Chebyshev Polynomial Functions Research Articles

A simplified decision feedback Chebyshev function link neural network with intelligent initialization for underwater acoustic channel equalization

IntroductionIn shallow-water environments, the reliability of underwater communication links is often compromised by significant multipath effects. Some equalization techniques such as decision feedback equalizer, and deep neural network equalizer suffer from slow convergence and high computational complexity.MethodsTo address this challenge, this paper proposes a simplified decision feedback Chebyshev function link neural network equalizer (SDF-CFLNNE). The structure of the SDF-CFLNNE employs Chebyshev polynomial function expansion modules to directly and non-linearly transform the input signals into the output layer, without the inclusion of hidden layers. Additionally, it feeds the decision signal back to the input layer rather than the function expansion module, which significantly reduces computational complexity. Considering that, in the training phase of neural networks, the random initialization of weights and biases can substantially impact the training process and the ultimate performance of the network, this paper proposes a chaotic sparrow search algorithm combining the osprey optimization algorithm and Cauchy mutation (OCCSSA) to optimize the initial weights and thresholds of the proposed equalizer. The OCCSSA utilizes the Piecewise chaotic population initialization and combines the exploration strategy of the ospreywith the Cauchy mutation strategy to enhance both global and local search capabilities. RseultsSimulations were conducted using underwater multipath signals generated by the Bellhop Acoustic Toolbox. The results demonstrate that the performance of the SDFCFLNNE initialized by OCCSSA surpasses that of CFLNN-based and traditional nonlinear equalizers, with a notable improvement of 2-6 dB in terms of signal-to-noise ratio at a bit error rate (BER) of 10−4 and a reduced mean square error (MSE). Furthermore, the effectiveness of the proposed equalizer was validated using the lake experimental data, demonstrating lower BER and MSE with improved stability. DiscussionThis underscores thepromise of employing the SDFCFLNNE initialized by OCCSSA as a promising solution to enhance the robustness of underwater communication in challenging environments.

Influences of Meteorological Factors on Maize and Sorghum Yield in Togo, West Africa

This paper explores the effect of meteorological factors such as rainfall, temperature, sunshine, wind speed, and relative humidity on the yield of maize (Zea mays L.) and sorghum (Sorghum bicolor L.) at different growth stages in Togo’s Plateau, Central, and Savannah regions. For this purpose, data from 1990 to 2019 on weather variables and maize and sorghum yields were used. The study applied Fisher’s meteorological regression and Chebyshev polynomial function. Our findings revealed that rainfall had a more beneficial than detrimental effect on maize and sorghum yield across stages and regions. Contrariwise, temperature influence was as beneficial as detrimental and more significant across all growth stages of maize and sorghum in the Savannah and Plateau regions. Furthermore, the sunshine effect on maize yield was more significant in the Central and Savannah regions, while negative on sorghum yield in all the growth stages in the Central region. Similarly, the wind speed was also beneficial and detrimental to maize and sorghum yields, although it was more significant for sorghum in Plateau and Savannah regions. Lastly, relative air humidity positively and negatively influenced maize and sorghum yields in all the growth stages and regions for maize and the Plateau and Savannah regions for sorghum. Therefore, there is a need for real-time agricultural meteorological information to help farmers plan crop production more efficiently and increase crop yield.

Reliability Assessment of Power Distribution System Using Relative Customer Average Interruptions Duration Index Model

Reliability assessment of power distribution system deals with the adequacy of overall system supply and indicates the system behaviour and response at the customer end. In order to attain satisfactory degree of reliability in electric power distribution system, there is the need to develop a model that will improve the three major system reliability indices via Systems Average Interruptions Duration Index (SAIDI), Systems Average Interruptions Frequency Index (SAIFI) and Customer Average Interruptions Duration Index (CAIDI). This research paper therefore, developed a Relative CAIDI model for assessment of reliability indices of electrical power distribution system. In this paper, ten (10) years outage data from selected distribution feeders of Ibadan, Ikeja and Port-Harcourt distribution systems were obtained and analyzed using curve fitting tools in MATLAB. The system reliability indices served as input parameters for the development of Relative CAIDI model. The best curves that fitted the relationship between the Relative CAIDI and the number of feeders were obtained using Lagrange polynomial functions, Newton polynomial functions and Chebyshev polynomial functions as performance evaluation metrics. The results showed that Ibadan, Ikeja and Port-Harcourt distribution systems had Relative CAIDI of 0.718, 0.3976 and 0.5279 respectively and the validation results produced by Largrange polynomial function, Newton polynomial function and Chebyshev polynomial function were 0.717, 0.3979 and 0.5278 respectively. The model developed is a polynomial of order six which depends majorly on the power requirements of the distribution systems. The developed model can be used for effective reliability assessment of electrical power distribution systems as well as forming a base-line information for system planning and maintenance strategies. Keywords: Reliability Indices, SAIDI, SAIFI, CAIDI, Chebyshev Polynomial Function, Lagrange Polynomial Function, Newton Polynomial Function. DOI: 10.7176/NCS/14-01 Publication date: August 31 st 2022

Eigenvectors of Deformed Wigner Random Matrices

We investigate eigenvectors of rank-one deformations of random matrices $\boldsymbol B = \boldsymbol A + \theta \boldsymbol {uu}^{*}$ in which $\boldsymbol A \in \mathbb R^{N \times N}$ is a Wigner real symmetric random matrix, $\theta \in \mathbb R^{+}$ , and $\boldsymbol u$ is uniformly distributed on the unit sphere. It is well known that for $\theta > 1$ the eigenvector associated with the largest eigenvalue of $\boldsymbol B$ closely estimates $\boldsymbol u$ asymptotically, while for $\theta the eigenvectors of $\boldsymbol B$ are uninformative about $\boldsymbol u$ . We examine $\mathcal O({1}/{N})$ correlation of eigenvectors with $\boldsymbol u$ before phase transition and show that eigenvectors with larger eigenvalue exhibit stronger alignment with deforming vector through an explicit inverse law ${1}/{\theta ^{*} - x}$ with $\theta ^{*}:= \theta + ({1}/{\theta })$ . This distribution function will be shown to be the ordinary generating function of Chebyshev polynomials of the second kind. These polynomials form an orthogonal set with respect to the semicircle weighting function. This law is an increasing function in the support of semicircle law for eigenvalues $(-2\:,+2)$ . Therefore, most of energy of the unknown deforming vector is concentrated in a $cN$ -dimensional ( $c ) known subspace of $\boldsymbol B$ . We use a combinatorial approach to prove the result. We also extend the result to constant rank- $r$ deformations.

Spectrally accurate approximate solutions and convergence analysis of fractional Burgers\u2019 equation

In this paper, a new numerical technique implements on the time-space pseudospectral method to approximate the numerical solutions of nonlinear time- and space-fractional coupled Burgers’ equation. This technique is based on orthogonal Chebyshev polynomial function and discretizes using Chebyshev–Gauss–Lobbato (CGL) points. Caputo–Riemann–Liouville fractional derivative formula is used to illustrate the fractional derivatives matrix at CGL points. Using the derivatives matrices, the given problem is reduced to a system of nonlinear algebraic equations. These equations can be solved using Newton–Raphson method. Two model examples of time- and space-fractional coupled Burgers’ equation are tested for a set of fractional space and time derivative order. The figures and tables show the significant features, effectiveness, and good accuracy of the proposed method.

A Numerical Solver for First Order Initial Value Problems of Ordinary Differential Equation Via the Combination of Chebyshev Polynomial and Exponential Function

Purpose: The purpose of this study is to derive a numerical solver for first order initial value problems of ordinary differential equation via the combination of Chebyshev polynomial and exponential function.Methodology: A new numerical method for solving Initial Value Problems of first order ordinary differential equation is developed. The method is based on finite difference method with a combination of Chebyshev polynomials and exponential function as interpolant. The accuracy, stability, consistency and convergence of the derived scheme were investigated. Numerical experiment was carried out by solving some test problems using the derived scheme.Findings: Results of the numerical experiment revealed that the derived method compared favourably with exact solutions and also performs better than some existing methods for solving initial value problems of first order. Unique Contribution to theory, practice and policy: The study therefore concludes that the method solves problems to expected level of accuracy and can thus be considered among the numerous methods suitable for solving IVPs of first order.

Hybrid reconstruction method for indirect monitoring of an ice load of a steel gate in a cold region

The steel gate of a hydraulic complex may be subjected to ice loads during freezing periods in cold regions, threatening the gate safety. The ice load on the gate is usually affected by several factors, including the ice thickness, snow cover, and changes in water level and temperature. The ice pressure distribution on the gate cannot be readily estimated by theoretical analysis or empirical formulae. Therefore, structural strain and local ice pressure data were collected over 140 days during the winter of 2016–2017 to investigate the structural deformation and local ice pressure distribution. A hybrid reconstruction method (HCM) was developed for establishing the ice pressure distribution using the monitoring data, and the effectiveness of the HCM was analysed based on several uniform load patterns and the Chebyshev polynomial functions. The ice pressure distributions on the gate were reconstructed during the entire monitoring period, considering the collected data for the lowest temperature of each day. The reconstructed ice pressure distribution, i.e., the equivalent and uniform ice pressure within every individual cell, was lower than 0.1 MPa in most parts of the gate.

Wideband Balanced-to-Unbalanced Bandpass Filters Synthetically Designed With Chebyshev Filtering Response

In this paper, two wideband balanced-to-unbalanced (balun) bandpass filters (BPFs) are synthetically designed with three- and four-pole filtering responses, respectively. Based on the simplified even- and odd-mode equivalent circuits of the proposed three-pole balun BPF, the exact impedance value of the cascaded microstrip sections connected with two balanced ports is first determined in order to attain a perfectly matched S-matrix. By the virtue of the modified two-port transmission-line network with asymmetrical port impedances, a quasi-Chebyshev filtering response for this three-pole balun filter is theoretically presented. Next, to improve the in-band performance, especially at the high frequency, a four-pole balun BPF designed with a folded slotline resonator is proposed. Via the virtually short-ended microstrip stubs of the modified two-port equivalent network, the four-pole frequency response can be directly analyzed with the Chebyshev polynomial function. For these two proposed wideband balun filters, their theoretical and simulated results are well confirmed by the circuit model responses with the same impedance values of the simulated microstrip and slotline resonators and the extracted turns ratios of the employed transformers. Finally, two prototype wideband balun BPFs are fabricated and measured to verify our design predictions.

The generating function of bivariate Chebyshev polynomials associated with the Lie algebra G 2

The generating function of the second kind bivariate Chebyshev polynomials associated with the simple Lie algebra $G_2$ is constructed by the method proposed in \cite{DKS} and \cite{DKS1}.

Analysis and Design of Wideband Microstrip-to-Microstrip Equal Ripple Vertical Transitions and Their Application to Bandpass Filters

This paper first presents two types of wideband microstrip-to-microstrip equal ripple vertical transitions designed with the uniform and folded slotline structures, respectively. By solving the characteristic function with differential calculation, the condition for realizing the constant equal ripple frequency response but different bandwidths (BWs) for the proposed transition is determined with resorting to the characteristic impedances of the coupled open-circuited microstrip line and slotline. Then, a vertical transition with the uniform slotline is synthetically analyzed and designed with Chebyshev polynomial function and another similar vertical transition with the same in-band ripple level but different BWs is also comparatively presented. By modifying the normalized impedances of the proposed transition to make the theoretical result close to the simulated one with the initial normalized impedances, the transformer turns ratios of the microstrip-to-slotline intersection is exactly calculated by the impedance ratios between two corresponding sets of normalized impedances. Due to the degraded high-frequency performance caused by the radiation loss of the uniform slotline, two vertical transitions with the folded slotline structure are further analyzed and the related turn ratios are extracted with the proposed approach. To verify the effectiveness on the reduction of radiation loss for the folded slotline structure, a six-pole wideband bandpass filter is presented with high-frequency performance greatly improved. In final, wideband vertical transitions with different slotline structures and a six-pole wideband bandpass filter are synthetically designed, simulated, fabricated, and measured.

Chebyshev polynomial functions based locally recurrent neuro-fuzzy information system for prediction of financial and energy market data

In this paper Chebyshev polynomial functions based locally recurrent neuro-fuzzy information system is presented for the prediction and analysis of financial and electrical energy market data. The normally used TSK-type feedforward fuzzy neural network is unable to take the full advantage of the use of the linear fuzzy rule base in accurate input–output mapping and hence the consequent part of the rule base is made nonlinear using polynomial or arithmetic basis functions. Further the Chebyshev polynomial functions provide an expanded nonlinear transformation to the input space thereby increasing its dimension for capturing the nonlinearities and chaotic variations in financial or energy market data streams. Also the locally recurrent neuro-fuzzy information system (LRNFIS) includes feedback loops both at the firing strength layer and the output layer to allow signal flow both in forward and backward directions, thereby making the LRNFIS mimic a dynamic system that provides fast convergence and accuracy in predicting time series fluctuations. Instead of using forward and backward least mean square (FBLMS) learning algorithm, an improved Firefly-Harmony search (IFFHS) learning algorithm is used to estimate the parameters of the consequent part and feedback loop parameters for better stability and convergence. Several real world financial and energy market time series databases are used for performance validation of the proposed LRNFIS model.

Free vibration of carbon nanotube reinforced composite plate on point Supports using Lagrangian multipliers

Chebyshev polynomial functions are used in the Lagrangian multipliers method to study the free vibration characteristics of rectangular moderately thick composite plates reinforced with carbon nanotubes (CNTs). Plate is resting on point supports. Distribution of CNTs across the plate thickness is considered to be either uniform or functionally graded. Properties of the plate are obtained using a refined rule of mixtures approach which includes the efficiency parameters to capture the size dependent characteristics of the composite plate. Using a Ritz solution method, an eigenvalue problem is established which results in natural frequencies and mode shapes of the plate. Based on the developed solution method, number and position of point supports are arbitrary and also various boundary conditions may be assumed for the four edges of the plate. After performing comparison studies for isotropic homogeneous plates on point supports, parametric studies are provided to explore the vibration characteristics of the carbon nanotube reinforced composite plates on point supports. It is shown that, frequencies of the plate increase as the volume fraction of CNTs increases.

Age-Related Changes in Corneal Deformation Dynamics Utilizing Scheimpflug Imaging

PurposeTo study age-related changes in corneal deformation response to air-puff applanation tonometry.MethodsFifty healthy subjects were recruited for a prospective study and divided into two equal age groups (≤ 28 and ≥ 50 years old). Up to three measurements by a corneal deformation analyser based on the Scheimpflug principle were performed on the left eye of each subject. Raw Scheimpflug images were used to extract changes in anterior and posterior corneal profiles, which were further modelled by an orthogonal series of Chebyshev polynomial functions. Time series of the polynomial coefficients of even order exhibited a dynamic behavior in which three distinct stages were recognized. A bilinear function was used to model the first and the third stage of corneal dynamics. Slope parameters of the bilinear fit were then tested between the two age groups using Wilcoxon rank sum test and two-way non-parametric ANOVA (Friedman) test.ResultsStatistically significant changes (Wilcoxon test, P<0.05) between the age groups were observed in the phase of the second applanation dynamics for the posterior corneal profile. In a two-way comparison, in which the corneal profile was used as a dependent variable, statistically significant changes (ANOVA/Friedman test, P = 0.017) between the groups were also observed for that phase.ConclusionCorneal biomechanics depend on age. The changes in corneal deformation dynamics, which correspond to mostly free return of the cornea to its original shape after the air pulse, indicate that the age related differences in corneal biomechanics are subtle but observable with high speed imaging.

Performance analysis for a chaos‐based code‐division multiple access system in wide‐band channel

Code-division multiple access technology is widely used in telecommunications and its performance has been extensively investigated in the past. Theoretical results for the case of wide-band transmission channel were not available until recently. The novel formulae which have been published in 2014 can have an important impact on the future of wireless multiuser communications, but limitations come from the Gaussian approximations used in their derivation. In this Letter, the authors obtain more accurate expressions of the bit error rate (BER) for the case when the model of the wide-band channel is two-ray, with Rayleigh fading. In the authors’ approach, the spreading sequences are assumed to be generated by logistic map given by Chebyshev polynomial function of order two. Their theoretical and experimental results show clearly that the previous results on BER, which rely on the crude Gaussian approximation, are over-pessimistic.

Model-free nonlinear restoring force identification for SMA dampers with double Chebyshev polynomials: approach and validation

The initiation and propagation of damage such as cracks in an engineering structure under dynamic loadings is a nonlinear process. Strictly speaking, conventional eigenvalue and eigenvector extraction-based damage identification approaches are suitable for linear systems only. Due to the unique nonlinearities associated with each civil engineering structure, it would be inefficient to attempt to express the nonlinear restoring force (NRF) of an engineering structure such as a reinforced concrete structure in a parametric form. Consequently, it is highly desirable to develop a general nonlinear identification approach to achieve structural damage detection in both qualitative and quantitative ways without the assumption on the parametric model of the hysteretic behavior. In this paper, based on a double Chebyshev polynomial function, a time-domain identification approach is proposed for identifying both the structural NRF and the mass distribution for multi-degree-of-freedom (MDOF) structures under incomplete dynamic loadings. As a typical nonlinear material, shape memory alloy (SMA) is introduced to a MDOF structural model to mimic the nonlinear behavior under dynamic loadings. The feasibility and robustness of the proposed approach is validated via (1) a numerical MDOF structure model equipped with a SMA damper whose restoring force is described by a double-flag-shaped nonlinear model, and (2) an experimental study on a four-story shear building structure model equipped with an in-house design of a SMA damper used to mimic the structural NRF under incomplete impact loadings. The identified NRF is compared with the theoretical values and the measurements in experiment. Both numerical and experimental results show that the proposed approach is capable of identifying both the mass distribution and structural nonlinearity under dynamic loadings, and could be potentially used for damage initiation and propagation monitoring during vibration of engineering structures under dynamic excitations.

Classification of Microarray Data using Functional Link Neural Network

Abstract Microarray dataset often contains a huge number of insignificant and irrelevant features that might lead to loss of useful information. The classes with both high relevance and having high significance feature sets are generally preferred for selecting the features, which determines the sample classification into their respective classes. This property has gained a lot of significance among the researchers and practitioners in DNA micro array classification. In this paper, Functional link neural network (FLNN) with four different basis functions named as Power series polynomial, Trigonometric, Chebyshev polynomial and Legendre polynomial functions have been considered to classify microarray data sets using t -test as a feature selection method. Further, this paper presents a comparative analysis on the obtained classification accuracy by coupling FLNN with different basis function and other existing models available in the literature. Performance parameters available in literature such as precision, recall, specificity, F-Measure, ROC curve and accuracy are applied in this comparative analysis to analyze the behavior of the classifiers. From the proposed approach, it is apparent that FLNN using Legendre polynomial is the suitable classification model among FLNN using different basis functions and other classifiers.

A Six-Dimensional Hyperchaotic System Selection and Its Application in DS-CDMA System

In this paper, we analyze the hyperchaotic properties of cellular neural network (CNN) systems based on Lyapunov exponents. A four-step algorithm for selecting the hyperchaotic system is proposed. Firstly, calculate the maximum absolute value of time sequences, which is one of the properties of chaotic system, noted as boundedness. Secondly, check another property of chaotic system: sensitive dependence on the initial conditions. Several systems can be identified as non-chaotic systems through above two steps, which saves lots of time to calculate the Lyapunov exponents. Then, calculate the Lyapunov exponents. If the largest Lyapunov exponent is negative, the system is identified as non-chaotic system. Finally, determine the given system is chaotic or not by the strange attractors and the time sequences. The binary hyperchaotic spread spectrum sequences can be generated by a six- dimensional CNN system. Some of the binary hyperchaotic sequences can be used in a direct sequence code division multiple access (DS-CDMA) system after selecting through a special rule. Simulation results prove the effectiveness of the six-dimensional CNN hyperchaotic sequences, compared with the m-sequence and second-order Chebyshev polynomial function.

Bit error rate improvement for chaos shift keying chaotic communication systems

The bit error rate (BER) performance of chaos shift keying is improved for an additive white Gaussian noise channel by applying a trimming operation to the Chebyshev polynomial function of order 2 and the piecewise linear map to make the input source more Gaussian distributed. Analytical expressions for the improved BER are also derived based on the curve-fitting approximation. Numerical results show that the proposed method can achieve performance gains of up to 3.5 dB in signal-to-noise ratio over the conventional method without trimming.

Vibration of Nonlocal Euler Beams Using Chebyshev Polynomials

This paper is concerned with the free vibration problem for micro/nano beams modelled after Eringen’s nonlocal elasticity theory and Euler beam theory. The small scale effect is taken into consideration in the former theory. The natural frequencies are obtained using the Hamilton’s principle and Chebyshev polynomial functions. The present method, which uses Rayleigh–Ritz technique in this paper, provides an efficient and extremely accurate vibration solution of micro/nano beams where the effects of small scale are significant. Numerical results for a variety of some micro/nano beams with various boundary conditions are given and compared with the available results wherever possible. Through the comparison of the results and the appropriate discussion, the knowledge of the level of capability of the developed method is promoted.

Energy approach vibration analysis of nonlocal Timoshenko beam theory

The aim of this paper is to present an efficient numerical method to analyze the vibration behavior of nano Timoshenko beams based on Eringen's nonlocal elasticity theory. The vibration frequencies of beams are firstly obtained using the theorem of minimum total potential energy and Chebyshev polynomial functions. The present method provides an efficient and extremely accurate vibration solution. Numerical results for a variety of some micro- and nano-beams with various boundary conditions are given and compared with the available results wherever possible. The small scale effects on the vibration frequencies of beams are determined and discussed.