Published on: 2026-06-17
Source: Novosibirsk State University –
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This study was carried out at the intersection of applied mathematics, nonlinear physics, and modern telecommunication technologies. Its results were published in the article “Numerical Approaches in Nonlinear Fourier Transform-Based Signal Processing for Telecommunications,” which appeared in the journal Studies in Applied Mathematics. It was prepared by graduates of the Physics and Mechano-Mathematical faculties of Novosibirsk State University over several years.Egor Sedov, Igor Chesov, Mikhail FedorukиSergey Turitsyn.
—The idea of the work can be explained through the familiar Fourier transform. This mathematical tool allows representing a complex signal as a set of simple frequencies — approximately the same way that a musical chord can be decomposed into individual notes. Such an approach is widely used in the processing of sound, images, radio signals, and data in communication systems., — explained the scientific director of the Artificial Intelligence Center (AI Center) of the NRU, director of the scientific and educational center “Machine Learning and Big Data Analysis”, academician of the Russian Academy of Sciences, doctor of physical and mathematical sciences, professor Mikhail Fedoruk.
However, the optical signal in the fiber does not always behave like a simple set of independent frequencies. At high transmission speeds and sufficient power, nonlinear effects appear: the signal begins to influence its own propagation through the medium. Moreover, different frequency components, due to different group velocities, propagate through the fiber at different speeds, causing a short pulse to stretch in time. This phenomenon is called dispersion. As a result, the signal shape is distorted, and recovering the original data becomes more difficult.
—Conventional digital processing methods work well as long as the optical signal distortions can be considered almost linear. But in long and high-speed fiber-optic lines, this is often insufficient. Therefore, researchers are looking for ways to take into account the physics of optical signal propagation not as noise, but as part of a mathematical model that can be worked with, improving signal quality., — explained the senior research fellow of the Laboratory of Numerical and Experimental Modeling of New Photonics Devices of the MMF NSU, candidate of physical and mathematical sciences Igor Chesovskoy.
One of such transformations is associated with nonlinear Fourier transform, or NFT. It is a more complex analogue of the usual Fourier transform, adapted for certain nonlinear wave systems. While the usual Fourier transform decomposes a signal into linear frequencies, the NFT describes it through a nonlinear spectrum, where dispersion and nonlinearity are taken into account simultaneously.
In nonlinear optics, solitons play an important role — optical pulses that under certain conditions can propagate almost without changing their shape. This behavior is possibly due to the balance of two effects: dispersion tends to stretch the pulse in time, while the nonlinearity of the medium counteracts this stretching.
—In the nonlinear spectrum of the signal, a continuous and a discrete part are distinguished. The continuous part describes components that behave more like radiation during propagation and can blur. The discrete part is associated with soliton components. The longer and more powerful the optical signal, the more such components may appear, and the more difficult the task of accurate numerical signal recovery becomes.By the way, it should be noted that the inverse scattering method (a nonlinear Fourier transform) for solving the nonlinear Schrödinger equation, which serves as the basis for describing the evolution of optical pulses in a fiber-optic communication line, was proposed in 1971 in Novosibirsk by V.E. Zakharov and A.B. Shabat. This is precisely what makes NFT simultaneously an attractive and complex tool for optical communication lines. In the ideal mathematical model, the change in the nonlinear spectrum along the fiber is described much more simply than the change of the signal itself in time.Therefore, it is possible to switch to a special spectral representation, perform compensation there, and then restore the signal back. In the future, such an approach may help to combat nonlinear distortions that limit data transmission at high speeds and over long distances.— clarified Mikhail Fedoruk.
In the published work, the authors considered not abstract model impulses, but signals close to those used in modern coherent optical communication systems. In particular, they analyzed telecommunication signals with quadrature amplitude modulation 16-QAM. This format encodes information simultaneously in the amplitude and phase of the optical wave and allows transmitting four bits per one symbol.
—One of the practical difficulties is that the most developed numerical NFT algorithms are more conveniently applied not to an infinite data stream, but to individual finite fragments of signals. Therefore, the authors considered transmission of information in blocks: useful parts of the signal are separated by protective intervals so that adjacent blocks do not interfere with each other after propagation through the fiber. Here a compromise arises. Excessively large protective intervals reduce the transmission efficiency because part of the channel time does not carry useful information.Too long blocks complicate processing instead: more nonlinear spectral components appear in the signal, and algorithms find it harder to accurately restore the original signal. Therefore, it is important to understand which block lengths, powers, and transmission distances still allow reliable use of NFT processing.— told Igor Chelovsky.
As an example, the authors considered the transmission of a 16-QAM signal over a line consisting of 12 sections of a standard single-mode fiber, each 80 km long. The total length of the line was 960 km. The quality of restoration was evaluated through characteristics typical for digital communication: bit error rate and rejection of received symbols from ideal values. This approach allows discussing NFT not only as a mathematical construction but also as a tool whose applicability can be tested on telecommunication scenarios.
Separate attention in the work is given to numerical methods. Nonlinear Fourier transform includes a direct problem — converting from a signal to a nonlinear spectrum — and an inverse problem, that is, restoring the signal based on this spectrum. For practical applications, both parts must be accurate and sufficiently fast. Therefore, a significant part of the research is devoted to algorithms for computing the nonlinear spectrum, searching for its discrete components, and stable signal restoration.
—The interest in the article is related to the fact that it lies on the boundary of several fields at once. For mathematicians, this is the development of ideas of soliton theory and the inverse scattering problem. For physicists, it is a way to describe complex nonlinear wave processes. For communication specialists, it is a possible path to new methods of digital signal processing under conditions where ordinary linear approximations become insufficient., — noted Mikhail Fedoruk.
The authors clarified that NFT is not a ready-made universal replacement for existing technologies. Their work shows both the possibilities of the new approach and its limitations. This is especially important for engineering applications, as there is always a large layer of numerical algorithms, checks, and limitations between a mathematical idea and a real technology.
A scientific article involving employees of Novosibirsk State University was among the top 10% most viewed works published in 2024 in the journal Studies in Applied Mathematics. The number of views was evaluated over the first 12 months after publication. The publisher Wiley informed the authors about this.
—For us, this means that the topic of our work is in demand not only among specialists in nonlinear waves, but also in the broader scientific community interested in mathematical methods for modern telecommunications technologies, — summarized Mikhail Fedoruk.
The work was supported by the Russian Scientific Foundation and the EPSRC TRANSNET project. The article is published in the journal Studies in Applied Mathematics by Wiley publishing house in open access.
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