Joint linear and nonlinear noise monitoring techniques based on spectrum analysis

Jianing Lu; Gai Zhou; Jing Zhou; Chao Lu; Chao Lu

doi:10.1364/OE.411522

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 [1–3]. 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 [23–25], 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 SNR_linear and SNR_nonlinear 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 SNR_linear and SNR_nonlinear 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 1^st and 2^nd order statistical properties (i.e. the mean value and autocorrelation function) are defined as:

(1)$${\bar{\mu }_x} = E[x(t)]$$

(2)$${\bar{R}_x}(\tau ) = E[x(t\textrm{ + }\frac{\tau }{\textrm{2}}) \cdot {x^{\ast }}(t\textrm{ - }\frac{\tau }{\textrm{2}})]$$

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

(3)$${R_x}^\alpha (\tau ) \equiv E[x(t + \tau /2) \cdot {x^\ast }(t - \tau /2) \cdot \textrm{exp} ( - j2\pi \alpha t)]$$

and then ${S_x}^\alpha (f)$ is

(4)$${S_x}^\alpha (f) \equiv \int_{\textrm{ - }\infty }^\infty {{R_x}^\alpha (\tau ) \cdot \textrm{exp} ( - j2\pi f\tau )} d\tau$$

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

(5)$${S_x}^\alpha (f) \equiv E[{X_T}(t,f + \alpha /2) \cdot {X_T}^{\ast }(t,f\textrm{ - }\alpha /2)]$$

where

(6)$${X_T}(t,v) = \int_{t - T/2}^{t + T/2} {x(u) \cdot \textrm{exp} ( - j2\pi vu)du}$$

and T is integration time much larger than ${T_s}$. Then, we normalize the SCDF to limit its range within [−1, 1], as

(7)$${\hat{S}_x}^\alpha (f) \equiv \frac{{E[{X_T}(t,f + \alpha /2) \cdot {X_T}^{\ast }(t,f\textrm{ - }\alpha /2)]}}{{\sqrt {E[{{|{{X_T}(t,f + \alpha /2)} |}^2}]} \sqrt {E[{{|{{X_T}(t,f - \alpha /2)} |}^2}]} }}$$

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:

(8)$${\tilde{X}_T}(t,v) = \int_{t - T/2}^{t + T/2} {(x(u) + n(u)) \cdot \textrm{exp} ( - j2\pi vu)du = } {X_T}(t,v) + {N_T}(t,v)$$

Then, the normalized SCDF can be expressed as

(9)$${\hat{S}_x}^\alpha (f)\textrm{ = }\frac{{E[{X_T}(t,f + \alpha /2) \cdot {X_T}^{\ast }(t,f\textrm{ - }\alpha /2)]}}{\begin{array}{l} \sqrt {E[{{|{{X_T}(t,f + \alpha /2)} |}^2}\textrm{ + }{{|{{N_T}(t,f + \alpha /2)} |}^2}]} \cdot \\ \sqrt {E[{{|{{X_T}(t,f - \alpha /2)} |}^2}\textrm{ + }{{|{{N_T}(t,f - \alpha /2)} |}^2}]} \end{array}}$$

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

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

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

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

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

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

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

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

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

(14)$$SNR\textrm{ = }\frac{{\sum\limits_{{f_i} \in {B_{meas}}} {E[{P_S}(t,{f_i})]} }}{{E[{P_N}(t)] \cdot {B_{meas}}}} = \frac{{\sum\limits_{{f_i} \in {B_{meas}}} {E[{P_S}(t,{f_i}) + {P_N}(t)]} }}{{E[{P_N}(t)] \cdot {B_{meas}}}} - 1$$

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

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

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

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

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.

Abstract

1. Introduction

2. Operating principle

2.1 SNR estimation using signal spectral correlation

2.2 Impact of nonlinearity on SNR estimation using signal spectral correlation

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

2.4 Principle of joint linear and nonlinear noise monitoring

3. Simulation results

4. Experimental results

5. Conclusion

Funding

Disclosures

References

Cited By

Figures (18)

Equations (19)

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