Fourier series

About: Fourier series is a research topic. Over the lifetime, 16548 publications have been published within this topic receiving 322486 citations.

Combined FEM and Green's function analysis of periodic SAW structure, application to the calculation of reflection and scattering parameters

Abstract: Because of more and more stringent requirements on SAW filter performances, it is important to compute, with very good accuracy, the SAW propagation characteristics, which include the calculation of reflection and scattering parameters For that reason, the analysis of periodic structures on a semi-infinite piezoelectric substrate is one of the most important problems being investigated by SAW researchers For infinite periodic grating modeling, we developed numerical mixed FEM/BEM (finite element method-boundary element method) models using an efficient interpolation basis function that takes into account the singularity at both edges of each electrode In this paper, a review of the numerical program that has been developed during the past few years will be presented For an infinite periodic grating, it is convenient to solve the propagation problem in the Fourier domain (wave number space and harmonic excitation), and important efforts have been spent to properly integrate the so-called periodic harmonic Green function Using this numerical model together with the general P-matrix formalism, it is possible to compute all of the basic parameters with a very good accuracy These consist of the single strip reflectivity, acoustic wave-phase velocity, and position offset between reflection and transduction centers Simulations and comparisons with experiments are shown for each model

A series solution for the in-plane vibration analysis of orthotropic rectangular plates with non-uniform elastic boundary constraints and internal line supports

Abstract: In this investigation, the free in-plane vibration analysis of orthotropic rectangular plates with non-uniform boundary conditions and internal line supports is performed with a modified Fourier solution. The exact solution for the problem is obtained using improved Fourier series method, in which both two in-plane displacements of the orthotropic rectangular plates are represented by a double Fourier cosine series and four supplementary functions, in the form of the product of a polynomial function and a single cosine series expansion, introduced to remove the potential discontinuities associated with the original displacement functions along the edges when they are viewed as periodic functions defined over the entire x–y plane. The unknown expansion coefficients are treated as the generalized coordinates and determined using the Rayleigh–Ritz procedure. The change of the boundary conditions can be easily achieved by only varying the stiffness of the two sets of the boundary springs at the all boundary of the orthotropic rectangular plates without the need of making any change to the solutions. The excellent accuracy of the current result is validated by comparison with those obtained from other analytical approach as well as the finite element method (FEM). Numerical results are presented to illustrate the current method that is applied not only to the homogeneous boundary conditions but also to other interesting and practically important boundary restraints on free in-plane vibrations of the orthotropic rectangular plates, including varying stiffness of boundary springs, point supported, partially supported boundary conditions and internal line supports. In addition to this, the effects of locations of line supports are also investigated and reported. New results for free vibration of moderately orthotropic rectangular plates with various edge conditions and internal line supports are presented, which may be used for benchmarking of researchers in the field.

Circular nodes in neural networks

Abstract: In the usual construction of a neural network, the individual nodes store and transmit real numbers that lie in an interval on the real line; the values are often envisioned as amplitudes. In this article we present a design for a circular node, which is capable of storing and transmitting angular information. We develop the forward and backward propagation formulas for a network containing circular nodes. We show how the use of circular nodes may facilitate the characterization and parameterization of periodic phenomena in general. We describe applications to constructing circular self-maps, periodic compression, and one-dimensional manifold decomposition. We show that a circular node may be used to construct a homeomorphism between a trefoil knot in â??3 and a unit circle. We give an application with a network that encodes the dynamic system on the limit cycle of the Kuramoto-Sivashinsky equation. This is achieved by incorporating a circular node in the bottleneck layer of a three-hidden-layer bottleneck network architecture. Exploiting circular nodes systematically offers a neural network alternative to Fourier series decomposition in approximating periodic or almost periodic functions.

Simulation and experimental results of multiharmonic least-squares fitting algorithms applied to periodic signals

Abstract: A new generation of multipurpose measurement equipment is transforming the role of computers in instrumentation. The new features involve mixed devices, such as analog-to-digital and digital-to-analog converters and digital signal processing techniques, that are able to substitute typical discrete instruments like multimeters and analyzers. Signal-processing applications frequently use least-squares (LS) sine-fitting algorithms. Periodic signals may be interpreted as a sum of sine waves with multiple frequencies: the Fourier series. This paper describes an algorithm that is able to fit a multiharmonic acquired signal, determining the amplitude and phase of all harmonics. Simulation and experimental results are presented.

Exact boundary controllability of 1-d parabolic and hyperbolic degenerate equations ∗

Abstract: The boundary controllability of a class of one-dimensional degenerate equations is studied in this paper. The novelty of this work is that the control acts through the part of the boundary where degeneracy occurs. First we consider a class of degenerate hyperbolic equations. Then, we prove sharp observability estimates for these equations using nonharmonic Fourier series. The transmutation method yields a result for the corresponding class of degenerate parabolic equations.