1. Introduction

Performance estimation consists in knowing ahead-of-time the performance of a communication system in terms of a meaningful metric. In optical communications, a fast and accurate assessment of system performance challenges the academic world and is strategic for the telecommunications industries. Indeed, we use performance estimation in many scenarios with different constraints in terms of time and accuracy. For instance, we can use performance estimation in routing algorithms to evaluate the light-path feasibility when establishing or restoring data streams. This operation requires a stringent response time, usually in the order of milliseconds or sometimes microseconds, possibly at the expense of prediction accuracy [1]. Furthermore, we can use performance estimation in engineering and planning tools, e.g., for call-for-bids network design, where the accuracy becomes the fundamental objective while relaxing the response-time requirements.

Estimation becomes harder and harder as optical network technologies grow in complexity to cope with modern data-transmission needs. A race to a universal performance prediction model or algorithm has, therefore, begun [2–8]. In this context, the reference theory is the Gaussian-noise (GN) model [2], which is a sufficiently accurate approximate tool able to estimate the transmission quality of dispersion uncompensated (or unmanaged) (DU) coherent optical systems. One of the fundamental assumptions at the GN-model footings is that the informative signal behaves as Gaussian noise. Namely, in uncompensated links, the signal gets quickly much dispersed, eventually assuming a Gaussian-like distribution. This assumption excludes from the GN-model scope all the slightly dispersive systems, like dispersion-managed (DM) system or short DU links with zero (or small) dispersion fiber spans. These systems, however, are deployed and are of concern for telecommunications operators. Therefore, in engineering and planning tools, two distinct estimators were used to assess performance. Specifically, [9] and [10] were used for legacy systems, i.e., intensity-modulated direct-detection 10G dispersion-managed transmissions, and the GN-model and its derivatives were used for modern dispersion uncompensated coherent systems. These estimators assess performance adequately within their scopes, but they fail to provide a comprehensive model where dispersion compensated and uncompensated regimes mix up. This is cumbersome in a scenario where computational complexity and simplicity are crucial, like in wavelength switched optical network routing algorithms.

Considering these reasons, in recent years, semi-analytical alternatives have been developed and published, [6] and [11]. These models predict the system SNR assessing the nonlinear interference (NLI) power in a semi-analytical manner, specifically using a two-step procedure comprising a profiling calibration to be later combined with an analytical formula. This could potentially address multi-span heterogeneous links, i.e., with uncompensated and compensated dispersion regimes and fiber types mixed up [6]. Nevertheless, although being able to provide meaningful performance assessment of the optical channels in mixed conditions, they do not take analytically into account the performance alterations coming from modern transmitter-receiver (TRX) digital signal processing (DSP), such as carrier-phase estimation (CPE) [12] or nonlinear compensation, like digital back-propagation (DBP) and many others [13].

Indeed, to satisfactorily predict transmission performance, the calibration should profile all possible DSP techniques used in the system. It is for this reason that we developed the estimator presented in our conference contribution [14]. The reasoning behind this novel estimator is depicted with a block diagram in Fig. 1. In this figure, we highlight that this model not only predicts the performance of the optical channel, i.e., transducer-to-transducer, but accounts for the implemented signal processing, providing as a performance metric the signal-to-noise ratio (SNR) before channel decoding. This predictor expands the scope of [6] and [11] from the NLI power estimation to the NLI power spectral density (PSD) estimation, i.e., how nonlinear effects distributes in the frequency domain, distinguishing between quadrature (Q) and in-phase (I) components [12]. This new knowledge enables analytical accounting of linear filtering (e.g., optical filtering) and, in some cases, also nonlinear signal processing (e.g., CPE), eventually overcoming the need for numberless calibrations.

 

Fig. 1. Mapper-to-demapper performance estimation concept. Looking at the source-to-destination communication line, the model not only estimates optical channel performance, i.e., from transducer-to-transducer, but also accounts for the digital signal processing unit, i.e., from mapper-to-demapper.

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Fig. 2. One-span profiling calibration setup. ECL: external-cavity laser; DAC: digital-to-analog converter: ADC: analog-to-digital converter; PDM: polarization division multiplexer; I/Q: in-phase/quadrature; LEAF: large effective-area fiber; SSMF: standard single-mode fiber; LO: local oscillator; DSP: digital signal processing; EDFA erbium-doped fiber amplifier; WSS: wavelength selective switch; WDM: wavelength-division multiplex.

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The price we pay using this estimator instead of [6] and [11] is an increased computational complexity and memory demand. Because, rather than a dataset of variances, i.e., a dataset of scalar variables to combine in the analytical formula, we use a dataset of spectral densities, i.e., arrays of scalar variables. However, at the present day, we have not yet performed a computational performance assessment.

This paper is an extension of [14], where we report in detail the mathematical foundations, experimental methodologies, and results. The paper is structured as follows. First, in Section 2, we recall the theoretical background, and we present the PSD estimation principles. Second, in Sections 3 and 4, we present the experimental results, focusing firstly on the profiling calibration and subsequentially to the estimation of heterogeneous four span-long transmission links. Third, in Section 5, we present how an adaptive CPE can be considered in the estimation process without harshening the calibration procedure. Finally, in Section 6, we report proper conclusions.

2. Theoretical background

2.1 Model assumptions

We developed the performance estimator under several assumptions that are reported in this subsection. First, as common in optical communication scenarios, we assume the transmission to be degraded by three main noise contributions [15]: the amplified spontaneous emission (ASE) coming from erbium-doped fiber amplifiers (EDFA), the Kerr-induced NLI, and the transmitter-receiver (TRX) imperfection noise. Furthermore, we assume them independent one from another, neglecting interaction among them. Hence, their contributions sum up to yield a meaningful estimate of the overall noise. Without loss of generality, we assume the amplifiers to share the same data-sheet, i.e., we consider the ASE contribution equal at every span, and the optical power at the span input to be homogeneous.

Second, we assume that the NLI contribution sums up independently span after span, cumulatively, neglecting the impact of span correlation. This assumption is rational in dispersion-uncompensated systems, where the correlation between different spans becomes negligible, as observed in [6]. Moreover, similarly to [6], we assume that the power-normalized NLI spectral density generated by the k-th span, $S_{NLI}^{NSR}({f,D_k^{in},{F_k}} )$, is a function of the span input dispersion pre-distortion, $D_k^{in}$, and the fiber type, ${F_k}$.

Third, we assume the in-phase (I) and quadrature (Q) components for both linear and nonlinear contributions independent; thus, we can separately sum them up span after span. At the end of the link, we have two distinct estimates for the in-phase and quadrature components yielding a prediction of the overall noise from their sum. In the context of this investigation, this is probably the most delicate assumption. Indeed, in-phase and quadrature components may be correlated. However, though this could be a limiting factor for the investigated methodology, thanks to the accurate results in the following sections, we observe that this assumption has no significant detrimental impact on the targeted links.

2.2 Power spectral density estimation principles

Eventually, to get the overall performance, we evaluate the power spectral densities (PSDs) of the noise-to-signal ratio (NSR) at a given power $P$, $S_{TOT}^{NSR}(f )$, as

The spectral densities associated with the linear sources of noise, $S_{ASE}^{NSR}(f )$ and $S_{TRX}^{NSR}(f )$, are practically assumed to be white, i.e., flat in the channel bandwidth. Regarding ASE contribution, we evaluate the noise power in the absence of fiber, with an equivalent optical attenuator to avoid silica-related nonlinear Kerr-effects. We assess the optical SNR, and then we normalize it to the channel bandwidth. Regarding TRX-imperfection, we evaluate the back-to-back SNR, in the absence of fiber and optical attenuation. Both techniques to extract linear noise contributions have been used and detailed in [11]. Finally, the two linear effects contributions, $S_{ASE}^{NSR}(f )$ and $S_{TRX}^{NSR}(f )$, are assumed to split equally into the in-phase and quadrature components.

The nonlinear effects spectral densities ($S_{{a_{NL}}}^{I/Q}({f,D_k^{in},{F_k}} )$) are obtained operating at high power levels, i.e., in the strongly nonlinear regime, securing that all the other noise contributions are as small as possible compared to NLI. Experiment outcomes are re-scaled according to the cubic power-law dependence of silica-fiber Kerr-induced nonlinearities [3]. Finally, thanks to the following formula, that accounts for single-span linear contributions, we get the NLI spectral density of each component

3. Profiling calibration

3.1 Calibration experiment setup

Figure 2 depicts the setup for the targeted profiling one-span calibration experiment. We transmit 21 coherent polarization division multiplexed (PDM) quadrature phase-shift keying (QPSK) channels. We investigate the central channel, which is generated by passing light from an external cavity laser (ECL), at 1545.32 nm, into a PDM I/Q modulator. The central channel is then multiplexed with two 50 GHz-shifted combs of ten 100 GHz-spaced distributed feedback lasers, each modulated with dedicated I/Q modulators, and followed by PDM emulation. The channels ordering and spectrum is represented as an inset of Fig. 2. Each modulator is driven by digital-to-analog converters (DACs) to produce optical data at 65 Gbit/s per polarization from independent 2¹⁶-long pseudo-random quaternary sequences and with a root-raised-cosine pulse shaping with 0.2 roll-off factor. All channels are recombined via a wavelength selective switch (WSS) that selects every other channel from each input port.

The multiplex is amplified to the desired launch power, e.g., 20 dBm (6.77 dBm/ch) when assessing the nonlinear impairments, and then sent to the line that can either be one span of the desired investigated fiber type, a variable optical attenuator, or back-to-back. We profile two span-types, a 100 km standard single-mode fiber (SSMF) with 16.7 ps/nm/km dispersion at 1550 nm and a large effective-area fibers (LEAF) with 4.3 ps/nm/km dispersion at 1550 nm. Before the receiver, the multiplex is pre-amplified to approximately the same power as the input of all fiber spans.

At the receiver end, the central channel is extracted with a 50 GHz optical filter. The last amplifier ensures a constant optical power to feed the coherent mixer, equipped with four balanced photo-diodes and an ECL as a local oscillator. The photo-currents are sampled at 40 GS/s by an oscilloscope with 20 GHz electrical bandwidth. Five acquisitions of four million symbols are stored and processed offline. After that, we complete skew adjustments, normalization, polarization demultiplexing with blind equalization based on the constant-modulus algorithm, and frequency offset compensation. Moreover, in order not to alter the spectral density shape, as we will observe in the following sections, we use a large number of CPE taps (half-window K=50) [17].

The investigation, and the calibration procedure, is performed accordingly to the QPSK modulation format. We see no obstacles for the presented model and its profiling calibration to work for other modulation formats. Nevertheless, the calibration should be correctly performed and targeted on the different considered formats. However, further investigations on the accuracy of the model and on the possibility of adapting the profiling calibration are necessary and are left for further studies.

3.2 Calibration experiment results

As stated previously, we assume the linear contributions spectral densities, $S_{ASE}^{NSR}(f )$ and $S_{TRX}^{NSR}(f )$, flat in the frequency domain, having a PSD with a constant magnitude given by the calibration experiments in the absence of the optical fiber. Similarly to [11], we get an $OSN{R_{0.1nm}}$ of about 30 dB and $SNR_{TRX}$ of about 21 dB. Nonlinear contributions are evaluated at high power levels, i.e., ${P_{WDM}} = 20dBm$, to consider the noise dominated by NLI only. We repeat the calibration for both SSMF and LEAF at several different digitally emulated values of input dispersion pre-distortion. The ${S^{I/Q}}({f,D_k^{in},{F_k}} )$ in Eq. (3) are obtained with the periodogram method using 2⁷ points rectangular window [18]. In Fig. 3, we plot a subset of the totality of spectral-densities estimated during the calibration process. In Fig. 3(a), we have different PSDs varying the input distortion pre-dispersion from 860 ps/nm to 3770 ps/nm for the LEAF fiber case, while in Fig. 3(b), we have the SSMF fiber case instead.

 

Fig. 3. Single-span calibrations spectral-densities for large effective area fiber (LEAF) (a) and standard single-mode fiber (SSMF) (b) at various input dispersion pre-distortion (D^in). As the pre-dispersion increases, the nonlinear effects become whiter, i.e. the central lobe narrows down, and stronger, i.e., the height of the curve rises.

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Looking at each plot singularly, we observe that the nonlinear contribution increases by increasing the input dispersion pre-distortion. Indeed, passing from 860 to 3770 ps/nm the nonlinear effects steadily increase their power as the input pre-dispersion increases. This behavior identically verifies for both fiber types. This behavior confirms the previous findings of [6] and [11]. Moreover, by increasing the input dispersion pre-distortion the spectra-density becomes whiter, i.e., flat, in the channel bandwidth. Indeed, for both cases reported in Fig. 3, as the pre-dispersion runs from 860 to 3170 ps/nm, the PSD tails start to flatten out, moving from a remarkably colored curve with a conspicuous central lobe for the 860 ps/nm curves to an almost flat PSD for the 3770 ps/nm cases, where the lobe sensitively narrows down.

Continuing looking at Fig. 3, we can make an additional comment on fiber-type related nonlinear performance. The SSMF has a lower nonlinear coefficient than the LEAF, hence, a lower degradation due to NLI. Indeed, as depicted in Fig. 3, the two fibers behave similarly, to the point that the calibration spectral-densities of the SSMF seem lower-power shifted replicas of the LEAF PSDs, i.e., with less impact on the signal quality.

4. Transmission experiments and results

4.1 Transmission testbed setup

In this section, we present the experimental testbeds implemented to replicate a heterogeneous metro-network condition and the experimental results in terms of spectral-densities prediction and SNR accuracy. Pragmatically, we compare the estimated PSD, obtained from the semi-analytical procedure presented in Section 2, to the actual one for four different heterogeneous dispersion unmanaged four-span long links. Specifically: LLSS, SSLL, LLLS, and SLLL, where L stands for LEAF and S stands for SSMF. The experimental testbeds are similar to the simulated target systems of [6]. In Fig. 4, we report schematically the four investigated links, where we note that the span is a cascade of an erbium-doped fiber amplifier, assuring a constant power at the input of the span, and a transmission fiber. Being the fiber Kerr-induced nonlinear effects related to the cube of the power [19], assuring a constant power at the span input assures a constant nonlinear contribution. The receiver and transmitter are the same as the one-span calibration experiment of Fig. 3.

 

Fig. 4. Heterogeneous four-span transmission setups. The experiment replicates the target system of [6] and uses the same transceiver of the setup of Fig. 2. A span is a concatenation of an erbium-doped fiber amplifier (EDFA), assuring a constant power at the input of the span, and a transmission fiber – either a large effective-area fiber (LEAF) or a standard single-mode fiber (SSMF).

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4.2 Transmission results

In Fig. 5, we report the estimated and actual spectral-densities at ${P_{WDM}} = 20dBm$, i.e., in the nonlinear regime, for all the targeted heterogeneous transmission systems. On the right-hand side, we depict the total NSR PSD at the end of the four-span link. The actual PSD (purple) is evaluated at the receiver and indeed accounts for all the optical and electrical transmission impairments. The estimated spectral density (green) is evaluated through the results of the single-span calibration experiment and Eq. (1), i.e., accounting for NLI, ASE, and TRX-impairments as detailed in Section 3. On the right-hand side of the figure, we depict the in-phase (dashed) and quadrature (solid) NSR components. For all the figures on the right-hand side, the red curves represent the experiments while the blue the estimations.

 

Fig. 5. Power spectral densities of the investigated heterogeneous transmission links. On the left-hand side we have the total spectral density, while on the right-hand side we have the in-phase and quadrature components separated. PSD: power spectral density; S: standard single-mode fiber; L: large effective-area fiber;

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In Fig. 5, we observe two behaviors: SSLL (Fig. 5(c)) and SLLL (Fig. 5(g)) transmissions are mainly white, and they present just a very narrow central lobe; LLSS (Fig. 5(a)) and LLLS (Fig. 5(e)) are colored, presenting a remarkably large central lobe. Indeed, SSMF accumulates dispersion faster than LEAF, e.g., for 100 km of SSMF we experience a cumulated distortion of about 1670 ps/nm while after LEAF it is only 430 ps/nm. Now, if we have SSMF fibers at the beginning of the link, we move faster towards where the nonlinear contributions are stronger and whiter (see Fig. 3). For this reason, the two links with the whitest noises are the ones that start with SSMF. Furthermore, the whitest transmission signature is the SSLL, which moves faster towards higher pre-dispersions after two SSMF spans. We observe that we predict well the PSDs in these two cases; this is sound since we are working in a regime where the links fulfill reasonably all model assumptions.

Nevertheless, the most insightful results can be found at the output of the links with more remarkable colored noises, i.e., LLSS (Fig. 5(a)) and LLLS (Fig. 5(e)). First, we observe that we can well predict the PSD shape with a little inaccuracy on the PSD spectral tails, where probably the calibration noise acts stronger, resulting in a noise overestimation. However, the estimated PSD describes well the central lobe characterizing the color. The worst performance, in these cases, is given by the LLSS (Fig. 5(a)) link, which, nevertheless, predicts both PSD and SNR reasonably good.

Looking at the right-hand side of Fig. 5, we can make the following observations: a low-pass noise characterizes the quadrature component while the in-phase component is band-pass. Moreover, the quadrature component has a dominant impact onto the low-pass shape of the overall noise. This finding confirms the significant impact of CPE on system performance, since, according to [17], CPE approximates with a small number of taps to a notch-filter. In the links characterized by colored noise, i.e., LLSS (Fig. 5(a)) and LLLS (Fig. 5(e)), we observe that the total noise is mostly due to the quadrature component, which dominates in the central frequencies the in-phase component by up to about 8 dB/Hz. In the systems characterized by white noise, i.e., SSLL (Fig. 5(c)) and SLLL (Fig. 5(g)), we observe that the band-pass shape of the in-phase (I) component and low-pass shape of the quadrature (Q) component tend to compensate and give a white spectrum.

We can conclude that, overall, Fig. 5 demonstrates the capability of the presented estimator of predicting the spectral density of the NSR in heterogeneous highly and moderately dispersive regimes, which are characteristic of metro networks. Moreover, the plots show that the quadrature component, in moderately dispersive links, contributes consistently to the overall noise, and it is responsible for the characteristic low-pass shape. Even though we can see that the estimator works by looking at the PSD prediction of these figures, in the next sections, we quantify the estimation accuracy to our target metric: the SNR.

In Fig. 6, we plot the experimental (markers) and estimated (solid lines) performance in terms of SNR versus total WDM launch power, i.e., the so-called bell curves, in the nonlinear regime. In solid line, we have the total SNR estimation, accounting for both linear contributions and nonlinear effects, i.e., Eq. (1), while in dashed, we have the nonlinear asymptote, i.e., the estimation of the SNR due to nonlinear effects only. From the figures, we understand that the model can fit the actual system performance for all the investigated cases. Precisely, in the remainder of this chapter, we report the model accuracy at 20 dBm. The LLSS link, depicted in blue on the left-hand side of Fig. 6, provides the worst SNR estimation accuracy, namely, $\Delta SNR \simeq 0.29dB$, which may be considered still reasonably accurate. Recall that, among all the investigated links, LLSS is one of the most colored. Furthermore, the SLLL link, depicted in red on the right-hand side of Fig. 6, has the best estimation accuracy, namely $\Delta SNR \simeq 0.12dB$. Hence, we ultimately state that the model predicts the performance in every targeted link accurately, capping the error below 0.3 dB.

 

Fig. 6. Experimental and estimated performance in terms of SNR versus wavelength-division multiplex (WDM) launch power for SSLL and LLSS (a) and SLLL and LLLS links (b). SNR: signal-to-noise ratio; NL: nonlinear.

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Moreover, leaving aside the estimator accuracy assessment and observing the experimental SNRs of the four bell-curves of Fig. 6, we remark a result that confirms previous findings summarized in [6]. The curves associated with the transmissions beginning with a LEAF fiber and ending with an SSMF fiber have an overall better SNR performance compared to the curves beginning with SSMF and ending with LEAF. This is because the SSMF fiber accumulates dispersion distortion faster than LEAF. So, placing an SSMF fiber at the beginning of the link allows for reaching the higher plateau of [6] quickly. Moreover, leaving LEAF fibers at the end of the link increases the amount of nonlinear effects affecting the link, since they contribute more to nonlinear impairments than SSMF fiber and they can add contributions relative to their higher dispersion region of functioning.

5. Performance prediction with carrier-phase estimation

5.1 Carrier-phase estimator agnostic predictor

In the introduction, we state that the main goal for the development of a semi-analytical method to assess the full power spectrum of the noise rather than just the power is the possibility of accounting for linear filtering, such as optical filters, and, in some cases, also nonlinear signal processing, eventually overcoming the need for numberless calibrations necessary with the estimators reported in [6] and [11]. We now focus on the LEAF-LEAF-SSMF-SSMF (LLSS) transmission, adjusting the Viterbi-Viterbi (VV) CPE algorithm at the receiver to enhance system performance [12]. Specifically, we address in the following paragraphs the impact of changing the number of CPE averaging taps [20].

The CPE is a nonlinear manipulation of the received samples, which, according to [17], can be approximated as linear filtering of the phase or, by small-angle approximation, the quadrature component. In Fig. 7(a), we plot the magnitude-squared frequency impulse responses, $H_K^{VV}(f )$, of the quadrature component linear filter approximation, according to [17], in case of 2, 5, 20 and 50 half-window taps. We observe that the shape varies a lot. When K=2, the estimator cuts the central frequencies abundantly while enhancing the lateral ones. When K=5 and K=20, we observe a shrinking of the central frequencies notch, and we start noticing an oscillation moving towards the lateral frequencies. This behavior becomes more evident when K=50, where only the closest frequencies around zero are cut and then frequency response oscillates around one. This behavior sharpens with the number of taps increasing. Eventually, with K big enough, the filter does not alter the incoming PSD anymore, apart from a very narrow central frequency cut. This is in accordance with Fig. 3 and Fig. 5, where a narrow central notch is always present (in these cases, we used K=50).

 

Fig. 7. On the left-hand side, the magnitude-squared frequency response for different phase-estimation averaging taps (a). On the right-hand side, the actual (red) and the estimated by filtering of the quadrature component (blue) spectral densities for the LEAF-LEAF-SSMF-SSMF case using carrier-phase estimation with K=10 (b). K: half-window number of taps; Q: quadrature; PSD: power spectral density. Normalized Frequency = (Relative Frequency*2) / Symbol Rate.

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Thanks to this observation, without increasing the complexity of the calibration, we can update the step procedure estimation as follows. First, we perform the single-span calibration experiment, and we store the quadrature and in-phase components PSDs of the NLI at different input dispersion pre-distortions. At this point, we use a large number of taps, K, which according to [17] does not alter the noise spectrum color significantly. Hence, we can write the following adapted equation

According to Eq. (4), we build an adaptive estimator that can assess performance for all possible CPE equalizing windows, using only the single-span calibration experiment with many averaging taps, K=50. In short, we prove the concept that the investigated semi-analytical model, thanks to the ability to predict the PSD of the quadrature and in-phase component separately, can efficiently estimate the system performance accounting for some signal-processing techniques performed at the transceiver without complicating the calibration procedure.

5.2 Experimental results with adaptive carrier-phase estimator

We propose SNR predictions with a half-window number of taps, K, equal to 3, 5, and 10 along with the long-average K=50 used in the profiling calibration. As a matter of instance, in Fig. 7(b), we represent the actual and estimated quadrature component spectral-densities with K=10 taps. For the actual quadrature PSD, in red, we can recognize the central frequency attenuation and the sidelong frequencies oscillations expected when using a CPE with ten half-window taps. In blue, we observe the estimated quadrature PSD, extrapolated combining the profiling calibration with K=50 and Eq. (4), for which the linearized-CPE phase-filter fingerprint remains, and it is even sharper since in this latter case comes from analytical formula manipulations. Indeed, recall that the actual experimental spectral density is evaluated straightforwardly after the CPE, thus it suffers from periodogram averaging window smoothing effects, which are more observable in the central notch.

Moreover, in Fig. 8(a), we present the bell-curves in the nonlinear regime. We report the actual experimental and the estimated performance for varying the half-window number of taps among K=3, 5, 10, and 50. We observe that the performance chances changing the taps number, eventually producing a performance gap between 3 and 50 taps of about 1.5 dB at 20 dBm. Moreover, we see that the estimated performance, thanks to the quadrature component filtering described in Eq. (4), adjusts with the number of taps according to the actual system behavior, capping the total SNR error, $\Delta SNR$ below 0.5 dB. In Fig. 8(a), markers indicate the actual performance, while solid curves represent the estimation. Different colors categorize different equalization taps number.

 

Fig. 8. On the left-hand side, the experimental and estimated performance curves for the LLSS link for different number of phase-estimation half-window averaging taps, K (a). On the right-hand side, the estimation error at 20 dBm in terms of the SNR difference between estimated and real ΔSNR for the LLSS link with filtering and without filtering the quadrature component (b). At the bottom, the model estimation error for each investigated number of taps in terms of ΔSNR versus power (c). WDM: wavelength division multiplex; SNR: signal-to-noise ratio.

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Moreover, in Fig. 8(b), we outline the estimation error accounting for the linearized-CPE phase-filtering (red) or not-accounting (blue) at 20 dBm. We can observe that if we consider the filtering of the quadrature component according to Eq. (4), we cap the error below 0.5 dB of SNR difference, while, if we do not account for filtering, the error rises to 1 dB of misprediction. In a receiver, where equalization taps are set accordingly to the noise characteristics, without the knowledge provided by the spectral density, we would not be able to assess the performance impact of the CPE without having performed a profiling calibration for every possible receiver configuration.

Finally, in Fig. 8(c), we report the estimation error vs. WDM launch power for the proposed model relative to every investigated number of CPE averaging taps, i.e., the difference between the actual experimental SNR and the one predicted according to the filtered model. We observe that the model always overestimates the performance, and the error is bounded, in every investigated case, below 0.6 dB.

5.3 Considerations on the power spectral density knowledge

Before moving to the conclusions of this paper, we would like to stress a further remark on what the knowledge of the spectral density may imply in the context of system performance design.

Lately, various efforts have been carried out trying to characterize the temporal correlation of the NLI [5,21,22]. These works are of interest because it can be shown that the information of the NLI temporal characterization, i.e., the autocorrelation, would produce significant advantages when designing a system with an adaptive receiver [22,23]. In this paper, we examined a method to semi-analytically asses the spectral density of a given transmission. This model gives equivalent information to the models providing temporal correlations because of the one-to-one relationship between the time-correlation and the PSD. It has also been observed that the number of taps of the CPE, influences the performance, by basically filtering and counteracting the temporal correlation on the quadrature component.

The presented solution may be considered as a fast and reliable candidate to provide a spectral (end equivalently temporal) characterization of the received noise to drive adaptive receivers. To deliver an exhaustive assessment of this possibility, several further investigations should be performed, both on the accuracy of the model in different scenarios, the employment of different modulation formats, and the actual methodology complexity in terms of computation and calibration. However, all those investigations are left for further work.

6. Conclusions

In this paper, we extended our conference contribution [14] by presenting the founding concepts of the semi-analytical spectral density estimation technique extensively and the experimental results that, for the sake of space, was not possible to report in a three-page contribution [14]. The ensemble of the experimental results confirms once more the proposed estimation model and highlights the spectral features of the noise.

First, after specifying the ideas behind the semi-analytical spectral density estimator, we reported in detail the results of the single-span calibration in the case of a standard single-mode fiber and a large effective-area fiber. We confirmed that the nonlinear contribution is fiber dependent and it grows and whitens increasing the input pre-dispersion of the span. Indeed, at small pre-dispersions, the nonlinear contribution is colored. Second, we experimentally assessed and estimated the noise spectral density in heterogeneous four-span long transmissions, confirming that the noise contribution in a link which does not rapidly cumulate dispersion can be sensibly colored due to nonlinear effects. Finally, we detailed and confirmed that, by accounting for the noise spectral-densities, it is possible to include in the performance estimation process, through linear filtering, even nonlinear manipulations performed at the transceiver, for instance, the Viterbi-Viterbi carrier-phase estimator.