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S. Chimmalgi
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Riemann theta functions play a crucial role in the field of nonlinear Fourier analysis, where they are used to realize inverse nonlinear Fourier transforms for periodic signals. The practical applicability of this approach has however been limited since Riemann theta functions are multi-dimensional Fourier series whose computation suffers from the curse of dimensionality. In this paper, we investigate several new approaches to compute Riemann theta functions with the goal of unlocking their practical potential. Our first contributions are novel theoretical lower and upper bounds on the series truncation error. These bounds allow us to rule out several of the existing approaches for the high-dimension regime. We then propose to consider low-rank tensor and hyperbolic cross based techniques. We first examine a tensor-train based algorithm which utilizes the popular scaling and squaring approach. We show theoretically that this approach cannot break the curse of dimensionality. Finally, we investigate two other tensor-train based methods numerically and compare them to hyperbolic cross based methods. Using finite-genus solutions of the Korteweg–de Vries (KdV) and nonlinear Schrödinger equation (NLS) equations, we demonstrate the accuracy of the proposed algorithms. The tensor-train based algorithms are shown to work well for low genus solutions with real arguments but are limited by memory for higher genera. The hyperbolic cross based algorithm also achieves high accuracy for low genus solutions. Its novelty is the ability to feasibly compute moderately accurate solutions (a relative error of magnitude 0.01) for high dimensions (up to 60). It therefore enables the computation of complex inverse nonlinear Fourier transforms that were so far out of reach.
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Riemann theta functions play a crucial role in the field of nonlinear Fourier analysis, where they are used to realize inverse nonlinear Fourier transforms for periodic signals. The practical applicability of this approach has however been limited since Riemann theta functions are multi-dimensional Fourier series whose computation suffers from the curse of dimensionality. In this paper, we investigate several new approaches to compute Riemann theta functions with the goal of unlocking their practical potential. Our first contributions are novel theoretical lower and upper bounds on the series truncation error. These bounds allow us to rule out several of the existing approaches for the high-dimension regime. We then propose to consider low-rank tensor and hyperbolic cross based techniques. We first examine a tensor-train based algorithm which utilizes the popular scaling and squaring approach. We show theoretically that this approach cannot break the curse of dimensionality. Finally, we investigate two other tensor-train based methods numerically and compare them to hyperbolic cross based methods. Using finite-genus solutions of the Korteweg–de Vries (KdV) and nonlinear Schrödinger equation (NLS) equations, we demonstrate the accuracy of the proposed algorithms. The tensor-train based algorithms are shown to work well for low genus solutions with real arguments but are limited by memory for higher genera. The hyperbolic cross based algorithm also achieves high accuracy for low genus solutions. Its novelty is the ability to feasibly compute moderately accurate solutions (a relative error of magnitude 0.01) for high dimensions (up to 60). It therefore enables the computation of complex inverse nonlinear Fourier transforms that were so far out of reach.
Fiber-Optic Communications Using Nonlinear Fourier Transforms
Algorithms and a Bound
Due to the ever increasing global connectivity, the demand on the fiber-optic communication infrastructure is projected to keep increasing rapidly. A major factor currently limiting transmission capacity is the fiber nonlinearity. Some researchers have suggested the application of nonlinear Fourier transforms to exploit the fiber nonlinearity rather than ignoring or mitigating it. Nonlinear Fourier transforms allow us to solve certain nonlinear partial differential equations by transforming the complex evolution of the solution in the time-domain to a simple multiplication with a nonlinear frequency response in the nonlinear Fourier domain. This method is analogous to solving linear partial differential equations using the Fourier transform. The nonlinear Schrödinger equation is a suitable model for the propagation of light through a single-mode optical fiber. Its lossless version is solvable through a nonlinear Fourier transform. In recent years, several nonlinear Fourier transform based communication systems have been proposed. Such systems require numerical algorithms to compute the nonlinear Fourier transforms as nonlinear Fourier spectra are known analytically for only a handful of signals, and linear superposition cannot be used to compute the spectrum of a more complex signal. Computationally efficient algorithms are therefore not only essential for the real-time operation of nonlinear Fourier transform based communication systems, but are also important for their simulation. One common way to improve the spectral efficiency of a communication system is to increase the signal power in order to reduce the impact of noise. Another is to increase the signal duration in order to reduce the impact of information-free guard intervals that are inserted between transmissions to deal with the channel memory. Longer signals however require more resources to process them. The numerical problem of computing nonlinear Fourier transforms furthermore gets harder for both higher power and longer durations. Hence in the literature, we observe that the inability to perform efficient communication in these regimes is typically attributed to numerical problems of existing algorithms. In this dissertation we develop new algorithms that require shorter computation times for achieving similar accuracies as existing algorithms. Furthermore, we theoretically investigate whether some of the problems that are commonly attributed to numerical difficulties could occur in the absence of numerical effects.
The nonlinear Fourier transform for signals that decay sufficiently fast is currently the most commonly used transform in nonlinear Fourier transform based communication systems. We developed new algorithms for computing the continuous nonlinear Fourier spectrum which is one part of the nonlinear Fourier spectrum for decaying signals. We demonstrated significant improvements over existing algorithms in multiple numerical benchmarks, and implemented the algorithms in the open source software library FNFT. We also developed NFDMLab, which is a Python based open source simulation environment for nonlinear Fourier transform based communication systems that relies on FNFT. The developed forward nonlinear Fourier transform algorithms are fast higher-order methods with a complexity of O(D log2D) for computing the continuous nonlinear Fourier spectrum from D samples of a decaying signal. In the numerical benchmarks, we introduced the trade-off between accuracy and computation time as a new way to compare nonlinear Fourier transform algorithms and found that the newly proposed algorithms perform significantly better than prior work in this regard. We also provided the first counting analysis of a fast nonlinear Fourier transform algorithm.
There is also interest in using the nonlinear Fourier transform for periodic signals, as it is closer to the method used in conventional orthogonal frequency division multiplexing communication systems. The definition of the nonlinear Fourier transform for periodic signals is different from that of decaying signals. Communication systems based on nonlinear Fourier transforms for periodic signals make use of so-called finite-genus solutions of the nonlinear Schrödinger equation. Riemann theta functions are the traditional way to realize inverse nonlinear Fourier transforms that are used to synthesize finite-genus solutions. They are multi-dimensional Fourier series and their numerical computation suffers from the curse of dimensionality. This limits the genus of the signals used in the communication systems and is seen as a major bottleneck. We derived new bounds on the series truncation error and proposed two tensor-train based and a hyperbolic cross index set based algorithms for computing high-dimensional Riemann theta functions. We compared them to existing algorithms in multiple numerical benchmarks. The bounds that we derived on the truncation error of the Riemann theta functions allowed us to rule out several of the existing approaches for the high-dimension regime. We demonstrated that the algorithm based on the hyperbolic cross can compute Riemann theta functions upto 60 dimensions with moderate accuracy which is significantly higher than what was previously feasible.
We also tried to improve the performance of nonlinear Fourier transform based communication systems known as b-modulators in the highly nonlinear regime using improved numerical algorithms. When we did not see improvements, we conducted a theoretical analysis of b-modulation systems. The analysis allowed us to prove theoretically that nonlinear bandwidth, signal duration and power are coupled when singularities in the nonlinear spectrum are avoided. When the nonlinear bandwidth is fixed, the coupling results in an upper bound on the transmit power. The power bound decreases with increasing signal duration which consequently decreases the signal-to-noise ratios for long signals, which explains the observed performance degradation in this regime without resorting to numerical difficulties. This result is the first of its kind as such a behaviour is not known from conventional linear systems. We also demonstrated numerically that the transmit powers achieved by an exemplary b-modulated system are close to its theoretical limits.
Fiber-optic communication systems based on nonlinear Fourier transforms have been proposed to potentially tackle fiber nonlinearity, which is a major factor currently limiting transmission capacity. Efficient numerical algorithms are essential for real-time operation as well as efficient simulations of nonlinear Fourier transform based fiber-optic communication systems. The algorithms presented in this dissertation potentially make already published nonlinear Fourier transform based communication systems more practical and also allow for development of new designs which were previously infeasible. In this dissertation furthermore a limitation on communication system design imposed by the structure of the nonlinear Fourier transform was identified. It can be used to explain the inability to perform efficient communication with long duration signals, which was previously attributed to numerical problems, and guide the design of future systems.
...
Due to the ever increasing global connectivity, the demand on the fiber-optic communication infrastructure is projected to keep increasing rapidly. A major factor currently limiting transmission capacity is the fiber nonlinearity. Some researchers have suggested the application of nonlinear Fourier transforms to exploit the fiber nonlinearity rather than ignoring or mitigating it. Nonlinear Fourier transforms allow us to solve certain nonlinear partial differential equations by transforming the complex evolution of the solution in the time-domain to a simple multiplication with a nonlinear frequency response in the nonlinear Fourier domain. This method is analogous to solving linear partial differential equations using the Fourier transform. The nonlinear Schrödinger equation is a suitable model for the propagation of light through a single-mode optical fiber. Its lossless version is solvable through a nonlinear Fourier transform. In recent years, several nonlinear Fourier transform based communication systems have been proposed. Such systems require numerical algorithms to compute the nonlinear Fourier transforms as nonlinear Fourier spectra are known analytically for only a handful of signals, and linear superposition cannot be used to compute the spectrum of a more complex signal. Computationally efficient algorithms are therefore not only essential for the real-time operation of nonlinear Fourier transform based communication systems, but are also important for their simulation. One common way to improve the spectral efficiency of a communication system is to increase the signal power in order to reduce the impact of noise. Another is to increase the signal duration in order to reduce the impact of information-free guard intervals that are inserted between transmissions to deal with the channel memory. Longer signals however require more resources to process them. The numerical problem of computing nonlinear Fourier transforms furthermore gets harder for both higher power and longer durations. Hence in the literature, we observe that the inability to perform efficient communication in these regimes is typically attributed to numerical problems of existing algorithms. In this dissertation we develop new algorithms that require shorter computation times for achieving similar accuracies as existing algorithms. Furthermore, we theoretically investigate whether some of the problems that are commonly attributed to numerical difficulties could occur in the absence of numerical effects.
The nonlinear Fourier transform for signals that decay sufficiently fast is currently the most commonly used transform in nonlinear Fourier transform based communication systems. We developed new algorithms for computing the continuous nonlinear Fourier spectrum which is one part of the nonlinear Fourier spectrum for decaying signals. We demonstrated significant improvements over existing algorithms in multiple numerical benchmarks, and implemented the algorithms in the open source software library FNFT. We also developed NFDMLab, which is a Python based open source simulation environment for nonlinear Fourier transform based communication systems that relies on FNFT. The developed forward nonlinear Fourier transform algorithms are fast higher-order methods with a complexity of O(D log2D) for computing the continuous nonlinear Fourier spectrum from D samples of a decaying signal. In the numerical benchmarks, we introduced the trade-off between accuracy and computation time as a new way to compare nonlinear Fourier transform algorithms and found that the newly proposed algorithms perform significantly better than prior work in this regard. We also provided the first counting analysis of a fast nonlinear Fourier transform algorithm.
There is also interest in using the nonlinear Fourier transform for periodic signals, as it is closer to the method used in conventional orthogonal frequency division multiplexing communication systems. The definition of the nonlinear Fourier transform for periodic signals is different from that of decaying signals. Communication systems based on nonlinear Fourier transforms for periodic signals make use of so-called finite-genus solutions of the nonlinear Schrödinger equation. Riemann theta functions are the traditional way to realize inverse nonlinear Fourier transforms that are used to synthesize finite-genus solutions. They are multi-dimensional Fourier series and their numerical computation suffers from the curse of dimensionality. This limits the genus of the signals used in the communication systems and is seen as a major bottleneck. We derived new bounds on the series truncation error and proposed two tensor-train based and a hyperbolic cross index set based algorithms for computing high-dimensional Riemann theta functions. We compared them to existing algorithms in multiple numerical benchmarks. The bounds that we derived on the truncation error of the Riemann theta functions allowed us to rule out several of the existing approaches for the high-dimension regime. We demonstrated that the algorithm based on the hyperbolic cross can compute Riemann theta functions upto 60 dimensions with moderate accuracy which is significantly higher than what was previously feasible.
We also tried to improve the performance of nonlinear Fourier transform based communication systems known as b-modulators in the highly nonlinear regime using improved numerical algorithms. When we did not see improvements, we conducted a theoretical analysis of b-modulation systems. The analysis allowed us to prove theoretically that nonlinear bandwidth, signal duration and power are coupled when singularities in the nonlinear spectrum are avoided. When the nonlinear bandwidth is fixed, the coupling results in an upper bound on the transmit power. The power bound decreases with increasing signal duration which consequently decreases the signal-to-noise ratios for long signals, which explains the observed performance degradation in this regime without resorting to numerical difficulties. This result is the first of its kind as such a behaviour is not known from conventional linear systems. We also demonstrated numerically that the transmit powers achieved by an exemplary b-modulated system are close to its theoretical limits.
Fiber-optic communication systems based on nonlinear Fourier transforms have been proposed to potentially tackle fiber nonlinearity, which is a major factor currently limiting transmission capacity. Efficient numerical algorithms are essential for real-time operation as well as efficient simulations of nonlinear Fourier transform based fiber-optic communication systems. The algorithms presented in this dissertation potentially make already published nonlinear Fourier transform based communication systems more practical and also allow for development of new designs which were previously infeasible. In this dissertation furthermore a limitation on communication system design imposed by the structure of the nonlinear Fourier transform was identified. It can be used to explain the inability to perform efficient communication with long duration signals, which was previously attributed to numerical problems, and guide the design of future systems.
Nonlinear frequency division multiplexing (NFDM) techniques encode information in the so-called nonlinear spectrum which is obtained from the nonlinear Fourier transform (NFT) of a signal. NFDM techniques so far have been applied to the nonlinear Schrödinger equation (NLSE) that models signal propagation in a lossless fiber. Conventionally, the true lossy NLSE is approximated by a lossless NLSE using the path-average approach which makes the propagation model suitable for NFDM. The error of the path-average approximation depends strongly on signal power, bandwidth and the span length. It can degrade the performance of NFDM systems and imposes challenges on designing high data rate NFDM systems. Previously, we proposed the idea of using dispersion decreasing fiber (DDF) for NFDM systems. These DDFs can be modeled by a NLSE with varying-parameters that can be solved with a specialized NFT without approximation errors. We have shown in simulations that complete nonlinearity mitigation can be achieved in lossy fibers by designing an NFDM system with DDF if a properly adapted NFT is used. We reported performance gains by avoiding the aforementioned path-average error in an NFDM system by modulating the discrete part of the nonlinear spectrum. In this paper, we extend the proposed idea to the modulation of continuous spectrum. We compare the performance of NFDM systems designed with dispersion decreasing fiber to that of systems designed with a standard fiber with the path-average model. Next to the conventional path-average model, we furthermore compare the proposed system with an optimized path-average model in which amplifier locations can be adapted. We quantify the improvement in the performance of NFDM systems that use DDF through numerical simulations.
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Nonlinear frequency division multiplexing (NFDM) techniques encode information in the so-called nonlinear spectrum which is obtained from the nonlinear Fourier transform (NFT) of a signal. NFDM techniques so far have been applied to the nonlinear Schrödinger equation (NLSE) that models signal propagation in a lossless fiber. Conventionally, the true lossy NLSE is approximated by a lossless NLSE using the path-average approach which makes the propagation model suitable for NFDM. The error of the path-average approximation depends strongly on signal power, bandwidth and the span length. It can degrade the performance of NFDM systems and imposes challenges on designing high data rate NFDM systems. Previously, we proposed the idea of using dispersion decreasing fiber (DDF) for NFDM systems. These DDFs can be modeled by a NLSE with varying-parameters that can be solved with a specialized NFT without approximation errors. We have shown in simulations that complete nonlinearity mitigation can be achieved in lossy fibers by designing an NFDM system with DDF if a properly adapted NFT is used. We reported performance gains by avoiding the aforementioned path-average error in an NFDM system by modulating the discrete part of the nonlinear spectrum. In this paper, we extend the proposed idea to the modulation of continuous spectrum. We compare the performance of NFDM systems designed with dispersion decreasing fiber to that of systems designed with a standard fiber with the path-average model. Next to the conventional path-average model, we furthermore compare the proposed system with an optimized path-average model in which amplifier locations can be adapted. We quantify the improvement in the performance of NFDM systems that use DDF through numerical simulations.
The performance of various nonlinear frequency division multiplexed (NFDM) fiber-optic transmission systems has been observed to decrease with increasing signal duration. For a class of NFDM systems known as b-modulators, we show that the nonlinear bandwidth, signal duration, and power are coupled when singularities in the nonlinear spectrum are avoided. When the nonlinear bandwidth is fixed, the coupling results in an upper bound on the transmit power that decreases with increasing signal duration. Signal-to-noise ratios are consequently expected to decrease, which can help explain drops in performance observed in practice. Furthermore, we show that there is often a finite bound on the transmit power of b-modulators even if spectral singularities are allowed
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The performance of various nonlinear frequency division multiplexed (NFDM) fiber-optic transmission systems has been observed to decrease with increasing signal duration. For a class of NFDM systems known as b-modulators, we show that the nonlinear bandwidth, signal duration, and power are coupled when singularities in the nonlinear spectrum are avoided. When the nonlinear bandwidth is fixed, the coupling results in an upper bound on the transmit power that decreases with increasing signal duration. Signal-to-noise ratios are consequently expected to decrease, which can help explain drops in performance observed in practice. Furthermore, we show that there is often a finite bound on the transmit power of b-modulators even if spectral singularities are allowed
A numerical method for the generation of fiber inputs in nonlinear frequency division multiplexing (NFDM) systems based on b-modulation is provided. The method is parallelizable, does not suffer from error propagation, and converges exponentially.
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A numerical method for the generation of fiber inputs in nonlinear frequency division multiplexing (NFDM) systems based on b-modulation is provided. The method is parallelizable, does not suffer from error propagation, and converges exponentially.
NFDMLab
Simulating nonlinear frequency division multiplexing in Python
Fiber-optic transmission based on nonlinear frequency division multiplexing (NFDM) has received much attention in recent years. We introduce NFDMLab, an open source software package for simulating NFDM transmissions written in the Python language.
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Fiber-optic transmission based on nonlinear frequency division multiplexing (NFDM) has received much attention in recent years. We introduce NFDMLab, an open source software package for simulating NFDM transmissions written in the Python language.
The optimal transmit power in various nonlinear Fourier transform-based transmission systems has been observed to decrease with the signal duration when bandwidth is fixed. A new theoretical explanation for this behavior is provided for a specific b-modulator and validated in simulations.
...
The optimal transmit power in various nonlinear Fourier transform-based transmission systems has been observed to decrease with the signal duration when bandwidth is fixed. A new theoretical explanation for this behavior is provided for a specific b-modulator and validated in simulations.
The path-average approximation penalizes NFDM transmission over lumped amplified fiber links.We investigate suitably tapered lossy fibers to overcome the approximation error induced by the path average, making the NFDM transmission exact. Error vector magnitude gains up to 4.8 dB are observed.
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The path-average approximation penalizes NFDM transmission over lumped amplified fiber links.We investigate suitably tapered lossy fibers to overcome the approximation error induced by the path average, making the NFDM transmission exact. Error vector magnitude gains up to 4.8 dB are observed.
The nonlinear Fourier transform (NFT) has recently gained significant attention in fiber optic communications and other engineering fields. Although several numerical algorithms for computing the NFT have been published, the design of highly accurate low-complexity algorithms remains a challenge. In this paper, we present new fast forward NFT algorithms that achieve accuracies that are orders of magnitudes better than current methods, at comparable run times and even for moderate sampling intervals. The new algorithms are compared to existing solutions in multiple, extensive numerical examples.
...
The nonlinear Fourier transform (NFT) has recently gained significant attention in fiber optic communications and other engineering fields. Although several numerical algorithms for computing the NFT have been published, the design of highly accurate low-complexity algorithms remains a challenge. In this paper, we present new fast forward NFT algorithms that achieve accuracies that are orders of magnitudes better than current methods, at comparable run times and even for moderate sampling intervals. The new algorithms are compared to existing solutions in multiple, extensive numerical examples.
The conventional Fourier transform was originally developed in order to solve the heat equation, which is a standard example for a linear evolution equation. Nonlinear Fourier transforms (NFTs)1 are generalizations of the conventional Fourier transform that can be used to solve certain nonlinear evolution equations in a similar way (Ablowitz et al. 1974). An important difference to the conventional Fourier transform is that NFTs are equationspecific. The Korteweg-de Vries (KdV) equation (Gardner et al. 1967) and the nonlinear Schroedinger equation (NSE) (Shabat and Zakharov 1972) are two popular examples for nonlinear evolution equations that can be solved using appropriate NFTs.
...
The conventional Fourier transform was originally developed in order to solve the heat equation, which is a standard example for a linear evolution equation. Nonlinear Fourier transforms (NFTs)1 are generalizations of the conventional Fourier transform that can be used to solve certain nonlinear evolution equations in a similar way (Ablowitz et al. 1974). An important difference to the conventional Fourier transform is that NFTs are equationspecific. The Korteweg-de Vries (KdV) equation (Gardner et al. 1967) and the nonlinear Schroedinger equation (NSE) (Shabat and Zakharov 1972) are two popular examples for nonlinear evolution equations that can be solved using appropriate NFTs.
We present a nonlinear Fourier transform algorithm whose accuracy, at a comparable runtime and for moderate step sizes, is orders of magnitude better than that of the classical Boffetta-Osborne method.
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We present a nonlinear Fourier transform algorithm whose accuracy, at a comparable runtime and for moderate step sizes, is orders of magnitude better than that of the classical Boffetta-Osborne method.
A fast algorithm for constructing multi-solitons with linear complexity in the number of samples and eigenvalues is introduced. The algorithm is shown to be significantly faster than the conventional Darboux transform in a numerical example, with acceptable error.
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A fast algorithm for constructing multi-solitons with linear complexity in the number of samples and eigenvalues is introduced. The algorithm is shown to be significantly faster than the conventional Darboux transform in a numerical example, with acceptable error.