Abstract

As one of the key indicators of signal quality in fiber communication systems, optical signal-to-noise ratio (OSNR) needs to be accurately monitored to ensure reliable network planning, operation, and reconfiguration. OSNR monitoring techniques considering only accumulated amplified spontaneous emission (ASE) noise are no longer suitable for dispersion unmanaged long-haul and dense wavelength division multiplexing (WDM) systems, where the contribution of fiber nonlinearity to total SNR cannot be neglected. In this paper, we propose a modulation-format-transparent, accurate joint linear and nonlinear noise monitoring scheme based on calculation of correlation between two spectral components at the upper and lower sideband of the signal spectrum. Different characteristics of flat linear noise spectrum and non-flat nonlinear noise spectrum are used to distinguish the influences on the correlation value from both noise sources. Simulation results show that the proposed scheme can accurately monitor SNRlinear and SNRnonlinear within a wide launch power range from −5 dBm to 5 dBm per channel for multi-channel WDM systems with a 915-km single mode fiber (SMF) link. The performance of the proposed scheme is further experimentally verified in up-to-7 channel WDM systems over a 915 km SMF link.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

In next-generation flexible and dynamic optical networks, reliable network planning, operation and reconfiguration highly relies on the quality of optical path link [13]. Therefore, robust and efficient optical performance monitoring (OPM) becomes ever more important in order to extract the impairment information from the received signal [4]. The operators are able to evaluate the quality of transmission (QoT), obtain link failure warning, and identify the reasons of underperformance and take possible countermeasures according to monitored optical signal-to-noise ratio (OSNR) value. So far, there have been various OSNR monitoring techniques proposed for different development stages of optical transmission systems. Early OSNR monitoring was mainly realized in the optical domain. However, the most classic out-of-band interpolation method [5] cannot be employed for dense wavelength division multiplexing (WDM) system with small channel spacing, while polarization nulling method [6] cannot be used for polarization division multiplexing (PDM) system. Meanwhile, the robustness and reconfigurability of OSNR monitoring techniques based on traditional optical components are relatively unsatisfactory [7].

Thanks to the development of coherent detection and digital signal processing (DSP) techniques, one can make full use of optical field information to perform analysis in electrical domain [8]. The calculation of electrical SNR (ESNR) can be realized by powerful and flexible DSP algorithms, which provides great convenience for engineers. A number of DSP-based OSNR monitoring methods have been proposed, e.g. the ESNR can be calculated based on error vector magnitude (EVM) [9], Stokes-vector [10], cyclostationary property [11] and statistical moments [12] or aided by training symbols [13]. However, these methods do not take nonlinear noise into account. As analysed and verified extensively in previous works [14,15], in a long-haul transmission link with sufficient accumulated chromatic dispersion (CD) and fiber nonlinearity, the nonlinear noise can be approximately modelled as additive Gaussian noise in time domain. Therefore, it is difficult to distinguish amplified spontaneous emission (ASE) noise induced by Erbium-doped fiber amplifier (EDFA) and nonlinear noise. The presence of nonlinear noise not only leads to an underestimation of OSNR but also makes it difficult to accurately evaluate the link condition [4]. To address this problem, a number of schemes have been proposed to eliminate the OSNR estimation error induced by nonlinearity. For example, nonlinearity can be measured using inserted pilot signals [16,17]. However, the spectrum efficiency (SE) and system flexibility are sacrificed. In [18], EVM-based method is employed with the nonlinear interference (NLI) fitting using amplitude noise correlation between neighboring symbols after carrier phase recovery (CPR), while in [19,20], statistical moments-based SNR estimation method is used combining with amplitude correlation functions between two symbols of the signal to conduct NLI fitting. It should be noticed that such kind of methods should be conducted with the aid of several DSP steps (after CPR [18] or after adaptive equalization [19,20]). From the perspective of practical system, they not only exhibit relatively high power-consumption and unavoidable DSP-induced noise (e.g. the taps of adaptive equalizers do not converge perfectly) [8,21], but also are not transparent to modulation formats and hence unsuitable for flexible optical networks where multiple possible formats can be employed for data transmission [22]. On the other hand, it remains highly desirable to distinguish and monitor the linear and nonlinear noise simultaneously (not only OSNR monitoring), enabling comprehensive system analysis for operators. Machine learning (ML) based methods have shown some potential [2325], but it remains challenging to acquire huge amount of high-quality training dataset. A simpler and effective method to monitor both linear and nonlinear noise is still worthy studying.

In this paper, we propose a modulation-format-transparent, accurate joint linear and nonlinear noise monitoring scheme based on correlation calculation between two spectral components at upper and lower sideband of the signal spectrum after CD compensation (CDC). Inspired by different characteristics of flat linear noise spectrum and non-flat nonlinear noise spectrum, we carry out a semi-analytical investigation on their influences on the correlation value. Linear and nonlinear noise are found to have different contribution to SNR calculation and the decline rate of the correlation value with frequency offset (FO). The joint monitoring is successfully realized by building a system of bivariate first-order equations using both noises as variables. Simulation results show that the proposed scheme can accurately monitored SNRlinear and SNRnonlinear simultaneously, with a wide launch power range from −5 dBm to 5 dBm per channel for up-to-7-channel WDM systems. The proposed scheme is further experimentally verified in up-to-7 channel 16-quadrature amplitude modulation (QAM) WDM systems with a 915 km single mode fiber (SMF) link consisting of 12 spans with different length, supporting launch power ranging from −3 dBm to 3 dBm per channel. The experimental estimation errors of SNRlinear and SNRnonlinear are lower than 0.3 dB and 2.4 dB, respectively.

2. Operating principle

2.1 SNR estimation using signal spectral correlation

Our scheme operates on the basis of a cyclostationary property-based SNR estimation method [26]. Cyclostationary property of digitally modulated signal has been comprehensively investigated in [27,28]. Here, we provide a brief introduction of the basic principle as well as the derivation of the expression of SNR estimation. For commonly used digital modulation formats, such as amplitude-shift keying (ASK), binary phase-shift keying (BPSK), quaternary PSK (QPSK), and QAM, their pseudo-random amplitude and phase variations is difficult to distinguish from random amplitude and phase variations of additive Gaussian noise generated after transmission through optical amplified link. However, since the symbol durations are predetermined and time-invariant, the statistical property of signal should be distinguishable from that of Gaussian noise. The 1st and 2nd order statistical properties (i.e. the mean value and autocorrelation function) are defined as:

$${\bar{\mu }_x} = E[x(t)]$$
$${\bar{R}_x}(\tau ) = E[x(t\textrm{ + }\frac{\tau }{\textrm{2}}) \cdot {x^{\ast }}(t\textrm{ - }\frac{\tau }{\textrm{2}})]$$
where $x(t)$ is the signal, while $\bar{x}$ and ${x^{\ast }}$ represent the estimated value and the complex conjugate of x, respectively, and E is the expectation operator (averaging in the time domain). Although the mean values of signal and additive Gaussian noise are both zero, the autocorrelation function of digitally modulated signals is periodic in time (at periodic interval of symbol duration ${T_s}$) [27], as long as $\tau$ is less than ${T_s}$, while that of additive Gaussian noise does not exhibit such periodicity. Therefore, in [28], digital modulated signal is identified as a wide-sense cyclostationary process while additive Gaussian noise is wide-sense stationary. The periodicity of the autocorrelation function of digitally modulated signals is manifested in the signal's optical frequency spectrum, which exhibits strong correlations between time-varying amplitude and phase sequences of certain pairs of spaced apart spectral components, whereas such correlations do not exist in the optical spectrum of additive Gaussian noise. Correlations between various spectral components of a digital modulated signal can be described by a spectral correlation density function (SCDF), ${S_x}^\alpha (f)$, which is defined as the Fourier transformation of the cyclic autocorrelation function [28], ${R_x}^\alpha (\tau )$, of the time-varying signal sequence $x(t)$, as
$${R_x}^\alpha (\tau ) \equiv E[x(t + \tau /2) \cdot {x^\ast }(t - \tau /2) \cdot \textrm{exp} ( - j2\pi \alpha t)]$$
and then ${S_x}^\alpha (f)$ is
$${S_x}^\alpha (f) \equiv \int_{\textrm{ - }\infty }^\infty {{R_x}^\alpha (\tau ) \cdot \textrm{exp} ( - j2\pi f\tau )} d\tau$$

Equivalently, the SCDF may be expressed as a correlation function of the time-varying amplitudes of the spectral components of the modulated signal, as

$${S_x}^\alpha (f) \equiv E[{X_T}(t,f + \alpha /2) \cdot {X_T}^{\ast }(t,f\textrm{ - }\alpha /2)]$$
where
$${X_T}(t,v) = \int_{t - T/2}^{t + T/2} {x(u) \cdot \textrm{exp} ( - j2\pi vu)du}$$
and T is integration time much larger than ${T_s}$. Then, we normalize the SCDF to limit its range within [−1, 1], as
$${\hat{S}_x}^\alpha (f) \equiv \frac{{E[{X_T}(t,f + \alpha /2) \cdot {X_T}^{\ast }(t,f\textrm{ - }\alpha /2)]}}{{\sqrt {E[{{|{{X_T}(t,f + \alpha /2)} |}^2}]} \sqrt {E[{{|{{X_T}(t,f - \alpha /2)} |}^2}]} }}$$

It is known that for noiseless signals modulated with previously mentioned formats, QAM et.al., ${\hat{S}_x}^\alpha (f)\textrm{ = 1}$ when $\alpha \textrm{ = }\alpha {}_\textrm{0}\textrm{ = 1/}{T_s}$ and for all f within [$\textrm{ - }\alpha \textrm{/2}, \textrm{ }\alpha \textrm{/2}$]. In digital optical transmission, the bandwidth of a signal is shaped by both electrical and optical filters (e.g. raise-cosine filter, anti-aliasing filter, limited bandwidth of electronic or optics devices) to limit the occupied bandwidth. Therefore, at the receiver side, ${\hat{S}_x}^\alpha (f)\textrm{ = 1}$ holds under a slightly more stringent condition: Two frequencies $f - \alpha {}_\textrm{0}\textrm{/2}$ and $f\textrm{ + }\alpha {}_\textrm{0}\textrm{/2}$ should be within the bandwidth of the signal. In contrast to modulated signals, ASE noise is a random Gaussian process and does not exhibit any significant correlation between its spectral components. Therefore, when additive Gaussian noise is added to a modulated signal, the normalized SCDF is always smaller than unity, i.e. ${\hat{S}_x}^\alpha (f) < \textrm{1}$. Letting n denote time sequence of additive Gaussian noise and the relation between ${N_T}(t,v)$ and n be defined as Eq. (6), we have the following Fourier transform relation:

$${\tilde{X}_T}(t,v) = \int_{t - T/2}^{t + T/2} {(x(u) + n(u)) \cdot \textrm{exp} ( - j2\pi vu)du = } {X_T}(t,v) + {N_T}(t,v)$$
Then, the normalized SCDF can be expressed as
$${\hat{S}_x}^\alpha (f)\textrm{ = }\frac{{E[{X_T}(t,f + \alpha /2) \cdot {X_T}^{\ast }(t,f\textrm{ - }\alpha /2)]}}{\begin{array}{l} \sqrt {E[{{|{{X_T}(t,f + \alpha /2)} |}^2}\textrm{ + }{{|{{N_T}(t,f + \alpha /2)} |}^2}]} \cdot \\ \sqrt {E[{{|{{X_T}(t,f - \alpha /2)} |}^2}\textrm{ + }{{|{{N_T}(t,f - \alpha /2)} |}^2}]} \end{array}}$$

Noted that for the numerator, $E[{X_T}(t,f + \alpha /2) \cdot {N_T}^{\ast }(t,f\textrm{ - }\alpha /2)]$, $E[{N_T}(t,f\textrm{ + }\alpha /2) \cdot {X_T}^{\ast }(t,f\textrm{ - }\alpha /2)]$, $E[{N_T}(t,f\textrm{ + }\alpha /2) \cdot {N_T}^{\ast }(t,f\textrm{ - }\alpha /2)]$ are obviously all zero for all $\alpha > 0$. Considering the condition of $f\textrm{ = }{f_c}$, where ${f_c}$ is the carrier center frequency (after coherent detection without FO, ${f_c}$ is 0 in electrical spectrum), since the power spectra of aforementioned modulation formats are symmetric about ${f_c}$, then

$$E[{|{{X_T}(t,{f_c} + \alpha /2)} |^2}\textrm{] = }E[{|{{X_T}(t,{f_c}\textrm{ - }\alpha /2)} |^2}] = E[{P_S}({f_c} \pm \alpha /2)]$$

Meanwhile, the spectrum of additive Gaussian noise is flat over all frequency,

$$E[{|{{N_T}(t,{f_c} + \alpha /2)} |^2}\textrm{] = }E[{|{{N_T}(t,{f_c}\textrm{ - }\alpha /2)} |^2}] = E[{P_N}({f_c} \pm \alpha /2)]$$
where ${P_S}$ and ${P_N}$ are power of signal and noise, respectively. The normalized SCDF with $\alpha \textrm{ = }\alpha {}_\textrm{0}\textrm{ = 1/}{T_s}$ becomes
$${\hat{S}_x}^{{\alpha _\textrm{0}}}({f_c})\textrm{ = }\frac{{E[{P_S}(t,{f_c} \pm {\alpha _0}/2)]}}{{E[{P_S}(t,{f_c} \pm {\alpha _0}/2)] + E[{P_N}(t,{f_c} \pm {\alpha _0}/2)]}}$$

Then, the SNR of the two spectral components at ${f_c} \pm {\alpha _\textrm{0}}\textrm{/2}$ can be expressed as

$$SNR({f_c},\textrm{ }{\alpha _0})\textrm{ = }\frac{{E[{P_S}(t,{f_c} \pm {\alpha _0}/2)]}}{{E[{P_N}(t,{f_c} \pm {\alpha _0}/2)]}} = \frac{{{{\hat{S}}_x}^{{\alpha _\textrm{0}}}({f_c})}}{{\textrm{1 - }{{\hat{S}}_x}^{{\alpha _\textrm{0}}}({f_c})}}$$

Considering the SNR calculation of the whole signal, it needs to include whole signal power and noise power within the whole measurement bandwidth ${B_{meas}}$, as

$$SNR\textrm{ = }\frac{{\sum\limits_{{f_i} \in {B_{meas}}} {E[{P_S}(t,{f_i})]} }}{{E[{P_N}(t)] \cdot {B_{meas}}}} = \frac{{\sum\limits_{{f_i} \in {B_{meas}}} {E[{P_S}(t,{f_i}) + {P_N}(t)]} }}{{E[{P_N}(t)] \cdot {B_{meas}}}} - 1$$
where f in ${P_N}(t)$ is omitted due to the spectral flatness of Gaussian noise. In Eq. (14), $E[{P_N}(t)]$ can be calculated according to Eq. (13), as
$$E[{P_N}(t)] = E[{P_S}(t,\textrm{ }{f_c} \pm {\alpha _0}/2) \textrm{ + }{P_N}(t)] \cdot [1 - {\hat{S}_x}^{{\alpha _\textrm{0}}}({f_c})]$$

Replacing Eq. (15) into Eq. (14), we can obtain

$$SNR\textrm{ = }\frac{{\sum\limits_{{f_i} \in {B_{meas}}} {E[{P_S}(t,{f_i}) + {P_N}(t)]} }}{{E[{P_S}(t,\textrm{ }{f_c} \pm {\alpha _0}/2) + {P_N}(t)] \cdot {B_{meas}}}} \cdot \frac{1}{{[1 - {{\hat{S}}_x}^{{\alpha _\textrm{0}}}({f_c})]}} - 1$$

According to Eq. (16), the SNR estimation can be efficiently realized by a few simple steps:

  • (1) Measure the power of the whole received signal within ${B_{meas}}$, i.e. $\sum\limits_{{f_i} \in {B_{meas}}} {E[{P_S}(t,{f_i}) + {P_N}(t)]}$.
  • (2) Use a pair of narrowband filters with center frequency at ${f_c} - \alpha {}_\textrm{0}\textrm{/2}$ and ${f_c}\textrm{ + }\alpha {}_\textrm{0}\textrm{/2}$ to filter out two spectral components, as illustrated in Fig. 1(a). Measure the power of these two spectral components and scale it according to the relative size of ${B_{meas}}$ and the bandwidth of narrowband filters. Then we obtain $E[{P_S}(t,\textrm{ }{f_c} \pm {\alpha _0}/2) + {P_N}(t)] \cdot {B_{meas}}$.
  • (3) Calculated ${\hat{S}_x}^{{\alpha _\textrm{0}}}({f_c})$ using two spectral components obtained in step (2). Actually, the calculation of ${\hat{S}_x}^{{\alpha _\textrm{0}}}({f_c})$ is very straightforward. In step 2, from the perspective of time domain, after filtering, we obtain two time-varying sequences whose FFT transforms are two spectral components located at $f - \alpha {}_\textrm{0}\textrm{/2}$ and $f\textrm{ + }\alpha {}_\textrm{0}\textrm{/2}$ in frequency domain. Therefore, complicated steps from Eq. (5) to Eq. (7) which includes FFT and time averaging can be simplified by direct time domain correlation calculation between above-mentioned two filtered time-varying sequences, as long as the sequence length is long enough.

 

Fig. 1. (a) A power spectrum of 28 GBaud 16-QAM signal and spectral components used for measuring the correlation. Signal power is 0 dBm and the ROF is 0.5. (b) The calculated normalized SCDF and the estimated SNR versus reference SNR.

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The normalized SCDF and the SNR estimated from the normalized SCDF versus reference in-band SNR is plotted in Fig. 1(b), showing highly reliable and accurate SNR estimation performance. The bandwidth of the narrowband filters is fixed at 100 MHz [26].

The correlation calculation-based SNR technique have the following advantages:

  • (1) It is modulation format transparent, because only spectrum information is used without the need of demodulation by complicated DSP, i.e. Format-dependent equalization, CPR, and de-mapping. The only necessary pre-process is CDC, which does not depend on the format. It should be noticed that CDC is essential to avoid correlation degradation since a time delay between the two spectral components will be induced by CD.
  • (2) Benefiting from “quasi-DSP free” feature, DSP-induced noise and the impact of residual phase noise are avoided [8,21], while conventional DSP-based OSNR monitoring techniques (e.g. EVM calculation) depends significantly on the DSP performance.
  • (3) The performance is insensitive to random rotation in the polarization state and polarization mode dispersion (PMD) in PDM systems, since the impact of PMD exists in both the spectral components at the upper and lower sideband of the signal spectrum, which are mixtures of X- and Y-tributaries before polarization demultiplexing. Meanwhile, it is robust to FO, as explained after Eq. (7), although not insensitive.

2.2 Impact of nonlinearity on SNR estimation using signal spectral correlation

Like most of OSNR monitoring techniques, strong fiber nonlinearity will affect the above SNR estimation method. Plenty of previous research works have modelled the nonlinear noise as a Gaussian-like noise in the time domain. It means that, nonlinear noise can simply be added to ASE noise and its impact on system performance could be assessed through a modified SNR considering both noise sources. Therefore, the SNR calculated based on EVM actually reflects the “true (or effective)” SNR contributed by both linear and nonlinear noise. To be fair, this “true” SNR can be used to evaluate the QoT of the channel of interest (COI), since fiber nonlinearity compensation algorithms such as digital backpropagation (DBP) [29] is still too complex to be realistic at present. However, for the above signal spectral correlation based-SNR estimation method, different from methods like EVM calculation etc., the impact of fiber nonlinearity is more complex. Although acting as a Gaussian-like noise, nonlinear noise is a data-dependent noise generated by signal itself. Some of nonlinearity-insensitive OSNR methods have employed amplitude noise correlation calculation between adjacent symbols in time domain to calibrate the estimated OSNR value [18,19]. Therefore, it is foreseeable that the nonlinear noise at the upper and lower sideband of the signal spectrum will also show correlation to a certain extent, unlike the ASE noise. In addition, the correlation between nonlinear noise is unlikely to be as strong as that of the modulated signal. Here, we conduct simulations to study the cyclostationarity-induced correlation characteristic of nonlinear noise for up-to-7 channel 16-QAM signal. The simulation setup is a 915-km link consisting of 12 equal SMF spans without ASE noise loading. Different from the clear cyclostationary property of the signal and additive Gaussian noise, cyclostationarity-induced correlation characteristic of nonlinear noise is much more complicated since the NLI is not only generated by the COI itself, but also induced by other channels. Therefore, the number of WDM channels will certainly affect the overall correlation characteristic of NLI. Figure 2(a) shows the relationship between the reference and estimated SNRnonlinear using spectral correlation-based SNR estimation method. There are two noteworthy points in Fig. 2(a). Firstly, different from ASE noise, nonlinear noise components at the upper and lower sideband of the signal spectrum will also show correlation to a certain extent, resulting the overestimation of SNRnonlinear. Secondly, it is found that the gap between the estimated and reference SNRnonlinear decreases with the number of WDM channels. Such phenomenon can be explained by the strength of NLI induced correlation. The NLI generated by self-channel interference (SCI) is expected to have strongest correlation within the spectrum of the COI, while the cross-channel and multi-channel interference (XCI and MCI) induced NLI is of less correlation [14]. Overall, the correlation of the two spectral components and the SNRnonlinear overestimation decreases with the number of WDM channels. Such characteristics bring great trouble to our measurement: Our final measurement value is not the SNR solely caused by ASE, nor “combined” SNR caused by the combination of linear and nonlinear noise, but an intermediate value, as shown in Fig. 2(b), where single-channel 16-QAM signal is simulated and the simulation setup is the same as that in Fig. 2(a) with the noise figure (NF) of EDFAs set to 6 dB. The amount of nonlinear noise can be calculated by both simulation and analytical GN models [14]. It is clearly observed that the estimated SNR is always higher than combined SNR while lower than ASE-induced noise. Therefore, when there is strong NLI, the measurement result will become meaningless since it can not reflect any useful information of link condition to guide operators’ action.

 

Fig. 2. (a) The relationship between the reference and estimated SNRnonlinear using spectral correlation-based SNR estimation for up-to-7-channel WDM systems. (b) Combined SNR, ASE-induced SNR, and the estimated SNR of a single-channel 16-QAM signal.

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2.3 Different characteristics of linear and nonlinear noise in the frequency domain

In the time domain, the nonlinear noise act like Gaussian noise, making it difficult to be distinguished from ASE noise. However, the spectrum shapes of linear and nonlinear noise present different characteristics in the frequency domain, i.e., linear noise is flat while the nonlinear noise spectrum is non-flat. Figure 3(a) plots power spectrum density (PSD) of the signal and pure linear noise, while Fig. 3(b) plots PSD of signal spectrum and pure nonlinear noise spectrum, where the nonlinear noise spectrum is calculated using GN model [14]. The setup is the same as the point in Fig. 2 with launch power of 0 dBm. The noise part has been exaggerated to facilitate viewing, so the ordinate is omitted.

 

Fig. 3. (a) Signal spectrum of 28 GHz 16QAM signal and pure linear noise spectrum; (b) Signal spectrum of 28 GHz 16QAM signal and pure nonlinear noise spectrum. ROF=0.5

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Then, let’s consider the scenario in the presence of FO and its impact on spectral correlation calculation. In the correlation measurement step described previously, for a 28 Gbaud signal, if FO is absent, the ideal situation is to filter out two spectral components at the lower and upper sideband at a frequency of ± 14 GHz which show a central symmetry with the zero frequency. If FO is present, the positions of these two frequency components spaced at the interval of 28 GHz are no longer symmetrical about the center frequency, as shown in the Fig. 4 where the FO is 3 GHz. Here, the effect of FO is equivalently manifested in the position moving of the two narrowband filters instead of that of signal spectrum for convenience. It is worth noting that the power of two spectral components is no longer the same.

 

Fig. 4. Position moving process of the two spectral components at the upper and lower sideband with induced-FO of 3 GHz. The signal parameter is same as that in Fig. 1(b).

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Next, let's analyse the impact of FO on the calculation of correlation ${\hat{S}_x}^{{\alpha _\textrm{0}}}(f)$. For a noiseless signal, the ${\hat{S}_x}^{{\alpha _\textrm{0}}}(f)$ remain theoretically 1 as ${\hat{S}_x}^{{\alpha _\textrm{0}}}({f_c})$, as stated in Section 2.1. It is easy to understand because this process is equivalent to the scenario where there are two identical signals, and one of them is amplified and the other is attenuated. Obviously, ${\hat{S}_x}^{{\alpha _\textrm{0}}}(f)$ is still 1 when there is not any noise. But for a signal with flat ASE noise PSD as in Fig. 3(a), the situation is quite different. Although the PSD of signal has changed due to its position moving in the spectrum, the PSD of noise is constant cross all frequency. It means that, in the process shown in Fig. 4, the PSD of the spectral component at the upper sideband (in the right part of Fig. 4) decreases significantly, and the proportion of noise increases. Although for the lower sideband signal (in the left part of Fig. 4) the condition is the opposite, it should be noticed that for the correlation calculation, the result is much more dependent on the noisier component, i.e. the spectral component in the upper sideband in Fig. 4, which causes the drop of ${\hat{S}_x}^{{\alpha _\textrm{0}}}(f)$. Therefore, ${\hat{S}_x}^{{\alpha _\textrm{0}}}(f)$ will decreases with FO. On the other hand, for the nonlinear noise spectrum as shown in Fig. 3(b), we can find that the PSD of the nonlinear noise is not flat, but it becomes smaller at the edge of the spectrum. According to the above analysis, we can know that in the nonlinear case, when there is a certain amount of FO, ${\hat{S}_x}^{{\alpha _\textrm{0}}}(f)$ will drop, but not as fast as in the scenario of pure linear noise. This is because as the PSD of the signal spectral component at the upper sideband decreases, the PSD of nonlinear noise also drops, keeping its proportion relatively low. From the above analysis, it can be concluded that the presence of linear and nonlinear noise can both cause a decrease of ${\hat{S}_x}^{{\alpha _\textrm{0}}}(f)$ with FO. However, the contribution of linear noise is more significant than that of nonlinear noise. Then, it is possible to find a way to distinguish between them.

2.4 Principle of joint linear and nonlinear noise monitoring

We need to study the relation between ${\hat{S}_x}^{{\alpha _\textrm{0}}}(f)$ and FO through numerical simulation. Figures 5(a)–5(c) show the simulation result where ${\hat{S}_x}^{{\alpha _\textrm{0}}}(f)$ decreased with FO from 0GHz to 4.5GHz when launch power is 0 dBm, and the red lines in Figs. 5(a)–5(c) represent fitting curves using $y = A{x^\textrm{2}} + Z$, $y = A{x^3} + Z$, $y = A{x^\textrm{4}} + Z$. We can clearly observe that the cubic function fits better than the quadratic function. On the other hand, it seems that the cubic function fits also better than the quartic function, at least the cubic function can respond well to the decreasing tendency of ${\hat{S}_x}^{{\alpha _\textrm{0}}}(f)$ with FO. According to principle of Occam's Razor [30], in polynomial fitting, it is better to choose the simplest form of the lowest order on the premise of performance guarantee [31]. Meanwhile, in simulation, better monitoring performance is achieved with the A fitted by $y = A{x^3} + Z$ than that by the quadratic and quartic functions. Therefore, for the detected signal, we use frequency sweep method to obtain a series of ${\hat{S}_x}^{{\alpha _\textrm{0}}}(f)$ values with various FO, and we choose cubic function $y = A{x^3} + Z$ to carry out curve fitting and obtain A, which reflects the rate of decline of ${\hat{S}_x}^{{\alpha _\textrm{0}}}(f)$ with FO. Figure 6 shows the relation between A and launch power. Values of A are obtained using above described curve fitting process and the system setup is the same as that in Fig. 2. It can be seen that the absolute value of A shows a trend of falling first and then rising with launch power, which reflects a comprehensive effect of linear and nonlinear noise.

 

Fig. 5. Fitting ${\hat{S}_x}^{{\alpha _\textrm{0}}}(f)$ as a function of FO using (a) $y = A{x^\textrm{2}} + Z$, (b) $y = A{x^\textrm{3}} + Z$, and (c) $y = A{x^\textrm{4}} + Z$.

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Fig. 6. Relation between coefficient A obtained by cubic curve fitting and launch power.

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Based on the analysis in Section 2.3, it is known that the contribution of linear noise and nonlinear noise to A is different. Here we propose a simple relationship

$$A = {A_1}\ast \frac{1}{{SN{R_{linear}}}} + {A_2}\ast \frac{1}{{SN{R_{nonlinear}}}} + {A_3}$$
where ${A_1}$ and ${A_\textrm{2}}$ represent the contribution factor of linear and nonlinear noise to A, respectively, and ${A_\textrm{3}}$ represents the bias introduced by other inherent factors. Whether such a simple relationship is correct or not requires investigation through numerical simulation. In the simulation, we collect a series of (SNRlinear, SNRnonlinear) pairs of signals after 915 km fiber link by adjusting the launch power from −5 dBm to 5 dBm at a step size of 0.5 dBm. SNRlinear are calculated according to the traditional ASE noise accumulation model [32], and SNRnonlinear are calculated by GN-model. Figure 7(a) shows the fitting result for Eq. (17) when the ROF is equal to 0.5 and the FO is swept from 0 GHz to 4.5 GHz. The FO sweeping range should be optimized according to different ROF, which determines the signal bandwidth. The objective of optimization is to minimize the estimation error, and we will not explain it in detail here. The results show that the assumption of Eq. (17) is quite accurate. In addition, in the fitting process of Fig. 6, we find that ${A_1}$ = −0.04866 and ${A_\textrm{2}}$ = −0.00608, which confirms the previous inference that “linear noise contributes more to A than nonlinear noise”.

 

Fig. 7. Curve fitting result of (a) Eq. (17) and (b) Eq. (18) using 21 pairs of (SNRlinear, SNRnonlinear), with the launch power swept from −5 dBm to 5 dBm at a stepsize of 0.5 dBm. SNRmeas is measured using the spectral correlation-based SNR estimation method.

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Now we find that there are two unknowns, i.e. SNRlinear and SNRnonlinear, but there is only one bivariate first-order equation, i.e. Eq. (17). Obviously, we need to find an additional equation to build a system of bivariate first-order equations. Careful readers may have already found out that the second equation has appeared already in Fig. 2(b). In the previous analysis, we have concluded that the nonlinear noise existing in the spectral components at the upper and lower sideband are not as uncorrelated as the ASE noise, and also, they are unlikely to have a very strong correlation as the signal itself. Therefore, when using the spectral correlation-based method to estimate SNR, nonlinear noise has also some contribution to the noise part, although less than linear noise. Therefore, we can establish the second bivariate first-order equation:

$$\frac{\textrm{1}}{{SN{R_{meas}}}} = {B_1}\ast \frac{1}{{SN{R_{linear}}}} + {B_2}\ast \frac{1}{{SN{R_{nonlinear}}}} + {B_3}$$
where ${B_1}$ and ${B_\textrm{2}}$ represent the contribution factors of linear and nonlinear noise to the estimated noise using the spectral correlation-based SNR estimation method. Figure 7(b) shows the curve fitting result for Eq. (18), again with high accuracy.

Now, because we have two unknowns and a system of bivariate first-order equations, the problem becomes solvable. In the implementation of the scheme, we can firstly sweep launch power to get a series of (SNRlinear, SNRnonlinear) pairs and build a system of bivariate first-order equations by curve fitting. Then, given a signal sequence, we can calculate the linear noise and the nonlinear noise simultaneously. Although this scheme is based on dataset for curve fitting, the quantity requirement is much less than that for ML models since in our scheme, the curve fitting steps are very simple and straightforward, while for ML models the training and optimization process is much more complicated and depends highly on huge amounts of high-quality training dataset.

3. Simulation results

The simulation setup is a PDM WDM transmission system with 915 km link consisting of 12 equal SMF spans and EDFAs based on split-step Fourier method (SSFM) [33]. Single channel and 3-, 5-, and 7-channel condition are considered and monitor the center channel. The modulation formats used are QPSK, 16-QAM, and 64-QAM. We change SNRlinear and SNRnonlinear mainly by adjusting the launch power from −5 dBm to 5 dBm at a step size of 0.5 dBm. Meanwhile, we also provide results with different NF of EDFAs i.e. 4 dB, 5 dB, and 6 dB. Conditions of 6-span and 9-span are also simulated. Therefore, the adjustment of SNR is more flexible. Results with ROF of 0.5, 0.6, 0.7 are presented.

Figure 8(a) shows the monitoring performance of SNRlinear while Fig. 8(b) shows the monitoring performance of SNRnonlinear for single channel 16QAM signal with ROF of 0.5, 0.6, 0.7, respectively. Here, matched filtering is used so that the reference SNRlinear for signal with different ROF are basically the same (as confirmed by simulation). SNRnonlinear is calculated using incoherent GN-model with integration form. As the ROF changes, the corresponding nonlinearity spectrum shows a slightly difference so does the value of reference SNRnonlinear, as shown in Fig. 8(b). The proposed joint SNRlinear and SNRnonlinear monitoring scheme can accurately estimate SNR from both noise source. The estimation error curve of SNRlinear and SNRnonlinear are plotted in Fig. 9(a) and 9(b), respectively. It can be seen for both SNRlinear and SNRnonlinear, the estimation error is smaller than 1.5 dB within an ultrawide launch power range from −5 dBm to 5 dBm. Noted that for estimation of SNRlinear, the estimation error is no more than 0.7 dB with the launch power equal to or lower than 4 dBm, which corresponding to SNR value of 29.55 dB. For a 28 GBaud signal, SNR value of 29.55 dB is equivalent to OSNR value of about 33.05 dB with the 0.1 nm noise reference bandwidth [34]. Therefore, the estimation range of our proposed scheme is sufficiently wide to practical application. On the other hand, we can observe that the estimation error of SNRlinear increases with launch power, in opposite to SNRnonlinear. This phenomenon is due to that the proportion of linear noise is lower when the launch power is high, while SNRnonlinear decreases by 2 dB with 1 dB increment of launch power. Higher noise proportion is beneficial to estimation accuracy.

 

Fig. 8. (a) SNRlinear monitoring and (b) SNRnonlinear monitoring for single channel 16-QAM signal with ROF of 0.5, 0.6, and 0.7. The link is 915 km consisting of 12 equal SMF spans and EDFAs with NF of 6 dB.

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Fig. 9. (a) SNRlinear estimation error curve and (b) SNRnonlinear estimation error curve of the results in Fig. 8.

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The performance of the proposed scheme for the conditions with low ROFs is also studied. Since a FO sweeping process of the narrowband filters is operated to filter out the two spectral components, it is necessary to reserve some spectral margins. After raised cosine shaping, the signal bandwidth of can be calculated as ${B_{signal}}\textrm{ = Symbol rate} \times \textrm{(1 + ROF)}$. The residual spectral margin is ${B_{signal}}\textrm{/2 - Symbol rate/2 = Symbol rate} \times \textrm{ROF/2}$. For a 28 Gbaud signal, the residual spectral margin is 7 GHz with the ROF of 0.5, while it is only 1.4 GHz with the ROF of 0.1. For the signals with low ROFs, e.g. 0.2 and 0.1, the monitoring performance may degrade due to the limited FO sweeping range. Here, we provide monitoring results for single-channel 16-QAM signal with ROF of 0.1 and 0.2. As shown in Fig. 10, for the signal with ROF of 0.2, the monitoring performance is still acceptable with the estimation errors of SNRlinear and SNRnonlinear smaller than 1.1 dB and 1.5 dB, respectively. However, the estimation errors of SNRlinear and SNRnonlinear both exceed 2.2 dB with the ROF of 0.1. Moreover, there is large error fluctuation over the launch powers. It indicates that, the performance of our proposed method is limited for the conditions of ultra-low ROFs, where it is difficult to make full use of the unflatness of NLI over the signal spectrum.

 

Fig. 10. (a) SNRlinear monitoring and (b) SNRnonlinear monitoring for single-channel 16-QAM signal with ROF of 0.1 and 0.2. (c) SNRlinear estimation error curve and (d) SNRnonlinear estimation error curve of the results in Fig. 10(a) and 10(b), respectively.

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Next, we study multi-channel WDM conditions with channel number of 3, 5, and 7. The estimation performance is shown in Fig. 11. The results show that our proposed scheme functions well for linear and nonlinear noise monitoring for WDM systems because nonlinearity spectrum in multi-channel systems also show the characteristic of non-flatness, as can be observed in Fig. 3 in [14]. The estimation error of SNRlinear is smaller than 1.1 dB when the launch power equal to or lower than 4 dBm. For SNRnonlinear, the estimation error is always lower than 1.6 dB. If the numbers of channels on both sides of the COI are different, the NLI spectrum becomes asymmetric. However, we can still use principle of curve fitting by FO sweeping at upper and lower sideband separately. Then the nonlinear effect from both left and right side of the COI can be fitted. This issue is left for future study.

 

Fig. 11. (a) SNRlinear monitoring and (b) SNRnonlinear monitoring for multi-channel 16-QAM signal. The simulation setup is the same as that in Fig. 8.

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As analysed previously, the number of WDM channels will affect the correlation characteristic of nonlinear noise. Therefore, ${B_1}$ and ${B_\textrm{2}}$ in Eq. (18) vary with the WDM system scale. Figure 12 shows the variation trend of the values of contribution factors in Eq. (17) and Eq. (18) versus the number of WDM channels. Specifically, when the WDM system scale becomes larger, the correlation of the two spectral components spaced at one symbol rate. Therefore, ${B_\textrm{2}}$ increases with the number of WDM channels, while ${B_\textrm{1}}$ decreases. Meanwhile, the unflatness of NLI spectrum is weakened when there are more WDM channels. Therefore, the absolute value of the contribution factor ${A_\textrm{2}}$ of the nonlinear noise to A increases while the absolute value of ${A_\textrm{1}}$ decreases with the number of WDM channels.

 

Fig. 12. The values of contribution factors in Eq. (17) and Eq. (18) versus the number of WDM channels.

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Figure 13 shows the monitoring results for QPSK and 64-QAM signal. The linear and nonlinear estimation error for QPSK is lower than 0.8 dB and 1.5 dB while that for 64-QAM is lower than 1.4 dB and 1.8 dB, respectively. Benefited from the feature of format-dependent DSP-free processing, the proposed scheme is suitable for different commonly used modulation formats and is desired to be employed in flexible optical networks.

 

Fig. 13. (a) SNRlinear monitoring and (b) SNRnonlinear monitoring for single channel QPSK and 64-QAM signal. The simulation setup is the same as that in Fig. 8.

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Essentially, the importance of OSNR monitoring mainly lies in monitoring the link parameter variation. In practice, the NF of inline EDFAs may be time-varying due to temperature, aging etc. Meanwhile, the number of optical spans through which the signal has been transmitted can be different depending on network structure and transmission requirements in flexible optical networks. Above kinds of parameter variations can be reflected in the OSNR information and are required to be monitored. Therefore, we conduct a verification using dataset with different setting of number of spans and NFs. We consider 6-, 9-, and 12-span conditions with possible NF values of 4 dB, 5 dB and 6 dB. The dataset consisting of (SNRlinear, SNRnonlinear) pairs generated in different transmission conditions is used to do curve fitting and test. The monitoring results is shown in Fig. 14. The SNRlinear and SNRnonlinear estimation error are lower than 1.7 dB and 1.8 dB, respectively. It indicates that the proposed scheme can deal with practical time-varying environment and provides reliable optical performance tracking.

 

Fig. 14. (a) SNRlinear monitoring and (b) SNRnonlinear monitoring performance for mixed conditions with span number of 6, 9, and 12, and NF values of 4 dB, 5 dB and 6 dB.

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It is necessary to study the impact of narrow filtering effect of cascaded reconfigurable optical add and drop multiplexers (ROADMs) in meshed optical networks on the monitoring performance and the sensitivity of fitting parameters. Here, we provide results of the monitoring performance with up-to-12 ROADMs distributed in the transmission link. The filter transfer function $S(f)$ of commercial ROADMs can be modelled as in [35,36]

$$S(f)\textrm{ = }\frac{\textrm{1}}{\textrm{2}}\sigma \sqrt {\textrm{2}\pi } [erf(\frac{{{B_0}/2 - f}}{{\sqrt {\textrm{2}} \sigma }}) - erf(\frac{{ - {B_0}/2 - f}}{{\sqrt {\textrm{2}} \sigma }})],\textrm{ }\sigma = \frac{{{B_{OTF}}}}{{2\sqrt {\textrm{2}\ln 2} }}$$
where ${B_0}$ denotes the 6-dB bandwidth, and ${B_{OTF}}$ represents the steepness of the filter edge. In simulation, the ${B_0}$ and ${B_{OTF}}$ are set to 45.34 GHz and 8.8 GHz, which is close to the specification of current commercial ROADMs for 50 GHz grid [37]. Figure 15(a) depicts the evolution of the normalized power frequency response as the number of cascaded ROADMs. The dependence of estimation errors of SNRlinear and SNRnonlinear on the number of cascaded ROADMs with the LPs of −3 dBm, 0 dBm, and +3 dBm is shown in Fig. 15(b). It can be found that the proposed scheme is robust to the filtering effect of cascaded ROADMs. In Fig. 15(c), we show the optimum upper limit of FO sweeping decreases with the number of cascaded ROADMs due to the reduction of FO sweeping margin. We also study the relationship between values of (${A_1}$, ${A_\textrm{2}}$, ${B_1}$, ${B_\textrm{2}}$) and the number of cascaded ROADMs. As shown in Fig. 15(d), the absolute values of ${A_1}$ and ${A_\textrm{2}}$ increase with the number of cascaded ROADMs. This phenomenon is due to the reason that the steepness of the signal spectrum edge increases with narrow filtering, resulting a higher A in the left part of Eq. (17). On the contrary, ${B_1}$ and ${B_\textrm{2}}$ decreases with the number of cascaded ROADMs because after narrow filtering, the noise distributed at the signal spectrum edge is suppressed while the central part of the signal spectrum where the signal power is much larger than noise power, suffers less impact. Therefore, $SN{R_{meas}}$ increases and thus a lower $\textrm{1/}SN{R_{meas}}$ is resulted in the left part of Eq. (18). Overall, in order to adapt to more heterogeneous conditions and meshed optical links with various number of WDM channels and cascaded ROADMs, the parameters in the proposed method need refitting to maintain the monitoring performance. The generalizability of the proposed scheme should be further improved which is left for future study.

 

Fig. 15. (a) Normalized power frequency response with 1 to 12 cascaded ROADMs. (b) Estimation errors versus the number of cascaded ROADMs with the LPs of −3 dBm, 0 dBm, and +3 dBm. (c) The optimum upper limit of FO sweeping and (d) the values of contribution factors in Eq. (17) and Eq. (18) versus the number of cascaded ROADMs. Single-channel 16-QAM signal with the ROF of 0.5 is simulated.

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4. Experimental results

To further verify the performance of the proposed joint linear and nonlinear noise monitoring scheme, we conduct experiments for a 28 GBaud PDM-16-QAM WDM transmission system with a 915 km real link consisting of 12 SMF spans with different length and EDFAs, as shown in Fig. 16. Different from loop-based structure, such setup is very close to practical heterogeneous optical transmission link. We establish automated control system of the experiment setup with the abilities to set and monitor the system parameters, i.e. input/output (I/O) power at transceiver sides and each EDFA node fast and accurately. By such automated operations, the input power launched into each span of fiber can be accurately controlled. Meanwhile, EDFA’s gains and attenuations are accurately adjusted to compensate for the fiber loss. At the transmitter side (Tx), up-to-7 channels are divided into the odd and even channels. 2 independent 28 GBaud 16-QAM digital waveforms shaped by raised cosine filter with ROF of 0.5 are generated offline and loaded into 84 GSa/s arbitrary waveform generator (AWG). The generated waveforms from 4 output ports of the AWG then drive the odd and even channels separately. The launch power is adjusted from −3 dBm to 3 dBm per channel by the EDFA used as pre-amplifier. The 2 modulated tributaries are then interleaved and go into a PDM emulator before launching into the 915 km fiber link. Each span is followed by a 4 nm optical filter to filter out-band ASE noise.

 

Fig. 16. Experimental setup of PDM 28 GBaud 16-QAM WDM system. WSS: wavelength selective switch; LO: local oscillator; PDM Emu: polarization-division multiplexing emulator.

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At the receiver-side (Rx), after coherent detection, the linear and nonlinear noise estimation is operated in offline DSP. To correctly evaluate the performance of our proposed monitoring scheme, there are three steps need to be carefully followed:

  • (1) Reference linear noise is consisting of EDFAs-induced ASE noise and transceiver induced electrical noise [32]. Therefore, we firstly conduct optical back-to-back (B2B) transmission to obtain the reference background electrical SNR. For received signal in B2B condition, we perform “quasi-perfect” DSP, i.e. training symbol-assisted equalization and carrier phase recovery (CPR) to avoid DSP-induced noise as much as possible. For 915-km link transmission, at the Rx, we calculate EDFAs-induced SNR by conventional out-of-band measurement in spectrum from optical spectrum analyzer (OSA). Different from OSNR measurement criteria, the reference noise bandwidth should be equal to that of signal spectrum rather than 0.1 nm [34]. Reference SNRlinear can then be calculated according to above two SNR measurements.
  • (2) Reference nonlinear noise should be calculated using GN model in integration-form rather than closed-form ones because of the non-rectangular spectrum and non-uniform span lengths. We use incoherent GN model in [13] to calculate reference SNRnonlinear.
  • (3) In the experiment, the FO is unavoidable due to wavelength mismatch between LO and the transmitter laser [38]. Since our proposed scheme is based on FO sweeping within the spectrum, a coarse FO compensation (FOC) is required. Fortunately, after CDC, FFT-based FO estimation (FOE) scheme can be used to estimate FO accurately for most commonly used formats [39], and we adopt it in our scheme. Actually, based on the proposed FO sweeping principle, the FOE can also be realized by fitting the central axis according to the relation between ${\hat{S}_x}^{{\alpha _\textrm{0}}}(f)$ and FO, which can be efficiently integrated in our scheme.
Figures 17(a)–17(b) show SNRlinear and SNRnonlinear monitoring results compared with reference SNR curves, indicating that the feasibility of the proposed scheme is verified well by experiments. The estimation error curves are also provided in Fig. 18. For launch power from −3 dBm to 3 dBm, the estimation errors of SNRlinear are all less than 0.3 dB, which are quite accurate. On the other hand, the estimation errors of SNRnonlinear can be as large as 2.4 dB under the launch power of −3 dBm. We attribute it to inaccurate parameter information of whole 12 optical fibers. Some of used fibers are old and there may exists some parameter uncertainties, e.g. CD and nonlinear coefficients, which may cause inaccurate calculation of reference SNRnonlinear using GN model [40]. Overall, the estimation performance is satisfying for most conditions. It is noted that with accurate SNRlinear estimation, the equivalent OSNR monitoring with the same accuracy can be realized [41].

 

Fig. 17. Experimental (a) SNRlinear monitoring and (b) SNRnonlinear monitoring performance for 1-, 3-, 5-, and 7-ch conditions.

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Fig. 18. Experimental (a) SNRlinear monitoring and (b) SNRnonlinear monitoring estimation error for 1-, 3-, 5-, and 7-ch conditions.

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5. Conclusion

A joint linear and nonlinear noise estimation scheme is proposed based on spectrum analysis. The proposed scheme is modulation format-transparent and without DSP-induced noise. High estimation accuracy over a wide launch power range for single channel and WDM transmission systems with various parameter settings are verified by both simulations and experiments over a real long-reach optical path link.

Funding

National Key Research and Development Program of China (2018YFB1801701); National Natural Science Foundation of China (61435006, U1701661); Hong Kong PhD Fellowship Scheme; State Key Laboratory of Advanced Optical Communication Systems and Networks. (2019GZKF1).

Disclosures

The authors declare no conflicts of interest.

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References

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  1. I. Tomkos, S. Azodolmolky, J. Sole-Pareta, D. Careglio, and E. Palkopoulou, “A tutorial on the flexible optical networking paradigm: state of the art, trends, and research challenges,” Proc. IEEE 102(9), 1317–1337 (2014).
    [Crossref]
  2. K. Roberts, Q. Zhuge, I. Monga, S. Gareau, and C. Laperle, “Beyond 100 Gb/s: Capacity, flexibility, and network optimization,” J. Opt. Commun. Netw. 9(4), C12–C24 (2017).
    [Crossref]
  3. J. Zhao, L. Gan, L. Su, J. Zhang, H. He, W. Cai, J. Wang, S. Fu, and M. Tang, “Carrier Beating Impairment in Weakly Coupled Multicore Fiber-Based IM/DD Systems,” IEEE Access 8(1), 65699–65710 (2020).
    [Crossref]
  4. Z. Dong, F. N. Khan, Q. Sui, K. Zhong, C. Lu, and A. P. T. Lau, “Optical performance monitoring: A preview of current and future technologies,” J. Lightwave Technol. 34(2), 525–543 (2016).
    [Crossref]
  5. D. C. Kilper, S. Chandrasekhar, L. Buhl, A. Agarwal, and D. Maywar, “Spectral monitoring of OSNR in high speed networks,” in European Conference and Exhibition on Optical Communication (ECOC), 2002, paper 7.4.4.
  6. J. H. Lee, D. K. Jung, C. H. Kim, and Y. C. Chung, “OSNR monitoring technique using polarization nulling method,” IEEE Photonics Technol. Lett. 13(1), 88–90 (2001).
    [Crossref]
  7. F. N. Hauske, M. Kuschnerov, B. Spinnler, and B. Lankl, “Optical performance monitoring in digital coherent receivers,” J. Lightwave Technol. 27(16), 3623–3631 (2009).
    [Crossref]
  8. S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16(2), 804–817 (2008).
    [Crossref]
  9. R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error vector magnitude as a performance measure for advanced modulation formats,” IEEE Photonics Technol. Lett. 24(1), 61–63 (2012).
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2020 (1)

J. Zhao, L. Gan, L. Su, J. Zhang, H. He, W. Cai, J. Wang, S. Fu, and M. Tang, “Carrier Beating Impairment in Weakly Coupled Multicore Fiber-Based IM/DD Systems,” IEEE Access 8(1), 65699–65710 (2020).
[Crossref]

2019 (3)

2018 (2)

2017 (4)

2016 (2)

Z. Wang, Y. Qiao, and Y. Lu, “OSNR monitoring technique based on multi-order statistical moment method and correlation function for PM-16QAM in presence of fiber nonlinearities,” Opt. Commun. 380, 10–14 (2016).
[Crossref]

Z. Dong, F. N. Khan, Q. Sui, K. Zhong, C. Lu, and A. P. T. Lau, “Optical performance monitoring: A preview of current and future technologies,” J. Lightwave Technol. 34(2), 525–543 (2016).
[Crossref]

2015 (1)

2014 (2)

P. Poggiolini, G. Bosco, A. Carena, V. Curri, Y. Jiang, and F. Forghieri, “The GN-model of fiber non-linear propagation and its applications,” J. Lightwave Technol. 32(4), 694–721 (2014).
[Crossref]

I. Tomkos, S. Azodolmolky, J. Sole-Pareta, D. Careglio, and E. Palkopoulou, “A tutorial on the flexible optical networking paradigm: state of the art, trends, and research challenges,” Proc. IEEE 102(9), 1317–1337 (2014).
[Crossref]

2012 (4)

2011 (2)

2010 (2)

2009 (1)

2008 (1)

2007 (1)

2006 (1)

W. A. Gardner, A. Napolitano, and L. Paura, “Cyclostationarity: Half a century of research,” Signal Process. 86(4), 639–697 (2006).
[Crossref]

2001 (1)

J. H. Lee, D. K. Jung, C. H. Kim, and Y. C. Chung, “OSNR monitoring technique using polarization nulling method,” IEEE Photonics Technol. Lett. 13(1), 88–90 (2001).
[Crossref]

1999 (1)

P. Domingos, “The role of Occam's razor in knowledge discovery,” Data Min. Knowl. Discov. 3(4), 409–425 (1999).
[Crossref]

1987 (1)

A. Blumer, A. Ehrenfeucht, D. Haussler, and M. K. Warmuth, “Occam's razor,” Inf. Process. Lett 24(6), 377–380 (1987).
[Crossref]

Agarwal, A.

D. C. Kilper, S. Chandrasekhar, L. Buhl, A. Agarwal, and D. Maywar, “Spectral monitoring of OSNR in high speed networks,” in European Conference and Exhibition on Optical Communication (ECOC), 2002, paper 7.4.4.

Agrawal, G. P.

G. P. Agrawal, “Nonlinear fiber optics,” in Nonlinear Science at the Dawn of the 21st Century. Springer: Berlin, Germany, 2000.

Anderson, T.

Aoki, Y.

L. Dou, T Yamauchi, X. Su, Z. Tao, S. Oda, Y. Aoki, T. Hoshida, and J. Rasmussen, “An accurate nonlinear noise insensitive OSNR monitor,” in The Optical Fiber Communication Conference and Exhibition 2016 (OFC), paper W3A.5. pp. 1–3.

Azodolmolky, S.

I. Tomkos, S. Azodolmolky, J. Sole-Pareta, D. Careglio, and E. Palkopoulou, “A tutorial on the flexible optical networking paradigm: state of the art, trends, and research challenges,” Proc. IEEE 102(9), 1317–1337 (2014).
[Crossref]

Baxter, G. W.

Becker, J.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error vector magnitude as a performance measure for advanced modulation formats,” IEEE Photonics Technol. Lett. 24(1), 61–63 (2012).
[Crossref]

Blumer, A.

A. Blumer, A. Ehrenfeucht, D. Haussler, and M. K. Warmuth, “Occam's razor,” Inf. Process. Lett 24(6), 377–380 (1987).
[Crossref]

Borowiec, A.

Bosco, G.

Buhl, L.

D. C. Kilper, S. Chandrasekhar, L. Buhl, A. Agarwal, and D. Maywar, “Spectral monitoring of OSNR in high speed networks,” in European Conference and Exhibition on Optical Communication (ECOC), 2002, paper 7.4.4.

Caballero, F. J. V.

Cai, M.

Cai, W.

J. Zhao, L. Gan, L. Su, J. Zhang, H. He, W. Cai, J. Wang, S. Fu, and M. Tang, “Carrier Beating Impairment in Weakly Coupled Multicore Fiber-Based IM/DD Systems,” IEEE Access 8(1), 65699–65710 (2020).
[Crossref]

Careglio, D.

I. Tomkos, S. Azodolmolky, J. Sole-Pareta, D. Careglio, and E. Palkopoulou, “A tutorial on the flexible optical networking paradigm: state of the art, trends, and research challenges,” Proc. IEEE 102(9), 1317–1337 (2014).
[Crossref]

Carena, A.

Cartledge, J. C.

Chan, C. C. K.

J. Lu, S. Fu, L. Deng, M. Tang, Z. Hu, D. Liu, and C. C. K. Chan, “Blind and Fast Modulation Format Identification by Frequency-offset Loading for Hitless Flexible Transceiver,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper M2F.5.

Chandrasekhar, S.

D. C. Kilper, S. Chandrasekhar, L. Buhl, A. Agarwal, and D. Maywar, “Spectral monitoring of OSNR in high speed networks,” in European Conference and Exhibition on Optical Communication (ECOC), 2002, paper 7.4.4.

Chang, J. H.

H. G. Choi, J. H. Chang, Hoon Kim, and Y. C. Chung, “Nonlinearity-Tolerant OSNR Estimation Technique for Coherent Optical Systems,” in Optical Fiber Communication Conference (Optical Society of America, 2015), paper W4D.2.

Charlton, D.

Charlton, D. W.

Chen, S.

Chen, W.

Choi, H. G.

H. G. Choi, J. H. Chang, Hoon Kim, and Y. C. Chung, “Nonlinearity-Tolerant OSNR Estimation Technique for Coherent Optical Systems,” in Optical Fiber Communication Conference (Optical Society of America, 2015), paper W4D.2.

Chung, Y. C.

J. H. Lee, D. K. Jung, C. H. Kim, and Y. C. Chung, “OSNR monitoring technique using polarization nulling method,” IEEE Photonics Technol. Lett. 13(1), 88–90 (2001).
[Crossref]

H. G. Choi, J. H. Chang, Hoon Kim, and Y. C. Chung, “Nonlinearity-Tolerant OSNR Estimation Technique for Coherent Optical Systems,” in Optical Fiber Communication Conference (Optical Society of America, 2015), paper W4D.2.

Cigliutti, R.

Curri, V.

Deng, L.

J. Lu, S. Fu, L. Deng, M. Tang, Z. Hu, D. Liu, and C. C. K. Chan, “Blind and Fast Modulation Format Identification by Frequency-offset Loading for Hitless Flexible Transceiver,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper M2F.5.

Do, C.

Domingos, P.

P. Domingos, “The role of Occam's razor in knowledge discovery,” Data Min. Knowl. Discov. 3(4), 409–425 (1999).
[Crossref]

Dong, Z.

Dou, L.

L. Dou, T Yamauchi, X. Su, Z. Tao, S. Oda, Y. Aoki, T. Hoshida, and J. Rasmussen, “An accurate nonlinear noise insensitive OSNR monitor,” in The Optical Fiber Communication Conference and Exhibition 2016 (OFC), paper W3A.5. pp. 1–3.

Dreschmann, M.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error vector magnitude as a performance measure for advanced modulation formats,” IEEE Photonics Technol. Lett. 24(1), 61–63 (2012).
[Crossref]

Du, L.

Ehrenfeucht, A.

A. Blumer, A. Ehrenfeucht, D. Haussler, and M. K. Warmuth, “Occam's razor,” Inf. Process. Lett 24(6), 377–380 (1987).
[Crossref]

Etemad, S. A.

Fan, Q.

Q. Fan, J. Lu, G. Zhou, D. Zeng, C. Guo, L. Lu, J. Li, C. Xie, C. Lu, F. N. Khan, and A. P. T. Lau, “Experimental Comparisons between Machine Learning and Analytical Models for QoT Estimations in WDM Systems,” in Optical Fiber Communication Conference (Optical Society of America, 2020), paper M2J.2.

Feng, Z.

Forghieri, F.

Freude, W.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error vector magnitude as a performance measure for advanced modulation formats,” IEEE Photonics Technol. Lett. 24(1), 61–63 (2012).
[Crossref]

Frisken, S.

Fu, S.

J. Zhao, L. Gan, L. Su, J. Zhang, H. He, W. Cai, J. Wang, S. Fu, and M. Tang, “Carrier Beating Impairment in Weakly Coupled Multicore Fiber-Based IM/DD Systems,” IEEE Access 8(1), 65699–65710 (2020).
[Crossref]

J. Lu, Z. Tan, A. P. T. Lau, S. Fu, M. Tang, and C. Lu, “Modulation format identification assisted by sparse-fast-Fourier-transform for hitless flexible coherent transceivers,” Opt. Express 27(5), 7072–7086 (2019).
[Crossref]

Q. Wu, L. Zhang, X. Li, M. Luo, Z. Feng, H. Zhou, M. Tang, S. Fu, and D. Liu, “Training Symbol Assisted in-Band OSNR Monitoring Technique for PDM-CO-OFDM System,” J. Lightwave Technol. 35(9), 1551–1556 (2017).
[Crossref]

J. Lu, S. Fu, L. Deng, M. Tang, Z. Hu, D. Liu, and C. C. K. Chan, “Blind and Fast Modulation Format Identification by Frequency-offset Loading for Hitless Flexible Transceiver,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper M2F.5.

Fukuyama, H.

Gan, L.

J. Zhao, L. Gan, L. Su, J. Zhang, H. He, W. Cai, J. Wang, S. Fu, and M. Tang, “Carrier Beating Impairment in Weakly Coupled Multicore Fiber-Based IM/DD Systems,” IEEE Access 8(1), 65699–65710 (2020).
[Crossref]

Gardner, W. A.

W. A. Gardner, A. Napolitano, and L. Paura, “Cyclostationarity: Half a century of research,” Signal Process. 86(4), 639–697 (2006).
[Crossref]

Gareau, S.

Guo, C.

Q. Fan, J. Lu, G. Zhou, D. Zeng, C. Guo, L. Lu, J. Li, C. Xie, C. Lu, F. N. Khan, and A. P. T. Lau, “Experimental Comparisons between Machine Learning and Analytical Models for QoT Estimations in WDM Systems,” in Optical Fiber Communication Conference (Optical Society of America, 2020), paper M2J.2.

Guo, P.

Hashizume, Y.

Hauske, F. N.

Haussler, D.

A. Blumer, A. Ehrenfeucht, D. Haussler, and M. K. Warmuth, “Occam's razor,” Inf. Process. Lett 24(6), 377–380 (1987).
[Crossref]

He, H.

J. Zhao, L. Gan, L. Su, J. Zhang, H. He, W. Cai, J. Wang, S. Fu, and M. Tang, “Carrier Beating Impairment in Weakly Coupled Multicore Fiber-Based IM/DD Systems,” IEEE Access 8(1), 65699–65710 (2020).
[Crossref]

Heismann, F. L.

F. L. Heismann, “Determining in-band optical signal-to-noise ratio in polarization-multiplexed optical signals using signal correlations”, U.S. Patent US 2018/0138974 Al, 2018.

Hillerkuss, D.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error vector magnitude as a performance measure for advanced modulation formats,” IEEE Photonics Technol. Lett. 24(1), 61–63 (2012).
[Crossref]

Ho, K.

Hoshida, T.

L. Dou, T Yamauchi, X. Su, Z. Tao, S. Oda, Y. Aoki, T. Hoshida, and J. Rasmussen, “An accurate nonlinear noise insensitive OSNR monitor,” in The Optical Fiber Communication Conference and Exhibition 2016 (OFC), paper W3A.5. pp. 1–3.

Hu, W.

Hu, Z.

J. Lu, S. Fu, L. Deng, M. Tang, Z. Hu, D. Liu, and C. C. K. Chan, “Blind and Fast Modulation Format Identification by Frequency-offset Loading for Hitless Flexible Transceiver,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper M2F.5.

Huebner, M.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error vector magnitude as a performance measure for advanced modulation formats,” IEEE Photonics Technol. Lett. 24(1), 61–63 (2012).
[Crossref]

Ionescu, M.

M. Ionescu, M. Sato, and B. Thomsen, “Cyclostationarity-based joint monitoring of symbol-rate, frequency offset, CD and OSNR for Nyquist WDM superchannels,” Opt. Express 23(20), 25762–25772 (2015).
[Crossref]

M. Ionescu, “Digital Signal Processing for Sensing in Software Defined Optical Networks”, PhD’s thesis, the University College London, 2015.

Ip, E.

Ives, D. J.

Jiang, Y.

Josten, A.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error vector magnitude as a performance measure for advanced modulation formats,” IEEE Photonics Technol. Lett. 24(1), 61–63 (2012).
[Crossref]

Jung, D. K.

J. H. Lee, D. K. Jung, C. H. Kim, and Y. C. Chung, “OSNR monitoring technique using polarization nulling method,” IEEE Photonics Technol. Lett. 13(1), 88–90 (2001).
[Crossref]

Kashi, A.

Khan, F. N.

Z. Dong, F. N. Khan, Q. Sui, K. Zhong, C. Lu, and A. P. T. Lau, “Optical performance monitoring: A preview of current and future technologies,” J. Lightwave Technol. 34(2), 525–543 (2016).
[Crossref]

Q. Fan, J. Lu, G. Zhou, D. Zeng, C. Guo, L. Lu, J. Li, C. Xie, C. Lu, F. N. Khan, and A. P. T. Lau, “Experimental Comparisons between Machine Learning and Analytical Models for QoT Estimations in WDM Systems,” in Optical Fiber Communication Conference (Optical Society of America, 2020), paper M2J.2.

Kilper, D. C.

D. C. Kilper, S. Chandrasekhar, L. Buhl, A. Agarwal, and D. Maywar, “Spectral monitoring of OSNR in high speed networks,” in European Conference and Exhibition on Optical Communication (ECOC), 2002, paper 7.4.4.

Kim, C. H.

J. H. Lee, D. K. Jung, C. H. Kim, and Y. C. Chung, “OSNR monitoring technique using polarization nulling method,” IEEE Photonics Technol. Lett. 13(1), 88–90 (2001).
[Crossref]

Kim, Hoon

H. G. Choi, J. H. Chang, Hoon Kim, and Y. C. Chung, “Nonlinearity-Tolerant OSNR Estimation Technique for Coherent Optical Systems,” in Optical Fiber Communication Conference (Optical Society of America, 2015), paper W4D.2.

Koenig, S.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error vector magnitude as a performance measure for advanced modulation formats,” IEEE Photonics Technol. Lett. 24(1), 61–63 (2012).
[Crossref]

Koos, C.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error vector magnitude as a performance measure for advanced modulation formats,” IEEE Photonics Technol. Lett. 24(1), 61–63 (2012).
[Crossref]

Kuschnerov, M.

Lankl, B.

Laperle, C.

Lau, A. P. T.

Lee, J. H.

J. H. Lee, D. K. Jung, C. H. Kim, and Y. C. Chung, “OSNR monitoring technique using polarization nulling method,” IEEE Photonics Technol. Lett. 13(1), 88–90 (2001).
[Crossref]

Leuthold, J.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error vector magnitude as a performance measure for advanced modulation formats,” IEEE Photonics Technol. Lett. 24(1), 61–63 (2012).
[Crossref]

Li, J.

Q. Fan, J. Lu, G. Zhou, D. Zeng, C. Guo, L. Lu, J. Li, C. Xie, C. Lu, F. N. Khan, and A. P. T. Lau, “Experimental Comparisons between Machine Learning and Analytical Models for QoT Estimations in WDM Systems,” in Optical Fiber Communication Conference (Optical Society of America, 2020), paper M2J.2.

Li, X.

Liu, D.

Q. Wu, L. Zhang, X. Li, M. Luo, Z. Feng, H. Zhou, M. Tang, S. Fu, and D. Liu, “Training Symbol Assisted in-Band OSNR Monitoring Technique for PDM-CO-OFDM System,” J. Lightwave Technol. 35(9), 1551–1556 (2017).
[Crossref]

J. Lu, S. Fu, L. Deng, M. Tang, Z. Hu, D. Liu, and C. C. K. Chan, “Blind and Fast Modulation Format Identification by Frequency-offset Loading for Hitless Flexible Transceiver,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper M2F.5.

Liu, X.

Lowery, A. J.

Lu, C.

Lu, J.

J. Lu and C. Lu, “Frequency offset drift monitoring: enabling simultaneously optimum performance and minimum cost of frequency offset estimation,” Opt. Lett. 44(15), 3753–3756 (2019).
[Crossref]

J. Lu, Z. Tan, A. P. T. Lau, S. Fu, M. Tang, and C. Lu, “Modulation format identification assisted by sparse-fast-Fourier-transform for hitless flexible coherent transceivers,” Opt. Express 27(5), 7072–7086 (2019).
[Crossref]

J. Lu, S. Fu, L. Deng, M. Tang, Z. Hu, D. Liu, and C. C. K. Chan, “Blind and Fast Modulation Format Identification by Frequency-offset Loading for Hitless Flexible Transceiver,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper M2F.5.

Q. Fan, J. Lu, G. Zhou, D. Zeng, C. Guo, L. Lu, J. Li, C. Xie, C. Lu, F. N. Khan, and A. P. T. Lau, “Experimental Comparisons between Machine Learning and Analytical Models for QoT Estimations in WDM Systems,” in Optical Fiber Communication Conference (Optical Society of America, 2020), paper M2J.2.

Lu, L.

Q. Fan, J. Lu, G. Zhou, D. Zeng, C. Guo, L. Lu, J. Li, C. Xie, C. Lu, F. N. Khan, and A. P. T. Lau, “Experimental Comparisons between Machine Learning and Analytical Models for QoT Estimations in WDM Systems,” in Optical Fiber Communication Conference (Optical Society of America, 2020), paper M2J.2.

Lu, Y.

W. Wang, A. Yang, P. Guo, Y. Lu, and Y. Qiao, “Joint OSNR and Interchannel Nonlinearity Estimation Method Based on Fractional Fourier Transform,” J. Lightwave Technol. 35(20), 4497–4506 (2017).
[Crossref]

Z. Wang, Y. Qiao, and Y. Lu, “OSNR monitoring technique based on multi-order statistical moment method and correlation function for PM-16QAM in presence of fiber nonlinearities,” Opt. Commun. 380, 10–14 (2016).
[Crossref]

Lun, H.

Luo, M.

Marom, D. M.

D. M. Marom, “Progress report on the design and development of flexible elements for the ‘Drop’ and ‘Add’ function,” http://cordis.europa.eu/docs/projects/cnect/5/318415/080/deliverables/001-FOXCD43Ares20142424820.pdf.

Maywar, D.

D. C. Kilper, S. Chandrasekhar, L. Buhl, A. Agarwal, and D. Maywar, “Spectral monitoring of OSNR in high speed networks,” in European Conference and Exhibition on Optical Communication (ECOC), 2002, paper 7.4.4.

Meyer, J.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error vector magnitude as a performance measure for advanced modulation formats,” IEEE Photonics Technol. Lett. 24(1), 61–63 (2012).
[Crossref]

Mizuno, T.

Monga, I.

Muramoto, Y.

Murata, K.

Napolitano, A.

W. A. Gardner, A. Napolitano, and L. Paura, “Cyclostationarity: Half a century of research,” Signal Process. 86(4), 639–697 (2006).
[Crossref]

Nebendahl, B.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error vector magnitude as a performance measure for advanced modulation formats,” IEEE Photonics Technol. Lett. 24(1), 61–63 (2012).
[Crossref]

Nespola, A.

Nosaka, H.

O’Sullivan, M.

Oda, S.

L. Dou, T Yamauchi, X. Su, Z. Tao, S. Oda, Y. Aoki, T. Hoshida, and J. Rasmussen, “An accurate nonlinear noise insensitive OSNR monitor,” in The Optical Fiber Communication Conference and Exhibition 2016 (OFC), paper W3A.5. pp. 1–3.

Ogawa, I.

Palkopoulou, E.

I. Tomkos, S. Azodolmolky, J. Sole-Pareta, D. Careglio, and E. Palkopoulou, “A tutorial on the flexible optical networking paradigm: state of the art, trends, and research challenges,” Proc. IEEE 102(9), 1317–1337 (2014).
[Crossref]

Paura, L.

W. A. Gardner, A. Napolitano, and L. Paura, “Cyclostationarity: Half a century of research,” Signal Process. 86(4), 639–697 (2006).
[Crossref]

Pendock, G.

Plant, D. V.

Poggiolini, P.

Poole, S.

Pulikkaseril, C.

Qiao, Y.

W. Wang, A. Yang, P. Guo, Y. Lu, and Y. Qiao, “Joint OSNR and Interchannel Nonlinearity Estimation Method Based on Fractional Fourier Transform,” J. Lightwave Technol. 35(20), 4497–4506 (2017).
[Crossref]

Z. Wang, Y. Qiao, and Y. Lu, “OSNR monitoring technique based on multi-order statistical moment method and correlation function for PM-16QAM in presence of fiber nonlinearities,” Opt. Commun. 380, 10–14 (2016).
[Crossref]

Qiu, K.

Qiu, M.

Rasmussen, J.

L. Dou, T Yamauchi, X. Su, Z. Tao, S. Oda, Y. Aoki, T. Hoshida, and J. Rasmussen, “An accurate nonlinear noise insensitive OSNR monitor,” in The Optical Fiber Communication Conference and Exhibition 2016 (OFC), paper W3A.5. pp. 1–3.

Roberts, K.

Roelens, M. A.

Saida, T.

Sano, K.

Sato, M.

Savory, S. J.

Schmogrow, R.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error vector magnitude as a performance measure for advanced modulation formats,” IEEE Photonics Technol. Lett. 24(1), 61–63 (2012).
[Crossref]

Shen, T. S.

Shieh, W.

Skafidas, E.

Sole-Pareta, J.

I. Tomkos, S. Azodolmolky, J. Sole-Pareta, D. Careglio, and E. Palkopoulou, “A tutorial on the flexible optical networking paradigm: state of the art, trends, and research challenges,” Proc. IEEE 102(9), 1317–1337 (2014).
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Stewart, L. A.

Su, L.

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[Crossref]

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L. Dou, T Yamauchi, X. Su, Z. Tao, S. Oda, Y. Aoki, T. Hoshida, and J. Rasmussen, “An accurate nonlinear noise insensitive OSNR monitor,” in The Optical Fiber Communication Conference and Exhibition 2016 (OFC), paper W3A.5. pp. 1–3.

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Taiba, M.

A. Carena, G. Bosco, V. Curri, P. Poggiolini, M. Taiba, and F. Forghieri, “Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links,” in European Conference and Exhibition on Optical Communication (ECOC), 2010, paper 4.07.

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Tang, M.

J. Zhao, L. Gan, L. Su, J. Zhang, H. He, W. Cai, J. Wang, S. Fu, and M. Tang, “Carrier Beating Impairment in Weakly Coupled Multicore Fiber-Based IM/DD Systems,” IEEE Access 8(1), 65699–65710 (2020).
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J. Lu, Z. Tan, A. P. T. Lau, S. Fu, M. Tang, and C. Lu, “Modulation format identification assisted by sparse-fast-Fourier-transform for hitless flexible coherent transceivers,” Opt. Express 27(5), 7072–7086 (2019).
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Q. Wu, L. Zhang, X. Li, M. Luo, Z. Feng, H. Zhou, M. Tang, S. Fu, and D. Liu, “Training Symbol Assisted in-Band OSNR Monitoring Technique for PDM-CO-OFDM System,” J. Lightwave Technol. 35(9), 1551–1556 (2017).
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J. Lu, S. Fu, L. Deng, M. Tang, Z. Hu, D. Liu, and C. C. K. Chan, “Blind and Fast Modulation Format Identification by Frequency-offset Loading for Hitless Flexible Transceiver,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper M2F.5.

Tao, Z.

L. Dou, T Yamauchi, X. Su, Z. Tao, S. Oda, Y. Aoki, T. Hoshida, and J. Rasmussen, “An accurate nonlinear noise insensitive OSNR monitor,” in The Optical Fiber Communication Conference and Exhibition 2016 (OFC), paper W3A.5. pp. 1–3.

Thomsen, B.

Tomkos, I.

I. Tomkos, S. Azodolmolky, J. Sole-Pareta, D. Careglio, and E. Palkopoulou, “A tutorial on the flexible optical networking paradigm: state of the art, trends, and research challenges,” Proc. IEEE 102(9), 1317–1337 (2014).
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Tran, A.

Tucker, R. S.

Wang, J.

J. Zhao, L. Gan, L. Su, J. Zhang, H. He, W. Cai, J. Wang, S. Fu, and M. Tang, “Carrier Beating Impairment in Weakly Coupled Multicore Fiber-Based IM/DD Systems,” IEEE Access 8(1), 65699–65710 (2020).
[Crossref]

Wang, W.

Wang, Z.

Z. Wang, Y. Qiao, and Y. Lu, “OSNR monitoring technique based on multi-order statistical moment method and correlation function for PM-16QAM in presence of fiber nonlinearities,” Opt. Commun. 380, 10–14 (2016).
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A. Blumer, A. Ehrenfeucht, D. Haussler, and M. K. Warmuth, “Occam's razor,” Inf. Process. Lett 24(6), 377–380 (1987).
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Winter, M.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error vector magnitude as a performance measure for advanced modulation formats,” IEEE Photonics Technol. Lett. 24(1), 61–63 (2012).
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Wu, B.

Wu, Q.

Xiang, M.

Xie, C.

Q. Fan, J. Lu, G. Zhou, D. Zeng, C. Guo, L. Lu, J. Li, C. Xie, C. Lu, F. N. Khan, and A. P. T. Lau, “Experimental Comparisons between Machine Learning and Analytical Models for QoT Estimations in WDM Systems,” in Optical Fiber Communication Conference (Optical Society of America, 2020), paper M2J.2.

Yamamoto, S.

Yamauchi, T

L. Dou, T Yamauchi, X. Su, Z. Tao, S. Oda, Y. Aoki, T. Hoshida, and J. Rasmussen, “An accurate nonlinear noise insensitive OSNR monitor,” in The Optical Fiber Communication Conference and Exhibition 2016 (OFC), paper W3A.5. pp. 1–3.

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Yi, L.

Yi, X.

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Q. Fan, J. Lu, G. Zhou, D. Zeng, C. Guo, L. Lu, J. Li, C. Xie, C. Lu, F. N. Khan, and A. P. T. Lau, “Experimental Comparisons between Machine Learning and Analytical Models for QoT Estimations in WDM Systems,” in Optical Fiber Communication Conference (Optical Society of America, 2020), paper M2J.2.

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Zeolla, D.

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Zhang, J.

J. Zhao, L. Gan, L. Su, J. Zhang, H. He, W. Cai, J. Wang, S. Fu, and M. Tang, “Carrier Beating Impairment in Weakly Coupled Multicore Fiber-Based IM/DD Systems,” IEEE Access 8(1), 65699–65710 (2020).
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Zhao, J.

J. Zhao, L. Gan, L. Su, J. Zhang, H. He, W. Cai, J. Wang, S. Fu, and M. Tang, “Carrier Beating Impairment in Weakly Coupled Multicore Fiber-Based IM/DD Systems,” IEEE Access 8(1), 65699–65710 (2020).
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Q. Wu, L. Zhang, X. Li, M. Luo, Z. Feng, H. Zhou, M. Tang, S. Fu, and D. Liu, “Training Symbol Assisted in-Band OSNR Monitoring Technique for PDM-CO-OFDM System,” J. Lightwave Technol. 35(9), 1551–1556 (2017).
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Opt. Commun. (1)

Z. Wang, Y. Qiao, and Y. Lu, “OSNR monitoring technique based on multi-order statistical moment method and correlation function for PM-16QAM in presence of fiber nonlinearities,” Opt. Commun. 380, 10–14 (2016).
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Figures (18)

Fig. 1.
Fig. 1. (a) A power spectrum of 28 GBaud 16-QAM signal and spectral components used for measuring the correlation. Signal power is 0 dBm and the ROF is 0.5. (b) The calculated normalized SCDF and the estimated SNR versus reference SNR.
Fig. 2.
Fig. 2. (a) The relationship between the reference and estimated SNRnonlinear using spectral correlation-based SNR estimation for up-to-7-channel WDM systems. (b) Combined SNR, ASE-induced SNR, and the estimated SNR of a single-channel 16-QAM signal.
Fig. 3.
Fig. 3. (a) Signal spectrum of 28 GHz 16QAM signal and pure linear noise spectrum; (b) Signal spectrum of 28 GHz 16QAM signal and pure nonlinear noise spectrum. ROF=0.5
Fig. 4.
Fig. 4. Position moving process of the two spectral components at the upper and lower sideband with induced-FO of 3 GHz. The signal parameter is same as that in Fig. 1(b).
Fig. 5.
Fig. 5. Fitting ${\hat{S}_x}^{{\alpha _\textrm{0}}}(f)$ as a function of FO using (a) $y = A{x^\textrm{2}} + Z$, (b) $y = A{x^\textrm{3}} + Z$, and (c) $y = A{x^\textrm{4}} + Z$.
Fig. 6.
Fig. 6. Relation between coefficient A obtained by cubic curve fitting and launch power.
Fig. 7.
Fig. 7. Curve fitting result of (a) Eq. (17) and (b) Eq. (18) using 21 pairs of (SNRlinear, SNRnonlinear), with the launch power swept from −5 dBm to 5 dBm at a stepsize of 0.5 dBm. SNRmeas is measured using the spectral correlation-based SNR estimation method.
Fig. 8.
Fig. 8. (a) SNRlinear monitoring and (b) SNRnonlinear monitoring for single channel 16-QAM signal with ROF of 0.5, 0.6, and 0.7. The link is 915 km consisting of 12 equal SMF spans and EDFAs with NF of 6 dB.
Fig. 9.
Fig. 9. (a) SNRlinear estimation error curve and (b) SNRnonlinear estimation error curve of the results in Fig. 8.
Fig. 10.
Fig. 10. (a) SNRlinear monitoring and (b) SNRnonlinear monitoring for single-channel 16-QAM signal with ROF of 0.1 and 0.2. (c) SNRlinear estimation error curve and (d) SNRnonlinear estimation error curve of the results in Fig. 10(a) and 10(b), respectively.
Fig. 11.
Fig. 11. (a) SNRlinear monitoring and (b) SNRnonlinear monitoring for multi-channel 16-QAM signal. The simulation setup is the same as that in Fig. 8.
Fig. 12.
Fig. 12. The values of contribution factors in Eq. (17) and Eq. (18) versus the number of WDM channels.
Fig. 13.
Fig. 13. (a) SNRlinear monitoring and (b) SNRnonlinear monitoring for single channel QPSK and 64-QAM signal. The simulation setup is the same as that in Fig. 8.
Fig. 14.
Fig. 14. (a) SNRlinear monitoring and (b) SNRnonlinear monitoring performance for mixed conditions with span number of 6, 9, and 12, and NF values of 4 dB, 5 dB and 6 dB.
Fig. 15.
Fig. 15. (a) Normalized power frequency response with 1 to 12 cascaded ROADMs. (b) Estimation errors versus the number of cascaded ROADMs with the LPs of −3 dBm, 0 dBm, and +3 dBm. (c) The optimum upper limit of FO sweeping and (d) the values of contribution factors in Eq. (17) and Eq. (18) versus the number of cascaded ROADMs. Single-channel 16-QAM signal with the ROF of 0.5 is simulated.
Fig. 16.
Fig. 16. Experimental setup of PDM 28 GBaud 16-QAM WDM system. WSS: wavelength selective switch; LO: local oscillator; PDM Emu: polarization-division multiplexing emulator.
Fig. 17.
Fig. 17. Experimental (a) SNRlinear monitoring and (b) SNRnonlinear monitoring performance for 1-, 3-, 5-, and 7-ch conditions.
Fig. 18.
Fig. 18. Experimental (a) SNRlinear monitoring and (b) SNRnonlinear monitoring estimation error for 1-, 3-, 5-, and 7-ch conditions.

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

μ ¯ x = E [ x ( t ) ]
R ¯ x ( τ ) = E [ x ( t  +  τ 2 ) x ( t  -  τ 2 ) ]
R x α ( τ ) E [ x ( t + τ / 2 ) x ( t τ / 2 ) exp ( j 2 π α t ) ]
S x α ( f )  -  R x α ( τ ) exp ( j 2 π f τ ) d τ
S x α ( f ) E [ X T ( t , f + α / 2 ) X T ( t , f  -  α / 2 ) ]
X T ( t , v ) = t T / 2 t + T / 2 x ( u ) exp ( j 2 π v u ) d u
S ^ x α ( f ) E [ X T ( t , f + α / 2 ) X T ( t , f  -  α / 2 ) ] E [ | X T ( t , f + α / 2 ) | 2 ] E [ | X T ( t , f α / 2 ) | 2 ]
X ~ T ( t , v ) = t T / 2 t + T / 2 ( x ( u ) + n ( u ) ) exp ( j 2 π v u ) d u = X T ( t , v ) + N T ( t , v )
S ^ x α ( f )  =  E [ X T ( t , f + α / 2 ) X T ( t , f  -  α / 2 ) ] E [ | X T ( t , f + α / 2 ) | 2  +  | N T ( t , f + α / 2 ) | 2 ] E [ | X T ( t , f α / 2 ) | 2  +  | N T ( t , f α / 2 ) | 2 ]
E [ | X T ( t , f c + α / 2 ) | 2 ] =  E [ | X T ( t , f c  -  α / 2 ) | 2 ] = E [ P S ( f c ± α / 2 ) ]
E [ | N T ( t , f c + α / 2 ) | 2 ] =  E [ | N T ( t , f c  -  α / 2 ) | 2 ] = E [ P N ( f c ± α / 2 ) ]
S ^ x α 0 ( f c )  =  E [ P S ( t , f c ± α 0 / 2 ) ] E [ P S ( t , f c ± α 0 / 2 ) ] + E [ P N ( t , f c ± α 0 / 2 ) ]
S N R ( f c ,   α 0 )  =  E [ P S ( t , f c ± α 0 / 2 ) ] E [ P N ( t , f c ± α 0 / 2 ) ] = S ^ x α 0 ( f c ) 1 -  S ^ x α 0 ( f c )
S N R  =  f i B m e a s E [ P S ( t , f i ) ] E [ P N ( t ) ] B m e a s = f i B m e a s E [ P S ( t , f i ) + P N ( t ) ] E [ P N ( t ) ] B m e a s 1
E [ P N ( t ) ] = E [ P S ( t ,   f c ± α 0 / 2 )  +  P N ( t ) ] [ 1 S ^ x α 0 ( f c ) ]
S N R  =  f i B m e a s E [ P S ( t , f i ) + P N ( t ) ] E [ P S ( t ,   f c ± α 0 / 2 ) + P N ( t ) ] B m e a s 1 [ 1 S ^ x α 0 ( f c ) ] 1
A = A 1 1 S N R l i n e a r + A 2 1 S N R n o n l i n e a r + A 3
1 S N R m e a s = B 1 1 S N R l i n e a r + B 2 1 S N R n o n l i n e a r + B 3
S ( f )  =  1 2 σ 2 π [ e r f ( B 0 / 2 f 2 σ ) e r f ( B 0 / 2 f 2 σ ) ] ,   σ = B O T F 2 2 ln 2