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Mathematics
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Novel Mathematical Methods in Signal Processing and Its Applications
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Novel Mathematical Methods in Signal Processing and Its Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E: Applied Mathematics".

Deadline for manuscript submissions: closed (15 October 2023) | Viewed by 27925

Special Issue Editors

Prof. Dr. Gordon Morison

E-Mail Website
Guest Editor
Department of Computing, School of Computing, Engineering and Built Environment, Glasgow Caledonian University, Glasgow G4 0BA, UK
Interests: signal processing; data science; tensor decompositions; machine learning
Dr. Robert C. Atkinson

E-Mail Website
Guest Editor
Department of Electronic and Electrical Engineering, University of Strathclyde, Glasgow G1 1XW, UK
Interests: data science; machine learning; AutoML; explainable AI
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Signal processing and machine learning are the key enablers at the heart of the technological advancements to drive developments in mobile communications, Internet of Things, FinTech, predictive maintenance, digital health and biomedical engineering. These continually evolving fields offer huge potential for the application of mathematics and statistics to develop the next generation of algorithms to ensure they are ethical, transparent, robust and explainable. It is expected that these new algorithms will expand upon and exploit mathematical methods such as tensor analysis, stochastic calculus, information theory, complex analysis, optimization, group theory and geometric methods. This Special Issue looks for novel algorithms and mathematical methods applied to signal processing applications.

Prof. Dr. Gordon Morison
Dr. Robert C. Atkinson
Guest Editors

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • information theory
  • data science
  • tensor signal processing
  • graph signal processing
  • time-frequency decompositions
  • machine learning
  • Bayesian signal processing
  • image processing and computer vision
  • optimization methods
  • audio and acoustic signal processing
  • compressed sensing
  • sparse modelling
  • statistical signal processing
  • explainable AI (XAI)
  • causal AI

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Published Papers (7 papers)

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Research

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20 pages, 9583 KiB  
Article
Model for Choosing the Shape Parameter in the Multiquadratic Radial Basis Function Interpolation of an Arbitrary Sine Wave and Its Application
by Jian Sun, Ling Wang and Dianxuan Gong
Mathematics 2023, 11(8), 1856; https://doi.org/10.3390/math11081856 - 13 Apr 2023
Cited by 2 | Viewed by 2151
Abstract
In multiquadratic radial basis function (MQ-RBF) interpolation, shape parameters have a direct effect on the interpolation accuracy. The paper presents an MQ-RBF interpolation technique with optimized shape parameters for estimating the parameters of sine wave signals. At first, we assessed the impact of [...] Read more.
In multiquadratic radial basis function (MQ-RBF) interpolation, shape parameters have a direct effect on the interpolation accuracy. The paper presents an MQ-RBF interpolation technique with optimized shape parameters for estimating the parameters of sine wave signals. At first, we assessed the impact of basic sinusoidal parameters on the MQ-RBF interpolation outcomes through numerical experiments. The results indicated that the angular frequency of a sine wave is a crucial determinant of the corresponding MQ-RBF interpolation shape parameters. A linear regression method was then used to establish the optimal parameter selection formula for a single-frequency sine wave, based on a large volume of experimental data. For multi-frequency sinusoidal signals, appropriate interpolation shape parameters were selected using the random walk algorithm to create datasets. These datasets were subsequently used to train several regression models, which were then evaluated and compared. Based on its operational cost and prediction accuracy, the random forest algorithm was chosen to establish the shape parameter selection model for multi-frequency sinusoidal signals. The inclusion of the Bayesian optimizer resulted in a highly accurate model. The establishment of this model enabled the adaptive selection of the corresponding shape parameters for any sine wave signal, providing a convenient means of selecting MQ-RBF interpolation shape parameters. Furthermore, the paper proposes an MQ-RBF interpolation subdivision least squares method that significantly improves the estimation accuracy of sine wave parameters. The practicality of the method was validated by successfully applying it in the calibration of the clock delay mismatch of a time-interleaved analog-to-digital converter system. Full article
(This article belongs to the Special Issue Novel Mathematical Methods in Signal Processing and Its Applications)
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16 pages, 568 KiB  
Article
Error Probability of a Coherent M-ary PSK FSO System Influenced by Phase Noise
by Milica Petković, Goran T. Đorđević, Jarosław Makal, Zvezdan Marjanović and Gradimir V. Milovanović
Mathematics 2023, 11(1), 121; https://doi.org/10.3390/math11010121 - 27 Dec 2022
Cited by 5 | Viewed by 2829
Abstract
In this paper, we aim to develop an analytical framework for design and analysis of new generation mobile networks fronthaul/backhaul links based on the application of free-space optical (FSO) technology. Taking the receiver hardware imperfections into account, we present an efficient analytical approach [...] Read more.
In this paper, we aim to develop an analytical framework for design and analysis of new generation mobile networks fronthaul/backhaul links based on the application of free-space optical (FSO) technology. Taking the receiver hardware imperfections into account, we present an efficient analytical approach in analyzing average symbol error probability (SEP) of the coherent FSO system employing M-ary phase-shift keying (PSK). Optical signal transmission is influenced by pointing errors and atmospheric turbulence. The signal intensity fluctuations caused by atmospheric turbulence are modeled by general Málaga (M) distribution, which takes into account the effect of multiple scattered components. We estimate the range of the signal-to-noise ratio at which the SEP floor appears, as well as the value of this non-removable error floor. The results illustrate that the effect of imperfect phase error compensation on the SEP is more critical under weaker turbulence conditions and for higher order modulation formats. Based on the analytical tools presented here, it is possible to estimate tolerable value of standard deviation of phase noise for the given value of SEP. This value of standard deviation is an important parameter in designing the phase-locked loop filter in the receiver. Full article
(This article belongs to the Special Issue Novel Mathematical Methods in Signal Processing and Its Applications)
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20 pages, 431 KiB  
Article
A Novel Optical-Based Methodology for Improving Nonlinear Fourier Transform
by Julian Hoxha, Wael Hosny Fouad Aly, Erdjana Dida, Iva Kertusha and Mouhammad AlAkkoumi
Mathematics 2022, 10(23), 4513; https://doi.org/10.3390/math10234513 - 29 Nov 2022
Viewed by 1579
Abstract
The increasing demand for bandwidth and long-haul transmission has led to new methods of signal processing and transmission in optical fiber communication systems. The nonlinear Fourier transform is one of the most recent methods proposed, and is able to represent an integrable nonlinear [...] Read more.
The increasing demand for bandwidth and long-haul transmission has led to new methods of signal processing and transmission in optical fiber communication systems. The nonlinear Fourier transform is one of the most recent methods proposed, and is able to represent an integrable nonlinear Schrödinger equation (NLSE) channel in terms of its continuous and discrete spectrum, to overcome the limitation of the bandwidth imposed by the Kerr effect on silica fibers. In this paper, we will propose and investigate the Boffetta-Osburne method for the direct nonlinear Fourier implementation, and the Gel’fand-Levitan-Marchenko equation for the inverse nonlinear Fourier, as only the continuous part of the nonlinear spectrum will be used to encode information. A novel methodology is proposed to improve their numerical implementation with respect to the NLSE, and we analyze in detail how the improved algorithm can be used in a real optical system, by investigating three different modulation schemes. We report increased performance transmission and consistency in the numerical results when the proposed methodology is applied. Our results show that b-modulation will increase the Q-factor by 2 dB with respect to the other two modulations. The improvement results with our proposed methodology suggest that b-modulation applied only to a continuous part of the nonlinear spectrum is a very effective method for maximizing both the transmission bandwidth and distance in optical fiber communication systems. Full article
(This article belongs to the Special Issue Novel Mathematical Methods in Signal Processing and Its Applications)
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11 pages, 12258 KiB  
Article
Nonlinear Frequency-Modulated Waveforms Modeling and Optimization for Radar Applications
by Zhihuo Xu, Xiaoyue Wang and Yuexia Wang
Mathematics 2022, 10(21), 3939; https://doi.org/10.3390/math10213939 - 24 Oct 2022
Cited by 11 | Viewed by 3819
Abstract
Conventional radars commonly use a linear frequency-modulated (LFM) waveform as the transmitted signal. The LFM radar is a simple system, but its impulse-response function produces a −13.25 dB sidelobe, which in turn can make the detection of weak targets difficult by drowning out [...] Read more.
Conventional radars commonly use a linear frequency-modulated (LFM) waveform as the transmitted signal. The LFM radar is a simple system, but its impulse-response function produces a −13.25 dB sidelobe, which in turn can make the detection of weak targets difficult by drowning out adjacent weak target information with the sidelobe of a strong target. To overcome this challenge, this paper presents a modeling and optimization method for non-linear frequency-modulated (NLFM) waveforms. Firstly, the time-frequency relationship model of the NLFM signal was combined by using the Legendre polynomial. Next, the signal was optimized by using a bio-inspired method, known as the Firefly algorithm. Finally, the numerical results show that the advantages of the proposed NLFM waveform include high resolution and high sensitivity, as well as ultra-low sidelobes without the loss of the signal-to-noise ratio (SNR). To the authors’ knowledge, this is the first study to use NLFM signals for target-velocity improvement measurements. Importantly, the results show that mitigating the sidelobe of the radar waveform can significantly improve the accuracy of the velocity measurements. Full article
(This article belongs to the Special Issue Novel Mathematical Methods in Signal Processing and Its Applications)
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18 pages, 542 KiB  
Article
Proper ARMA Modeling and Forecasting in the Generalized Segre’s Quaternions Domain
by Jesús Navarro-Moreno, Rosa M. Fernández-Alcalá and Juan C. Ruiz-Molina
Mathematics 2022, 10(7), 1083; https://doi.org/10.3390/math10071083 - 28 Mar 2022
Cited by 4 | Viewed by 2151
Abstract
The analysis of time series in 4D commutative hypercomplex algebras is introduced. Firstly, generalized Segre’s quaternion (GSQ) random variables and signals are studied. Then, two concepts of properness are suggested and statistical tests to check if a GSQ random vector is proper or [...] Read more.
The analysis of time series in 4D commutative hypercomplex algebras is introduced. Firstly, generalized Segre’s quaternion (GSQ) random variables and signals are studied. Then, two concepts of properness are suggested and statistical tests to check if a GSQ random vector is proper or not are proposed. Further, a method to determine in which specific hypercomplex algebra is most likely to achieve, if possible, the properness properties is given. Next, both the linear estimation and prediction problems are studied in the GSQ domain. Finally, ARMA modeling and forecasting for proper GSQ time series are tackled. Experimental results show the superiority of the proposed approach over its counterpart in the Hamilton quaternion domain. Full article
(This article belongs to the Special Issue Novel Mathematical Methods in Signal Processing and Its Applications)
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Review

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50 pages, 1073 KiB  
Review
Matrix Factorization Techniques in Machine Learning, Signal Processing, and Statistics
by Ke-Lin Du, M. N. S. Swamy, Zhang-Quan Wang and Wai Ho Mow
Mathematics 2023, 11(12), 2674; https://doi.org/10.3390/math11122674 - 12 Jun 2023
Cited by 19 | Viewed by 11189
Abstract
Compressed sensing is an alternative to Shannon/Nyquist sampling for acquiring sparse or compressible signals. Sparse coding represents a signal as a sparse linear combination of atoms, which are elementary signals derived from a predefined dictionary. Compressed sensing, sparse approximation, and dictionary learning are [...] Read more.
Compressed sensing is an alternative to Shannon/Nyquist sampling for acquiring sparse or compressible signals. Sparse coding represents a signal as a sparse linear combination of atoms, which are elementary signals derived from a predefined dictionary. Compressed sensing, sparse approximation, and dictionary learning are topics similar to sparse coding. Matrix completion is the process of recovering a data matrix from a subset of its entries, and it extends the principles of compressed sensing and sparse approximation. The nonnegative matrix factorization is a low-rank matrix factorization technique for nonnegative data. All of these low-rank matrix factorization techniques are unsupervised learning techniques, and can be used for data analysis tasks, such as dimension reduction, feature extraction, blind source separation, data compression, and knowledge discovery. In this paper, we survey a few emerging matrix factorization techniques that are receiving wide attention in machine learning, signal processing, and statistics. The treated topics are compressed sensing, dictionary learning, sparse representation, matrix completion and matrix recovery, nonnegative matrix factorization, the Nyström method, and CUR matrix decomposition in the machine learning framework. Some related topics, such as matrix factorization using metaheuristics or neurodynamics, are also introduced. A few topics are suggested for future investigation in this article. Full article
(This article belongs to the Special Issue Novel Mathematical Methods in Signal Processing and Its Applications)
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22 pages, 523 KiB  
Review
Bright Soliton Solution of the Nonlinear Schrödinger Equation: Fourier Spectrum and Fundamental Characteristics
by Natanael Karjanto
Mathematics 2022, 10(23), 4559; https://doi.org/10.3390/math10234559 - 1 Dec 2022
Cited by 5 | Viewed by 2417
Abstract
We derive exact analytical expressions for the spatial Fourier spectrum of the fundamental bright soliton solution for the 1+1-dimensional nonlinear Schrödinger equation. Similar to a Gaussian profile, the Fourier transform for the hyperbolic secant shape is also shape-preserving. Interestingly, this [...] Read more.
We derive exact analytical expressions for the spatial Fourier spectrum of the fundamental bright soliton solution for the 1+1-dimensional nonlinear Schrödinger equation. Similar to a Gaussian profile, the Fourier transform for the hyperbolic secant shape is also shape-preserving. Interestingly, this associated hyperbolic secant Fourier spectrum can be represented by a convergent infinite series, which can be achieved using Mittag–Leffler’s expansion theorem. Conversely, given the expression of the series of the spectrum, we recover its closed form by employing Cauchy’s residue theorem for summation. We further confirm that the fundamental soliton indeed satisfies essential characteristics such as Parseval’s relation and the stretch-bandwidth reciprocity relationship. The fundamental bright soliton finds rich applications in nonlinear fiber optics and optical telecommunication systems. Full article
(This article belongs to the Special Issue Novel Mathematical Methods in Signal Processing and Its Applications)
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