Search Results (81)

This paper proposes a Hybrid Recursive Trigonometric (HRT) technique for FPGA-based direct digital frequency synthesizers. The HRT technique integrates a recursive cosine generator with periodic reinitialization via a second-order Taylor polynomial to reduce cumulative errors without requiring ROMs or iterative CORDIC units. A resource-efficient combinational architecture is implemented and validated on the Lattice iCE40HX1K FPGA. The effectiveness of the proposed HRT technique is evaluated through simulation and FPGA-based experiments, with respect to spectral accuracy and resource efficiency, particularly for fixed-point cosine waveform synthesis in low-resource digital systems. Simulation results show that the system has a spurious-free dynamic range (SFDR) of −86.09 dBc and signal-to-noise ratio of 52.74 dB using 16-bit fixed-point arithmetic. Experimental measurements confirm the feasibility, achieving −58.86 dBc SFDR. Full article

This paper presents a novel analytical approach for the efficient design of a particular class of 2D FIR filters, having a frequency response with an elliptically shaped support in the frequency plane. The filter design is based on a Gaussian shaped prototype filter, which is frequently used in signal and image processing. In order to express the Gaussian prototype frequency response as a trigonometric polynomial, we developed it into a Fourier series up to a specified order, given by the imposed approximation precision. We determined analytically a 1D to 2D frequency transformation, which was applied to the factored frequency response of the prototype, yielding directly the factored frequency response of a directional, elliptically shaped 2D filter, with specified selectivity and an orientation angle. The designed filters have accurate shapes and negligible distortions. We also designed a 2D uniform filter bank of elliptical filters, which was then applied in decomposing a test image into sub-band images, thus proving its usefulness as an analysis filter bank. Then, the original image was accurately reconstructed from its sub-band images. Very selective directional elliptical filters can be used in efficiently extracting straight lines with specified orientations from images, as shown in simulation examples. A computationally efficient implementation at the system level was also discussed, based on a polyphase and block filtering approach. The proposed implementation is illustrated for a smaller size of the filter kernel and input image and is shown to have reduced computational complexity due to its parallel structure, being much more arithmetically efficient compared not only to the direct filtering approach but also with the most recent similar implementations. Full article

Some classical texts on the Sturm–Liouville equation ${(p(x)y^{\prime })}^{\prime }-q(x)y+\lambda \rho (x)y=0$ are revised to highlight further properties of its solutions. Often, in the treatment of the ensuing integral equations, $\rho =const$ is assumed (and, further, $\rho =1$). Instead, here we preserve $\rho (x)$ and make a simple change only of the independent variable that reduces the Sturm–Liouville equation to $y^{″}-q(x)y+\lambda \rho (x)y=0$. We show that many results are identical with those with $\lambda \rho -q=const$. This is true in particular for the mean value of the oscillations and for the analog of the Riemann–Lebesgue Theorem. From a mechanical point of view, what is now the total energy is not a constant of the motion, and nevertheless, the equipartition of the energy is still verified and, at least approximately, it does so also for a class of complex $\lambda$. We provide here many detailed properties of the solutions of the above equation, with $\rho =\rho (x)$. The conclusion, as we may easily infer, is that, for large enough $\lambda$, locally, the solutions are trigonometric functions. We give the proof for the closure of the set of solutions through the Phragmén–Lindelöf Theorem, and show the separate dependence of the solutions from the real and imaginary components of $\lambda$. The particular case of $q(x)=\alpha \rho (x)$ is also considered. A direct proof of the uniform convergence of the Fourier series is given, with a statement identical to the classical theorem. Finally, the proof of J. von Neumann of the completeness of the Laguerre and Hermite polynomials in non-compact sets is revisited, without referring to generating functions and to the Weierstrass Theorem for compact sets. The possibility of the existence of a general integral transform is then investigated. Full article

This paper presents new fast algorithms for the fourth type discrete Hartley transform (DHT-IV) for input data sequences of lengths from 3 to 8. Fast algorithms for small-sized trigonometric transforms can be used as building blocks for synthesizing algorithms for large-sized transforms. Additionally, they can be utilized to process small data blocks in various digital signal processing applications, thereby reducing overall system latency and computational complexity. The existing polynomial algebraic approach and the approach based on decomposing the transform matrix into cyclic convolution submatrices involve rather complicated housekeeping and a large number of additions. To avoid the noted drawback, this paper uses a structural approach to synthesize new algorithms. The starting point for constructing fast algorithms was to represent DHT-IV as a matrix–vector product. The next step was to bring the block structure of the DHT-IV matrix to one of the matrix patterns following the structural approach. In this case, if the block structure of the DHT-IV matrix did not match one of the existing patterns, its rows and columns were reordered, and, if necessary, the signs of some entries were changed. If this did not help, the DHT-IV matrix was represented as the sum of two or more matrices, and then each matrix was analyzed separately, if necessary, subjecting the matrices obtained by decomposition to the above transformations. As a result, the factorizations of matrix components were obtained, which led to a reduction in the arithmetic complexity of the developed algorithms. To illustrate the space–time structures of computational processes described by the developed algorithms, their data flow graphs are presented, which, if necessary, can be directly mapped onto the VLSI structure. The obtained DHT-IV algorithms can reduce the number of multiplications by an average of 75% compared with the direct calculation of matrix–vector products. However, the number of additions has increased by an average of 4%. Full article

This study explores analytical soliton solutions for the cubic–quintic time-fractional nonlinear non-paraxial pulse transmission model. This versatile model finds numerous uses in fiber optic communication, nonlinear optics, and optical signal processing. The strength of the quintic and cubic nonlinear components plays a crucial role in nonlinear processes, such as self-phase modulation, self-focusing, and wave combining. The fractional nonlinear Schrödinger equation (FNLSE) facilitates precise control over the dynamic properties of optical solitons. Exact and methodical solutions include those involving trigonometric functions, Jacobian elliptical functions (JEFs), and the transformation of JEFs into solitary wave (SW) solutions. This study reveals that various soliton solutions, such as periodic, rational, kink, and SW solitons, are identified using the complete discrimination polynomial methods (CDSPM). The concepts of chaos and bifurcation serve as the framework for investigating the system qualitatively. We explore various techniques for detecting chaos, including three-dimensional and two-dimensional graphs, time-series analysis, and Poincarè maps. A sensitivity analysis is performed utilizing a variety of initial conditions. Full article

This paper proposes the generalized modified unstable nonlinear Schrödinger’s equation with applications in modulated wavetrain instabilities. The extended direct algebra and generalized Ricatti equation methods are applied to find innovative soliton solutions to the equation. The solutions are obtained in the form of elliptic, hyperbolic, and trigonometric functions. Moreover, a Galilean transformation is used to convert the problem into a dynamical system. We use the theory of planar dynamical systems to derive the equilibrium points of the dynamical system and analyze the Hamiltonian polynomial. We further investigate the bifurcation phase portrait of the system and study its chaotic behaviors when an external force is applied to the system. Graphical 2D and 3D plots are explored to support our mathematical analysis. A sensitivity analysis confirms that the variation in initial conditions has no substantial effect on the stability of the solutions. Furthermore, we give the modulation instability gain spectrum of the considered model and graphically indicate its dynamics using 2D plots. The reported results demonstrate not only the dynamics of the analyzed equation but are also conceptually relevant in establishing the temporal development of modest disturbances in stable or unstable media. These disturbances will be critical for anticipating, planning treatments, and creating novel mechanisms for modulated wavetrain instabilities. Full article

The stability of differential equations is one of the most important aspects to consider in dynamical system theory. Chaotic systems were classified according to stability as multi-stable systems; systems with a single stable equilibrium; bi-stable systems; and, recently, mega-stable systems. Mega-stability refers to the infinity countable nested attractors of a periodically forced non-autonomous system. Many researchers attempted to present a simple mega-stable system. In this paper, we investigated the mega-stability of periodically damped non-autonomous differential systems with the following different order cases: integer and fractional. In the case of the integer order, we generalize the mega-stable system, such that the velocity is multiplied by a trigonometrical polynomial, and we present the necessary and sufficient conditions to generated countable infinity nested attractors. In the case of the fractional order, we obtained that the fractional order of periodically damped non-autonomous differential systems has infinity countable nested unstable attractors for some orders. The mega-instability was illustrated for two examples, showing the order effect on the trajectories. In addition, and to further recent work presenting simple high dimensional mega-stable chaotic systems, we introduce a 4D mega-stable hyperchaotic system, examining chaotic and hyperchaotic behaviors through Lyapunov exponents and bifurcation diagrams. Full article

The fractional Schrödinger equation with time-dependent coefficients (FSE-TDCs) is taken into consideration here. The mapping method and the $(G^{\prime }/G)$-expansion method are applied to generate new bright solutions, kink solutions, dark optical solutions, singular solutions, periodic solutions, and others. Because the Schrödinger equation is widely employed in quantum computers, quantum mechanics, physics, engineering, and chemistry, the solutions developed can be utilized to examine a wide range of important physical phenomena. In addition, we illustrate the influence of the coefficients, when these coefficients have specific values, such as random, polynomial, trigonometric, and hyperbolic functions, on the exact solutions of FSE-TDCs. Also, we show the influence of fractional-order derivatives on the obtained solutions. Full article

This paper extends the well-known transfinite interpolation formula, which was developed in the late 1960s by the applied mathematician William Gordon at the premises of General Motors as an extension of the pre-existing Coons interpolation formula. Here, a conjecture is formulated, which claims that the meaning of the involved blending functions can be enhanced, such that it includes any linear independent and complete set of functions, including piecewise-linear, trigonometric functions, Bernstein polynomials, B-splines, and NURBS, among others. In this sense, NURBS-based isogeometric analysis and aspects of T-splines may be considered as special cases. Applications are provided to illustrate the accuracy in the interpolation through the L₂ error norm of closed-formed functions prescribed at the nodal points of the transfinite patch, which represent the solution of partial differential equations under boundary conditions of the Dirichlet type. Full article

This study presents a novel approach by implementing an active memristor in a hyperchaotic discrete system, based on a cubic map, which is implemented by using two different microcontrollers. The key contributions of this work are threefold. The use of two microcontrollers with improved characteristics, such as speed and memory, for faster and more accurate computations significantly improves upon previous systems. Also, for the first time, an active memristor is used in a discrete-time system, which is implemented by using a microcontroller. Furthermore, the system is compared with two different types of microcontrollers regarding the execution time and the quality of the produced bifurcation diagrams. The proposed memristive cubic map uses computationally efficient polynomial functions, which are well suited to microcontroller-based systems, in contrast to more resource-intensive trigonometric and exponential functions. Bifurcation diagrams and a Lyapunov exponent analysis from simulating the system in Mathematica revealed hyperchaotic behavior, along with other significant dynamical phenomena, such as regular orbits, chaotic trajectories, and transitions to chaos through mechanisms like period doubling and crisis phenomena. Experimental verification confirmed the consistency of the results across microcontroller platforms, underscoring the practicality and potential applications of active memristor-based chaotic systems. Full article

This study investigates the impact response of polymer foams commonly used in protective packaging, considering the fractal nature of their material microstructure. The research begins with static material characterization and impact tests on two low-density polyethylene foams. To capture the multiscale nature of the dynamic response behavior of two low-density foams to sustain impact loads, fractional differential equations of motion are used to qualitatively and quantitatively describe the dynamic response behavior, assuming restoring forces for each foam characterized, respectively, by a polynomial of heptic degree and by a trigonometric tangential function. A two-scale transform is employed to solve the mathematical model and predict the material’s behavior under impact loads, accounting for the fractal structure of the material’s molecular configuration. To assess the accuracy of the mathematical model, we performed impact tests considering eight dropping heights and two plate weights. We found good predictions from the mathematical models compared to experimental data when the fractal derivatives were between 1.86 and 1.9, depending on the cushioning material used. The accuracy of the theoretical predictions achieved using fractal calculus elucidates how to predict multiscale phenomena associated with foam heterogeneity across space, density, and average pore size, which influence the foam chain’s molecular motion during impact loading conditions. Full article

It is well known that the four ancient Greek triangle centers and others have homogeneous barycentric coordinates that are polynomials in the sidelengths $a,b,$ and c of a triangle $ABC$. For example, the circumcenter is represented by the polynomial $a(b^2+c^2-a^2)$. It is not so well known that triangle centers have barycentric coordinates, such as $\tan A : \tan B : \tan C$, that are also representable by polynomials, in this case, by $p(a, b, c) : p(b, c, a) : p(c, a, b)$, where $p(a, b, c)=a(a^2+b^2-c^2)(a^2+c^2-b^2)$. This paper presents and discusses the polynomial representations of triangle centers that have barycentric coordinates of the form $f(a, b, c) : f(b, c, a) : f(c, a, b)$, where f depends on one or more of the functions in the set $\{\cos , \sin , \tan , \sec , \csc , \cot \}$. The topics discussed include infinite trigonometric orthopoints, the n-Euler line, and symbolic substitution. Full article

This paper presents direct and inverse theorems concerning the approximation of functions of several variables with bounded p-fluctuation using Walsh polynomials. These theorems provide estimates for the best approximation of such functions by polynomials in the norm of the space under consideration. The paper investigates the properties of the Walsh system, which includes piecewise constant functions, and builds on earlier work on trigonometric and multiplicative systems. The results are theoretical and have potential applications in such areas as coding theory, digital signal processing, pattern recognition, and probability theory. Full article

In this work, an analytical method in the frequency domain is proposed for the design of two-dimensional digital FIR differentiators. This technique uses an approximation based on two methods: the Chebyshev series and the Fourier series, which, finally, lead to a trigonometric polynomial, which is a remarkably precise approximation of the transfer function of the ideal differentiator. The digital differentiator is applied to three test images, one greyscale image and two binary images, and simulation results show its performance in the processing task. Also, based on the fact that this 2D differentiator is separable on the two frequency axes, we propose an efficient implementation at the system level, using polyphase filtering. The designed digital differentiator is very accurate and efficient, having a high level of parallelism and reduced computational complexity. Full article

This paper solves a nonhomogeneous version of the pantograph equation. The nonhomogeneous term is taken as a polynomial of degree n with arbitrary coefficients. The nonhomogeneous pantograph equation is successfully converted to the standard homogeneous version by means of a simple transformation. An explicit formula is derived for the coefficients of the assumed transformation. Accordingly, the solution of the nonhomogeneous version is obtained in different forms in terms of power series, in addition to exponential functions. The obtained solution in power-series form is investigated to produce exact solutions for several examples under specific relationships between the involved parameters. In addition, exact solutions in terms of trigonometric and hyperbolic functions are determined at a certain value of the proportional delay parameter. The obtained results may be reported for the first time for the present nonhomogeneous version of the pantograph equation and can be further applied to include other versions with different nonhomogeneous terms. Full article