a}$ adjacent subcarriers can be performed PMD equalization together. As a result, the complexity of PMD equalization is reduced. Based on the consideration above, the measurement matrix is constructed as

$$h({\hat{{\textbf x}}} )= \sum\limits_{k = 1}^{{K_a}} {{{\textbf m}_k}} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\textbf m}_k} = \left[ {\begin{array}{{cc}} {\prod\limits_{g = 1}^G {({{{|{c_{1,n,k}^e} |}^2} - r_g^2} )} }\\ {\prod\limits_{g = 1}^G {({{{|{c_{2,n,k}^e} |}^2} - r_g^2} )} } \end{array}} \right],$$
where ${r_g}$ denotes the radii of different circles, and $G = 3$ for 16QAM constellation. The innovation is
$${\textbf e} = \left( {\begin{array}{{c}} 0\\ 0 \end{array}} \right) - h({\hat{{\textbf x}}} ).$$

The running process of the Kalman filter is given as [43]

$${\hat{{\textbf x}}_{n|n - 1}} = {\hat{{\textbf x}}_{n - 1|n - 1}},$$
$${{\textbf P}_{n|n - 1}} = {{\textbf P}_{n - 1|n - 1}} + {\textbf Q},$$
$${{\textbf G}_n}{\textbf = }{{\textbf P}_{n|n - 1}}{\textbf H}_n^T{({{{\textbf H}_n}{{\textbf P}_{n|n - 1}}{\textbf H}_n^T{\textbf + R}} )^{ - 1}},$$
$${\hat{{\textbf x}}_{n|n}}{\textbf = }{\hat{{\textbf x}}_{n|n - 1}}{\textbf + G}{}_n{{\textbf e}_n},$$
$${{\textbf P}_{n|n}}{\textbf = }({{\textbf I - }{{\textbf G}_n}{{\textbf H}_n}} ){{\textbf P}_{n|n - 1}},$$
where ${\hat{{\textbf x}}_{n|n - 1}}$ and ${\hat{{\textbf x}}_{n - 1|n - 1}}$ denote the priori state and posteriori state estimate, respectively. P denotes the state vector covariance matrix, ${\textbf H} = {{\partial \textrm{h}({\textbf x})} / {\partial {\textbf x}}}$, and ${\textbf G}$ represents the Kalman gain matrix. The matrices ${\textbf Q}$ and ${\textbf R}$ denote the covariance matrix of state noise and measurement noise.

Figure 4 shows the DSP flow of the PMD equalization and phase noise estimation scheme. K subcarriers are divided into multiple sub-blocks of the length of Ka. The symbols of sub-blocks are fed into the PMD equalization module and phase noise estimation module (blind phase search [44]) to recover the symbols information.

 figure: Fig. 4.

Fig. 4. PMD equalization and phase noise estimation (blind phase search)

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5.2 Effectiveness verification of the scheme

We performed the PMD equalization scheme verification in the 153.8Gbit/s NFDM system above. Here, the linewidth of lasers of the transceiver is set to 100kHz. The differential Gray coding is adopted to cancel the impact of cycle slip caused by phase noise, but it induces additional performance penalty [45] comparing with the Gray coding used in section 4. Based on the maximum value of PMD coefficient in the ITU-T G.652 recommendations [38], the fiber link PMD coefficient is set to 0.2ps/km1/2. The performance of the proposed PMD equalization scheme is dependent on the sub-block length. Figure 5 shows the Q-factor as a function of sub-block length, where the burst power of the signal is -4.2 dBm (the optimal burst power). It can be seen that the optimal sub-block length is 18, which is the result of the balance of phase noise and PMD. Using the optimal sub-block length, the proposed equalization scheme is compared with the training sequence (TS) method (50Gbaud Root-raised cosine (RRC) pulses with a roll-off factor of 0.1, QPSK modulation format) [23]. The oversampling factor of RRC pulses is 10, which is same with NFDM signals. With the help of TS, constant modulus algorithm (CMA) [23,46] with 15 taps and 1e-3 step size is used to estimate the link PMD. When CMA converges, it is used to equalize the PMD impairment of NFDM signals before NFT [23]. Figure 6 shows the Q-factor as a function of burst power after using different PMD equalization schemes. We can observe that the proposed scheme has better performance than TS within the range of soft-decision forward error correction (SD-FEC) threshold. It should be noted that the proposed scheme is blind. The NFDM signals can complete PMD equalization by themselves without TS.

 figure: Fig. 5.

Fig. 5. Q-factor(dB) as a function of the sub-block length, where the burst power of signal is -4.2 dBm.

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 figure: Fig. 6.

Fig. 6. Q-factor(dB) as a function of burst power after using different PMD equalization schemes.

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6. Conclusion

To the best of our knowledge, we first propose an NFD-PMD model using a linear approximation method. The model consists of a linear term and a nonlinear term. Firstly, we verified the effectiveness of the proposed NFD-PMD model in a 50Gbit/s NFDM transmission simulation along the lossless and noiseless channel with an ultra-long guard interval to avoid the interplay between PMD and processing noise. The results show that the NFD-PMD model is almost effective. Next, a 153.8Gbit/s NFDM transmission with common used guard interval was performed. Interestingly, in the presence of processing noise and ASE noise, the impact of the nonlinear term become ignorable, although in the large burst power and PMD scenario. The results demonstrate that the linear term plays a main role in the NFD-PMD model and the nonlinear term can be ignored. Furthermore, we design a blind NFD-PMD model-based PMD equalization scheme employing Kalman filter and verify its effectiveness. The results indicate that the proposed blind PMD equalization scheme has better performance than the TS method.

The proposed NFD-PMD model enriches the basic theory of physical impairments behaving in the NFD. It may provide theoretical guidance for the future PMD monitoring of channel links in the NFDM system.

Appendix: Proof of the corollary 1

If ${||{{\textbf q}({\tau ,0} )} ||_1} \ll 1$, utilizing Property 1, one can obtain

$${\hat{{\textbf q}}_{PMD}}(\lambda ,l) \to - {{\cal F}^{\ast }}\{{{{\textbf q}_{PMD}}(\tau ,l)} \}( - {\lambda / \pi }).$$

After substituting Eq. (14) into Eq. (17), the Eq. (15) can be demonstrated. Note that the ${\tilde{{\textbf U}}^\ast }({{ - \lambda } / \pi })$ is a unitary matrix for any nonlinear frequency. Employing the relationship between continuous spectrum and scattering data as Eq. (4) and Eq. (5), the Eq. (16) can be obtained.

Funding

National Key Research and Development Program of China (2018YFB1800901); Fundamental Research Funds for the Central Universities (2020XD-A05-2); National Natural Science Foundation of China (62071065); China Postdoctoral Science Foundation (2020M680463).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (6)
Fig. 1.
Fig. 1. NFDM system setup, where the “PMD Inversion”, and “PMD Equalization” plus “Phase Noise Estimation” modules are used in NFD-PMD model verification and model-based PMD equalization scheme verification, respectively.

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Fig. 2.
Fig. 2. Q-factor(dB) as a function of burst power for 50Gbits NFDM system (ultra-long GI) after 1600 km lossless and noiseless fiber transmission, where the channel PMD is assumed to be known and inversed in the NFD.

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Fig. 3.
Fig. 3. Q-factor(dB) as a function of burst power for a 153.8Gbit/s NFDM/OFDM system after 1600 km fiber transmission, where the channel PMD is assumed to be known and inversed in the NFD.

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Fig. 4.
Fig. 4. PMD equalization and phase noise estimation (blind phase search)

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Fig. 5.
Fig. 5. Q-factor(dB) as a function of the sub-block length, where the burst power of signal is -4.2 dBm.

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Fig. 6.
Fig. 6. Q-factor(dB) as a function of burst power after using different PMD equalization schemes.

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Tables (1)

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Table 1. The channel link parameters

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Equations (35)

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i ∂ Q ∂ z − β 2 2 ∂ 2 Q ∂ t 2 + 8 9 γ | | Q | | 2 Q = 0 ,
i ∂ q ∂ l + ∂ 2 q ∂ τ 2 + 2 | | q | | 2 q = 0 ,
q = Q / A S , τ = t / T S , l = z / Z S , Z S = 2 T S 2 / | β 2 | , A S = 2 / ( 8 9 γ Z S ) ,
q ^ ( λ ) = b ( λ ) / a ( λ ) ,
| a ( λ ) | = 1 1 + | q ^ 1 ( λ ) | 2 + | q ^ 2 ( λ ) | 2 , Arg ( a ( λ ) ) = H ( log ⁡ | a ( λ ) | ) ,
a ( λ , l ) = a ( λ , 0 ) , b ( λ , l ) = b ( λ , 0 ) e 4 j λ 2 l , q ^ ( λ , l ) = q ^ ( λ , 0 ) e 4 j λ 2 l .
i ∂ Q ∂ z − β 2 2 ∂ 2 Q ∂ t 2 + 8 9 γ | | Q | | 2 Q = − i β 1 ( z ) ∂ Q ∂ t ,
β 1 ( z ) = Δ β 1 ( z ) 2 σ 1 R ( z ) , σ 1 = ( 1 0 0 − 1 ) , R ( z ) = [ cos ⁡ κ z − sin ⁡ κ z e j η z sin ⁡ κ z e − j η z cos ⁡ κ z ] ,
U ( f ) = ∏ m = 1 M ⁡ U m , U m = [ e j 2 π f Δ β 1 , m z c 2 0 0 e − j 2 π f Δ β 1 , m z c 2 ] [ cos ⁡ κ m − sin ⁡ ( κ m ) e j η m sin ⁡ κ m e − j η m cos ⁡ κ m ] ,
i ∂ q ∂ l + i β ~ 1 ( l ) ∂ q ∂ τ + ∂ 2 q ∂ τ 2 + 2 | | q | | 2 q = 0 ,
β ~ 1 = Δ β ~ 1 2 σ 1 R ( z ) , Δ β ~ 1 = Δ β 1 2 T S | β 2 | .
U ~ ( f ~ ) = ∏ m = 1 M ⁡ U ~ m , U ~ m = [ e j 2 π f ~ Δ β ~ 1 , m l c 2 0 0 e − j 2 π f ~ Δ β ~ 1 , m l c 2 ] [ cos ⁡ κ m − sin ⁡ κ m e j η m sin ⁡ κ m e − j η m cos ⁡ κ m ] ,
q ( τ ) ↔ q ^ ( λ ) → − F ∗ { q ( τ ) } ( − λ / π ) = − ∫ − ∞ ∞ q ∗ ( τ ) e − j 2 λ τ d τ ,
q P M D ( τ , l ) → F − 1 { U ~ ( f ~ ) F { q ( τ , l ) } ( f ~ ) } ( τ ) ,
q ^ P M D ( λ , l ) → U ~ ( − λ / π ) ∗ q ^ ( λ , l ) = U ~ ( − λ / π ) ∗ q ^ ( λ , 0 ) e 4 j λ 2 l ,
b P M D ( λ , l ) → U ~ ( − λ / π ) ∗ b ( λ , l ) = U ~ ( − λ / π ) ∗ b ( λ , 0 ) e 4 j λ 2 l .
b P M D ( λ , l ) = ( T ( λ ) + ε ( λ , l ) ) b ( λ , 0 ) e 4 j λ 2 l , T ( λ ) = U ~ ∗ ( − λ / π ) = ∏ m = 1 M ⁡ T m ( λ ) ,
T m = [ e j λ Δ β ~ 1 , m l c 0 0 e − j λ Δ β ~ 1 , m z c ] [ cos ⁡ κ m − sin ⁡ κ m e − j η m sin ⁡ κ m e j η m cos ⁡ κ m ] ,
u i , n ( λ ) = A ∑ k = − K / 2 K / 2 − 1 c i , n , k t sin ⁡ ( λ T 0 / T S + k π ) λ T 0 / T S + k π , i ∈ { 1 , 2 } ,
b i , n ( λ ) = 1 − e − | u 1 , n | 2 − | u 2 , n | 2 | u 1 , n | 2 + | u 2 , n | 2 u i , n ( λ ) e − 2 j n λ T 1 / T S ,
u i , n ( λ ) = − log ⁡ ( 1 − | b 1 , n ( λ ) | 2 − | b 2 , n ( λ ) | 2 ) | b 1 , n ( λ ) | 2 + | b 2 , n ( λ ) | 2 b i , n ( λ ) e 2 j n λ T 1 / T S , i ∈ { 1 , 2 } ,
b P M D I ( λ ) = b P M D ( λ , l ) e − j 4 λ 2 l = ( T ( λ ) + ε ( λ , l ) ) b ( λ , 0 ) .
u P M D I ( λ ) = T ( λ ) u ( λ ) .
c n , k I = T n , k c n , k t  =  [ cos ⁡ κ n , k e j ξ n , k sin ⁡ κ n , k e j η n , k − sin ⁡ κ n , k e − j η n , k cos ⁡ κ n , k e − j ξ n , k ] c n , k t ,
c n , k p = e − j δ n c n , k I = e − j δ n [ cos ⁡ κ n , k e j ξ n , k sin ⁡ κ n , k e j η n , k − sin ⁡ κ n , k e − j η n , k cos ⁡ κ n , k e − j ξ n , k ] c n , k t ,
x ^ = ( κ s , ξ s , η s ) T .
c n , k e = [ cos ⁡ κ s e j ξ s − sin ⁡ κ s e j η s sin ⁡ κ s e − j η s cos ⁡ κ s e − j ξ s ] c n , k P .
h ( x ^ ) = ∑ k = 1 K a m k , m k = [ ∏ g = 1 G ( | c 1 , n , k e | 2 − r g 2 ) ∏ g = 1 G ( | c 2 , n , k e | 2 − r g 2 ) ] ,
e = ( 0 0 ) − h ( x ^ ) .
x ^ n | n − 1 = x ^ n − 1 | n − 1 ,
P n | n − 1 = P n − 1 | n − 1 + Q ,
G n = P n | n − 1 H n T ( H n P n | n − 1 H n T + R ) − 1 ,
x ^ n | n = x ^ n | n − 1 + G n e n ,
P n | n = ( I − G n H n ) P n | n − 1 ,
q ^ P M D ( λ , l ) → − F ∗ { q P M D ( τ , l ) } ( − λ / π ) .
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