Discrete-time Fourier transform

About: Discrete-time Fourier transform is a research topic. Over the lifetime, 5072 publications have been published within this topic receiving 144643 citations. The topic is also known as: DTFT.

On the Application of Spectral Filters in a Fourier Option Pricing Technique

Abstract: When Fourier techniques are applied to specific problems from computational finance with nonsmooth functions, the so-called Gibbs phenomenon may become apparent. This seriously affects the efficiency and accuracy of the numerical results. For example, the variance gamma asset price process gives rise to algebraically decaying Fourier coefficients, resulting in a slowly converging Fourier series. We apply spectral filters to achieve faster convergence. Filtering is carried out in Fourier space; the series coefficients are pre-multiplied by a decreasing function. This does not add any significant computational costs. Tests with different filters show how the algebraic index of convergence is improved.

General optical implementations of fractional Fourier transforms

Abstract: General optical setups that implement the fractional Fourier transforms are proposed by use of the impulse response theory. These architectures are demonstrated to be flexible in practical spatially variant filtering systems that employ cascaded multiple stages of fractional Fourier transforms, because of their capabilities of changing the standard focal length.

Simulation of digital earthquake accelerograms using the inverse discrete Fourier transform

Abstract: Acceleration time histories of horizontal earthquake ground motion are obtained by inverting the discrete Fourier transform, which is defined by modelling the probability distribution of the Fourier phase differences conditional on the Fourier amplitude. The Fourier amplitude spectrum is modelled as a scaled, lognormal probability density function. Three parameters are necessary to define the Fourier amplitude spectrum. They are the total energy of the accelerogram, the central frequency, and the spectral bandwidth. The Fourier phase differences are simulated conditional on the Fourier amplitudes. The amplitudes are classified into three categories: small, intermediate and large. For each amplitude category, a beta distribution or a combination of a beta distribution and a uniform distribution are defined for the phase differences. Seven parameters are needed to completely define the phase difference distributions: two for each of the three beta distributions, and the weight of the uniform distribution for phase differences corresponding to small Fourier amplitudes. Approximately 300 uniformly processed horizontal ground motion records from recent California earthquakes are used to develop prediction formulas for the model parameters, as well as to validate the simulation model. The moment magnitude of the earthquakes ranges from 5.8 to 7.3. The source to site distance for all the records is less than 100 km. Copyright © 2002 John Wiley & Sons, Ltd.

Signals, Systems, Transforms, and Digital Signal Processing with MATLAB

Abstract: Signals, Systems, Transforms, and Digital Signal Processing with MATLAB has as its principal objective simplification without compromise of rigor. Graphics, called by the author, "the language of scientists and engineers", physical interpretation of subtle mathematical concepts, and a gradual transition from basic to more advanced topics are meant to be among the important contributions of this book. After illustrating the analysis of a function through a step-by-step addition of harmonics, the book deals with Fourier and Laplace transforms. It then covers discrete time signals and systems, the z transform, continuous- and discrete-time filters, active and passive filters, lattice filters, and continuous- and discrete-time state space models. The author goes on to discuss the Fourier transform of sequences, the discrete Fourier transform, and the fast Fourier transform, followed by Fourier-, Laplace, and z-related transforms, including WalshHadamard, generalized Walsh, Hilbert, discrete cosine, Hartley, Hankel, Mellin, fractional Fourier, and wavelet. He also surveys the architecture and design of digital signal processors, computer architecture, logic design of sequential circuits, and random signals. He concludes with simplifying and demystifying the vital subject of distribution theory. Drawing on much of the authors own research work, this book expands the domains of existence of the most important transforms and thus opens the door to a new world of applications using novel, powerful mathematical tools.

A Fourier method for the numerical solution of Poisson’s equation

Abstract: A method for the solution of Poisson's equation in a rectangle, based on the relation between the Fourier coefficients for the solution and those for the right-hand side, is developed. The Fast Fourier Transform is used for the computation and its influence on the accuracy is studied. Error estimates are given and the method is shown to be second order accurate under certain general conditions on the smoothness of the solution. The accuracy is found to be limited by the lack of smoothness of the periodic extension of the inhomogeneous term. Higher order methods are then derived with the aid of special solutions. This reduces the problem to a case with sufficiently smooth data. A comparison of accuracy and efficiency is made between our Fourier method and the Buneman algorithm for the solution of the standard finite difference formulae.