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A simplified closed form expression for the noise power due to four-wave mixing in coherent OFDM systems is derived. The proposed model is in very good agreement with the exact model. The derived analytical expressions can be used in performance evaluation of systems employing CO-OFDM with any number of subcarriers and/or as an integral part of physical layer aware routing algorithms.
© 2014 Optical Society of America
1. Introduction
Coherent optical frequency division multiplexing (CO-OFDM) demonstrated the potential of increasing spectral efficiency, making possible optical transmission with per channel data rates beyond 400 Gb/s for future high-speed networks [1, 2]. In such systems, no dispersion compensation mechanisms are implemented whilst the subcarriers are very densely packed in order to maximize spectral efficiency. As a result, there is significant FWM noise [3–5] which in densely spaced CO-OFDM systems dominates over SPM and XPM.
In this paper, we provide a closed-form approximate expression for the noise power of FWM in OFDM systems the exact solution of which is given in [6]. For a large number of subcarriers the computational time of the model in [6] is significant, making it cumbersome in performance calculations whilst it is useless in Routing and Spectral Assignment (RSA) algorithms that take into account physical layer performance [7]. In prior art, approximations have been reported [3, 4]. However, in [3] there is a lower limit in the number of symbols/s or in the number of subcarriers for a given subcarrier spacing for which the model can be used whilst in [4] fiber dispersion and subcarrier spacing are not taken into account. The main advantages of our analytical model are: 1) there is no lower limit in the number of symbols/s used 2) the expression for the nonlinear noise holds for an arbitrary subcarrier spacing, 3) the closed form expression derived in this work allows to generalise results for the behavior of FWM and ease its introduction to planning/design tools.
2. Simplified FWM model derivation
The system under investigation is the same as in [3] which consists of a single polarization system with many wavelength channels, each of them OFDM modulated. This system can be evaluated as a huge single band OFDM channel provided that the bandwidth of each wavelength channel is much larger than the guard band between them. From [6], and following the same approach as [3] which is generally valid for the central subcarrier, the total noise power due to the FWM in a subcarrier i , is:
with the number of subcarriers, the power of subcarrier x, the degeneracy factor (equal to 6 for non-degenerate products and 3 for degenerate products), the nonlinear coefficient of the fiber, the effective core area of the fiber, D the local dispersion parameter, the effective length of the fiber, L the span length, the subcarrier spacing and the number of spans. Equation (1) for the worst case subcarrier frequency (which is the central one with i = 0) and , is simplified to:Equation (2) is obtained as follows. The phased array term due to the interference between the FWM products in spans is given asThe constructive interferences depend on the termsuch as Eq. (3) equals when Eq. (4) equals to . In this case principal maxima occur, producing strong contributions to the total FWM power. Moreover, for a typical OFDM system the majority of the produced FWM terms is around the first maximum. An example is shown in Fig. 1 where the statistical distribution of all of the produced FWM terms for 256 subcarriers is plotted against Eq. (4) for and . As it is obvious the main FWM power is produced by subcarriers that constructively interfere, with their interference located around the first maximum. In parallel, the termshows us how strong the interference between the subcarriers in a single span is and gets its maximum value which is 1 around the central maximum while around the kth principal maximum its value is much lower and equals . For example, in a system with and the value of the second principal maximum located in is 0.35 and around the third principal maximum located in is 0.118. This shows that the interference effect is much weaker away from the central principal maximum so it can be ignored. Figure 2 shows the FWM efficiency which is the product between the FWM coefficient for a single span calculated in Eq. (5) and the FWM coefficient between spans calculated in Eq. (3). The number of spans is 10. As a result, by taking into account the two facts that 1) the occurrence of the FWM terms is much higher around the central principal maximum and 2) the constructive interference is much stronger around the central principal maximum we can omit all other principal maxima by using the Taylor approximation forfor Eq. (4) around 0. As a result Eq. (3) can be simplified toFurthermore, the termcan be omitted for spans longer than 30 km.
Fig. 1 Statistical distribution of produced FWM terms for 256 subcarriers, Δf = 100MHz and L = 100 km.
Since in the previous step all principal maxima except the central have been eliminated we can also eliminate the subsidiary maxima located in positions where Eq. (4) equals. This elimination is justified by the fact that the strength of a FWM term located in the first and the second subsidiary maximum is less than 0.055 and 0.03 respectively of the strength of the central principal maximum for a system with more than 5 spans. So we can approximate the function with a Gaussian function of the form (for all combinations)
Variable takes values around 3 for good matching. By expanding Eq. (9) into Taylor series, we getThe next step is the conversion from summation to integration of Eq. (10) and the mathematical solution of the double integral. This conversion can be justified that as we can approximate integration with the rectangle rule, we can also approximate a sum using integration under similar conditions. In our approximation we assume , so we calculate the nonlinear noise for the central subcarrier which is the worst case subcarrier. We set a new variable and Eq. (10) becomesWe now convert the double summation to integration and by dividing with we write Eq. (11) in a more convenient form:By using the reduction formulaAnd knowing thatEquation (12) is integrated intoWe can further approximate Eq. (15) by taking the fact that and by removing k from the terms and with maximum absolute error of 0.63 dB within the range 1-20 spans, 100-800 MHz with step 100MHz and 16-1024 subcarriers. So we haveThe conversion from discrete summation of Eq. (11) to integration leading to Eq. (16) is not valid for the full range of values of k and system parameters. That is because after integration the peak values of the function inside the integral of Eq. (16) against k are not calculated as in discrete summation, resulting in significantly lower noise especially in higher number of subcarriers and large subcarrier spacing. This can be shown in Fig. 3 where inner sum of Eq. (11) and the function inside the integral of Eq. (16) are plotted against various values of k for spans, , , and km. For higher values of k the peak values of discrete summation of Eq. (11) converge to .To deal with this we calculate the limit when the following equation is lower than 1:Fig. 3 The red curve shows the inner sum of Eq. (11) and the green curve shows the function inside the integral of Eq. (16), both plotted for various values of k.
Firstly, we convert the function of Eq. (17) to complex function since and we get
Next we split the real from the imaginary part of the complex numbers and since we have converges to π for large x which is the range when Eq. (17) approaches 1, so we can writeThe first term of Eq. (20) is much smaller than the second and as a result so . Equation (16) can be rewritten asThe next step is the integration of Eq. (21). This integral can be solved analytically using Eq. (14) and the following equation:where function denotes the polylogarithm function with s = 2: and y a product of variables.As a result Eq. (21) is solved into
Knowing that Eq. (23) is simplified to: for converges to and for converges to x while converges to for and to x for . Since for all cases we can writeAs a result, the nonlinear noise for the worst case subcarrier or the central subcarrier with i = 0 is simplified to Eq. (2). The performance of the system is dominated by the central subcarriers giving us an upper bound for the expected BER. The approach of the central subcarrier is considered acceptable since only the few outer subcarriers have significantly improved performance.3. Results and discussions
The accuracy between the approximated model of Eq. (2) and the model in [3] against the model of Inoue in Eq. (1) can be shown in Fig. 4 where the absolute approximation error in dB is plotted against various values of subcarrier spacing and number of subcarriers. This figure depicts the maximum absolute error within the range of 1-20 spans with length of 100 km and. We can see from Fig. 4 that this error between 1 and 20 spans is less than 1.75 dB for total bandwidth and less than 1.1 dB for while exceeds 2 dB in higher system bandwidths. This happens because as the subcarrier spacing increases, the number of the produced FWM terms located in the second and third principal maxima is higher so they cannot be ignored. On the other hand, the accuracy of our model is much better than the well known model in [3] the inaccuracy of which within the range of 1-20 spans exceeds 2 dB for large subcarrier spacing while showing good agreement for subcarrier spacing up to 200 MHz in all cases. This inaccuracy is expected and justified by the fact that in model [3] in order to solve the double integral analytically the “dense subcarrier” assumption [3, Eq. (11)-(12)] is made. Practically, this assumption holds when the subcarrier spacing is lower than 200 MHz making the model unsuitable for larger subcarrier spacing. As a result, Eq. (2) is a fast and reliable formula for calculation of the total noise power due to FWM in coherent OFDM systems for a wide range of system parameters.
Fig. 4 Comparison of the Absolute Approximation Error (dB) of Eq. (2) and model in [3] against Eq. (1) for various number of subcarriers.
Finally, the validity of the approximation of Eq. (2) is tested for systems with fractional spans (with spans longer than 30 km) by averaging the span lengths and maintaining the number of spans:
In Fig. 5 we can see the comparison between [6, Eq. (10)] and our approximation in a system with 10 spans with variable lengths which in the current example was Ln = 3 spans of 40 km, 3 spans of 80 km and 4 spans of 100 km. As it is evident, a maximum deviation of 1.25 dB is observed which is acceptable. Furthermore, the results indicate that the number of spans is by far more important than the span length for FWM noise generation as expected since non-linear effects occur mainly about over the first twenty kilometres. After that length, the optical power is very low (due to fiber losses) for the non-linear effects to occur.Fig. 5 Comparison of our model against Inoue’s model with fractional spans [6, Eq. (10)] for 10 spans and 100 MHz and 200 MHz subcarrier spacing.
4. Conclusion
An analytical approximate formula for calculating the nonlinear noise in CO-OFDM systems has been derived. This formula is valid regardless of the number of subcarriers and has been benchmarked with the results produced from the exact solution against sub-carrier frequency spacing, number of spans and span length for the central (worst) subcarrier in the spectrum. This model gives us the advantage of the fast calculation of the nonlinear noise providing a maximum deviation of 1.75 dB from the exact model for total bandwidth of 100 GHz and less than 1.1 dB for total bandwidth of 50 GHz while showing lower inaccuracy than previous approximated models.
Acknowledgments
We would like to thank Dr. A. Hadjifotiou for valuable discussions and suggestions. This research was funded by the Operational Program “Education and Lifelong Learning” of the Greek National Strategic Reference Framework (NSRF) Research Funding Program: THALES PROTOMI, grant number MIS 377322.
References and links
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