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Abstract
In long-haul fiber-optic networks, precise modeling of physical-layer impairments (PLIs) is crucial to optimizing network resource usage while ensuring adequate transmission quality. In order to accurately estimate PLIs, many mathematical models have been proposed. Among them, the so-called Gaussian noise (GN) model is one of the most accurate and simple enough to use on complex continental-size networks. However, the closed-form GN model assumes that the signals can be represented as having rectangular spectra, leading to a significant estimation error in typical cases when this assumption is violated. We propose the component-wise Gaussian noise (CWGN) PLI model that can account for arbitrary spectral-shaped demands. The CWGN model is computationally simple and suitable for most network management approaches. Results indicate that the CWGN model can prevent as much as a 136% overestimation of the PLIs resulting from the closed-form GN model applied to network lightpaths containing cascaded filters.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Elastic optical networks (EONs) have been proposed as a solution to meet the growing demand for long-haul communications. They can efficiently accommodate heterogeneous traffic as well as provide reliable data transmission [1–3]. EONs must be designed so that physical-layer impairments (PLIs) consisting of accumulated noise and nonlinear fiber interference (NLI) do not unduly harm the quality of transmission (QoT) of signals [4,5]. PLIs limit the transmission reach of the signal [6,7], forcing the network to employ reconfigurable optical add-drop multiplexers (ROADMs) as signal regenerators. Estimating the extent of these impairments is difficult due to the dynamic and heterogeneous nature of EONs, yet essential for efficient network planning and configuration. PLI estimation must be both fast and accurate if it is to be used effectively within an EON dynamic resource allocation algorithm.
Many PLI estimation models have been proposed [8–10]. They are often either overly simplistic worst-case estimates or are unsuitable for use in multiple-layer continental-scale networks because of their high complexity [11]. The most widely-applied PLI estimation model for fiber-optic networks is the transmission reach (TR) model [12,13], which estimates the longest signal transmission distance that satisfies the QoT in the worst-case noise and interference scenario. The TR model is simple and fast to compute but overly conservative and ignores the actual network state, leading to resource over-provisioning. At the other extreme, there has been significant research in developing highly accurate PLI estimates, such as reported in [14,15]. While these approaches yield precise estimates, they are too computationally slow to be used in the dynamic management of large networks.
In between these two extremes lies a scalable and accurate state-dependent PLI estimation model, named the Gaussian noise (GN) model [16–18]. Researchers further simplified the GN model resulting in the closed-form expression given in [19], which is suitable for estimating noise and interference levels in complex fiber-optic networks, such as EONs. It estimates the instantaneous impairment state, including the amplified spontaneous emission (ASE) noise caused by the amplification process of Erbium-doped fiber amplifiers (EDFAs), the self-interference (SCI) caused by fiber nonlinearity on the channel of interest, and the cross-channel interference (XCI) caused by signals transmitted on other channels on the same fiber links [4,19]. Extensions to this model to account for other impairments such as inter-channel stimulated Raman scattering have also been proposed [20,21]. The so-called enhanced GN (EGN) model that has gained recent prominence was developed to correct several of the GN model’s estimation errors [22]. [23] proposes the use of machine learning techniques to improve the model. In this paper, we restrict our attention to [19], which has been widely applied to cross-layer network resource provisioning [6,13,24–27].
In the model described in [19] and the extensions just described, each demand is assumed to have a rectangular spectrum, making it unsuitable for estimating the nonlinear interference for demands with arbitrary, non-rectangular spectra. However, signals in long-haul networks often have non-rectangular spectra, such as root-raised-cosine (RRC) [28]. Signals are transmitted and received with matched RRC filters to minimize inter-symbol interference (ISI) caused by the temporal overlap of adjacent symbols in the same signal, which impairs the QoT [28]. Even when the signal transmitted has a nearly rectangular spectrum, it will be distorted as it travels through the fiber due to nonlinear effects and band-pass filtering within wavelength selective switches (WSS). This is especially so for signals subjected to concatenated filters from repeated ROADM crossings [29], as shown through examples in Fig. 1. Cascaded filters can narrow the spectrum, as seen from demands $A$ and $C$ in the figure. In addition, the shape of the spectrum can no longer be assumed rectangular, and doing so can lead to errors. Therefore, developing a fast yet accurate PLI estimate for arbitrary spectral-shaped demands is of significant interest. Note that a cascade of five WSS filters is similar in shape to an RRC signal spectrum with a roll-off factor (RoF) of about 0.9. Therefore, without loss of generality, we restrict our numerical results in this paper to signals with RRC spectra.
Fig. 1. Illustration of the optical spectrum in a link. We assume all transmitted signals are 32 Gbaud RRC pulse-shaped with RoF = 0.2. The channel of interest is centered at 50 GHz. Interfering demand $A$ centered at 125 GHz is shown going through five cascaded 37.5 GHz WSS filters. The filtered signal is shown using a dashed line and the filter itself using a solid line. Interfering demand $B$ centered at 200 GHz is shown going through one 37.5 GHz WSS filter. Interfering demand $C$ centered at 300 GHz is shown going through five cascaded 50 GHz WSS filters. All models and parameters are according to [29,30]
We propose a model that we call the component-wise Gaussian noise (CWGN) model able to calculate noise and interference for arbitrary spectral-shaped demands. The CWGN model decomposes the signal of interest into frequency components and then accumulates the SCI and XCI contributed by each component. It has a simple closed-form mathematical expression that can be easily computed. The complexity of the proposed model is low enough to be applied to mainstream network planning and resource deployment schemes. The CWGN model improves the PLI estimation accuracy, avoiding substantial resource over-provisioning compared to the closed-form GN model [19], which we show to be overly conservative in many cases. The CWGN model is general and comprehensive for any pulse-shaped demand, also achieving similar complexity compared to the closed-form GN model.
This paper is organized as follows. In Section 2, we introduce the dominant PLIs in fiber-optic networks. The GN model for NLI is introduced in Section 3. Section 4 then describes the proposed CWGN model. Section 5 presents our test system settings and numerical results. Finally, we conclude the paper in Section 6.
2. Physical-layer impairments
In long-haul transmission, the nonlinear Schrödinger equation, given by
All long-haul fiber systems use optical amplifiers to compensate for attenuation along the fiber; this paper focuses on lumped amplification using EDFAs. The fiber length between two EDFAs is called a fiber span, denoted as $L$, usually around 100 km. The transmitted power is attenuated by about 20 dB by the end of each span, given a fiber loss of about 0.2 dB/km [4]. The optical amplification process will add ASE noise, which is modeled as additive Gaussian noise with PSD per polarization in one span, $G_{ASE}$, given as [4]
where $\hbar$ is Planck’s constant, $n_{sp}$ is the spontaneous emission factor, and $\nu$ is the light frequency. We assume the gain of the EDFAs is frequency flat [33] and each EDFA exactly compensates for the span loss.The NLI effects for each span can be divided into self-interference effects, called SCI, and interference due to other co-propagating channels, called XCI:
$G_{NLI}$ represents the NLI PSD per span and per polarization; $G_{SCI}$ represents the SCI PSD, and $G_{XCI}$ represents the XCI PSD. The total PLI is computed by adding the ASE PSD in (2) to the NLI PSD in (3). In this paper, we model the PLI for one span and polarization; these can be accumulated over all spans to estimate the PLIs generated in each link for each polarization. For a particular channel of interest, the end-to-end light-path QoT can be approximated by adding the estimated PLIs over all links that comprise it. Note that other PLIs such as stimulated Raman scattering are not included in this paper and relegated to future work.3. Gaussian noise model of NLI
The GN model is often used to estimate the NLI PSD analytically and is valid based on several assumptions, as stated in [33]: the fiber links are dispersion uncompensated (i.e., the dispersion is purely compensated by digital signal processing); the signal PSD is the same for each polarization; the fiber loss is totally compensated; the NLI PSD is accumulated along the light-path independent from span to span; and the channels’ spectra are rectangular and non-overlapping. With the above assumptions, the GN model can be applied to accurately estimate the signal PLIs.
The SCI is caused by the channel itself, varying with the power and bandwidth of the channel of interest, denoted as channel $p$. The GN model approximates the PSD of the SCI as a constant given by
where $G_p$ represents the $p$th channel’s signal PSD, assumed to be constant over the entire signal bandwidth. The factor $F_{p,p}$ is a special case of the nonlinear efficiency factor $F_{p,q}$, given byThe XCI is caused by the interaction between channels. It depends on the difference in center frequencies and the bandwidths of neighboring channels that share the link with the channel of interest $p$. The GN model approximates the PSD of the XCI caused by channel $q$ sharing a link with channel $p$ as
utilizing $F_{p,q}$ from (5) when $p\neq q$. The XCI can be simplified and further approximated asThe GN model is often used to estimate the XCI for channels that do not have rectangular spectra. In these cases, $G_p$ and $G_q$ in (7) and (9) are replaced with the maximum of the actual PSD functions $G_p(f)$ and $G_q(f)$, respectively. The maximum of the PSD function is denoted as $G^{\max }_q = \max _{f} G_q(f)$.
4. Component-wise Gaussian noise model
The CWGN estimates the LPI experienced by an arbitrary spectral-shaped signal of interest caused by itself and other signals that also have arbitrary spectra, given the signal and fiber parameters. In this section, we derive the analytical expressions for the CWGN model of the expected interference. As in [11], we assume that PLIs consist of additive ASE noise and interference (also assumed independent), accumulating incoherently over all spans on the transparent fiber segment. The ASE noise only depends on the transmission length and not the shape of the signal spectrum, and thus we focus on estimating the NLI for arbitrary spectral-shaped demands. Without loss of generality and for notation simplicity, we assume the signal of interest is centered at frequency 0.
For demands with non-rectangular spectra (either transmitted as such or distorted through network propagation), the XCI is challenging to describe analytically since the closed-form GN model [19, Eq. (16)] is designed for demands with rectangular (flat) spectra.
To account for signals with an arbitrary spectral shape, we adopt the idea and theory proposed in [11] to decompose the spectrum along the frequency axis into rectangular components, as a Riemann sum, and then accumulate the overall XCI over these components. Using (9), we compute the XCI caused by channel $q$ assuming $\Delta _q$ is composed of frequency differentials $df$, where $\sum df=\Delta _q$. The contribution of each differential component to the XCI depends on the frequency difference between the channel of interest and the interfering signal component. Without violating the assumptions in [19, Eq. (16)], the PSD of the XCI contributed by signal $q$ to the channel of interest $p$ (located at frequency 0) equals the sum over the frequency differentials, written as
We thus conclude that the XCI can be written as a Riemann sum using the GN model for each differential term. When $df\rightarrow 0$, the discrete sum becomes an integral. Using the fact that when $dx \rightarrow 0$, $\ln \left (\frac {x+dx/2}{x-dx/2}\right ) \approx \frac {1}{x} \, dx,$ the XCI PSD in (10) can be further simplified as
The integral above can be directly computed or closely approximated in closed-form for many typical signal spectra.
To derive the SCI for a typical spectral-shaped channel of interest, the proposed CWGN divides the spectrum into three parts that are treated in different ways: the center-band (labeled $C$ in Fig. 1), and the two side-bands that describe the edges of the spectrum that are often significantly impacted by optical filtering (labeled $S$ in Fig. 1). The SCI contributed by the side-bands, which are typically far from the center frequency of the channel of interest, is estimated using an equation similar to the XCI above, even though they are a part of the channel of interest. The center-band with bandwidth $\Delta ^{C}_p$ is assumed to have a sufficiently flat spectrum to be modeled as a rectangle, contributing to the SCI according to (7). The SCI is thus written as the summation
According to [19], the maximum frequency in the spectrum of the signal must be at least 14 GHz away from the center frequency in order for the SCI to be accurately approximated using (7). Thus, the side-bands are assumed to start at 14 GHz, as shown in Fig. 1, i.e., $\Delta _p^{C}$ is chosen to be its minimal value, 28 GHz.
5. Numerical results
In this section, we estimate link-level PLIs for signals transmitted through a single-mode fiber with parameters listed in Table 1, unless noted otherwise. While the CWGN model can be applied to any spectral-shaped signal, we focus our numerical results on signals that use an RRC spectrum.
Table 1. System Parameters [4]
The proposed CWGN model utilizes information about the signal pulse-shaping to provide a more accurate interference estimate. We assume that the PLI estimator knows the center frequency, baud rate, and roll-off factor (RoF) used by both the signal of interest and the interfering signals.
We validate the proposed CWGN model by comparing it with the result of the double-integral GN model [16, Eq. (18)], which is used in this paper as the benchmark and is assumed to be an accurate model of the nonlinear interference [4,34]. The CWGN model is shown to yield almost identical results as the benchmark but has a much lower computational cost because of its closed-form expressions compared to the double-integral needed to compute the GN model in [16, Eq. (18)] for a signal with an arbitrary spectrum [31]. The estimation errors shown are defined as a percentage error, given by
Three versions of the closed-form GN model are compared depending on the assumptions governing the assumed rectangular spectrum. When the assumed rectangular spectrum has the same total bandwidth and peak power as the signal, we denote it as “GN w/ RRC BW Peak”; these are standard assumptions for the GN model for unknown interference. When the model uses the full bandwidth yet the same average power as the actual signal, we denote it as “GN w/ RRC BW Ave”. When it uses the RCC signal baud rate and peak power, we denote it as “GN w/ RRC Baud-rate”. BW in this paper is defined as the null-to-null bandwidth.
5.1 Physical-layer impairment estimation
In this section, we begin by showing the relative time taken to compute the proposed model compared with the benchmark. We then evaluate and compare the SCI and XCI estimates for the transmission of RRC signals with various baud rates ranging from 30 to 400 Gbaud, which are representative scenarios for current and future fiber-optic communications. The RoF is varied from 0 to 0.9 to simulate the effects of severe optical filtering.
Comparisons between the computational time required for the proposed CWGN model and the benchmark estimation models are listed in Table 2. In the table, a commonly-used long-haul optical backbone network, the NSF24 network, is simulated with 23$\times$24 demands (i.e., each node transmits to all other nodes). Each demand is an RRC signal with RoF = 0.1 with spectrum uniformly distributed from 30 to 200 GBaud. According to our previous results [13], the average number of hops per route is 4.9, and the average number of demands sharing the same link is 5.3, assuming a first-fit spectrum assignment and shortest path routing. The proposed CWGN model can calculate all demands’ PLIs within 3 seconds, similar to the GN model. However, for sophisticated PLI estimators requiring double integrals (such as [16, Eq. (18)]), it takes more than 3 hours to calculate all PLIs. Modern fiber-optic networks configure network resources dynamically and must predict the QoT in real-time. The sophisticated models are much too slow to be used for PLI estimation in long-haul fiber-optic networks.
Table 2. Running Time Comparisona
Figure 2 shows the SCI estimation error for the various GN models compared to the proposed CWGN model. From Fig. 2 (a), when the RoF is zero for a 100 Gbaud signal, all GN models and the CWGN model are identical since the RRC spectrum becomes a rectangle and the SCI is well-modeled by (7) for this large bandwidth signal. As the RoF increases, the normalized estimation error of the CWGN remains near zero, yet the error of the closed-form GN model using the full signal bandwidth increases up to $66\%$ at an RoF of 0.9. The GN model using the baud rate instead of the bandwidth to establish the extent of the spectrum has lower error but still overestimates the SCI by over 20% in this extreme case. The GN model using the average power of the RRC signal instead of the peak power significantly underestimates the SCI and should not be used.
Fig. 2. Normalized SCI estimation error for the CWGN model and the various versions of the closed-form GN model; (a) as a function of the RoF for a signal with a baud rate of 100 Gbaud, and (b) as a function of the symbol rate for signals with RoF of 0.3.
An optically filtered signal often has a narrowed spectrum compared with the signal transmitted, as for interfering demand $A$ illustrated in Fig. 1 [29]. If we estimate the SCI of demand $A$, according to “GN w/ RRC BW Peak” using the bandwidth of the filtered signal and its peak power, using the closed-form GN model leads to $51\%$ over-estimation; the “GN w/ RRC BW Ave” approach leads to $56\%$ under-estimation; and the “GN w/ RRC Baud-rate” approach leads to $24\%$ over-estimation similar to an RRC signal with an RoF = 0.8.
Figure 2 (b) shows the estimation error as the bandwidth of the channel of interest increases. The overall estimation error between the proposed CWGN model and the double-integral GN model is less than $1\%$ and is only noticeable for small bandwidths when the SCI for the center band cannot be accurately modeled using (7). In contrast, all versions of the GN model result in notable estimation error, even at high baud rates.
In Fig. 3, we show the XCI estimation error for the proposed CWGN model and the various GN models for several values of the bandwidth of the channel of interest and RoFs, as the baud rate of the interfering channel increases. The channel of interest is neighbored by one interfering signal separated by a guard-band of 12.5 GHz. The CWGN model’s estimate of the XCI has been verified to have less than 1% estimation error compared with the double-integral GN model in [16, Eq. (18)] for all cases shown. As seen in the figure, “GN w/RRC BW peak” severely over-estimates the XCI, up to 39% for an RoF of 0.2, a channel of interest bandwidth of 50 GHz, and an interfering signal bandwidth of 400 GHz. “GN w/RRC BW Ave” underestimates the XCI, up to $20\%$ for the same RoF and an interfering channel bandwidth of 50 GHz. Increasing the RoF strongly increases the estimation-error for “GN w/RRC BW Ave” and “GN w/RRC BW peak” because it makes the transmitted spectrum less rectangular; unsurprisingly, the XCI estimation error is zero when the RoF is zero. The bandwidth of the interfering demand has less impact on the estimation error (the curves are almost flat in each sub-figure) because both the absolute estimation error and the actual XCI increase as the bandwidth of the interfering channel increases. The bandwidth of the channel of interest, differentiated by rows, only slightly impacts the estimation error as well. “GN w/RRC baud rate” has less than 3% XCI estimation error for the low RoFs shown; however, performing this calculation requires information about the baud rate of the transmitted demand. Once this is known, one may as well compute the CWGN and reduce the error to zero.
Fig. 3. XCI estimation error for the closed-form GN model compared to the CWGN model for demands with an RRC spectrum as a function of the RoF, the bandwidth of the interfering channel, and the bandwidth of the channel of interest.
We also calculated the XCI caused by interfering demand $A$ of Fig. 1 showing the effects of cascaded filtering, assuming the interfering signal is 75 GHz away from the channel of interest (center frequency to center frequency). The “GN w/ RRC BW Peak” approach leads to $136\%$ XCI over-estimation; “GN w/ RRC BW Ave” leads to $29\%$ under-estimation; and “GN w/ RRC Baud-rate” leads to $90\%$ over-estimation. Even the better model, “GN w/ RRC Baud-rate”, leads to severe over-estimation since the actual filtered demand is considerably narrower than expected and its spectral shape is quite different from the assumed rectangle. Calculating the closed-form GN model using the peak power and 3 dB bandwidth of the filtered signal instead of the baud rate still leads to 32% over-estimation.
Since LPI estimation errors accumulate along the fiber during transmission through the network, the over-estimation error can become significant and then cost tremendous network resources due to over-provisioning, especially for continental-size networks. As XCI estimation errors increase with the number of channels sharing each link, the problem will exacerbate as the network load increases, which is inevitable in the near future, leading to significant wasted network devices and resources.
6. Conclusions
In conclusion, the proposed CWGN model accurately estimates the PLIs for arbitrary spectral-shaped demands. It has a simple mathematical expression that can be easily computed for use in any fiber network management system. Compared to the closed-form GN model, the CWGN model avoids significant PLI over-estimation with similar computational complexity. Depending on the assumptions made, the CWGN can provide accurate PLI estimates when the closed-form GN model might overestimate them by as much as 136%, such as when the signal traverses cascaded filters. Network-level simulations and applications based on the CWGN model will be investigated in future research.
Funding
National Science Foundation (CNS-1718130).
Acknowledgments
Portions of this work were presented at CLEO 2022.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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