On the Fourier transform of sequency limited signals with applications to delta modulation and medicine†

It is the binary nature of the Walsh basis functions that make the Walsh transform a potentially useful signal processing tool. Unlike the FFT, which produces a database in the easily interpreted frequency domain, a Walsh data base resides in the obscure sequency domain. Using a fast Walsh algorithm, an N-point transform can, however, be computed more rapidly than a FFT using a similar computer architecture. Even though the sequency space has been shown to be useful in coding and picture processing applications, the aperiodic behaviour of its basis functions make it. unsuitable for many traditional signal-processing problems. It would be desirable, for example, to be able to relate the computationally efficient Walsh spectra to the intuitively pleasing Fourier spectra. This work will address that problem and present new results.     

Research progress of the fractional Fourier transform in signal processing

While solving a heat conduction problem in 1807, a French scientist Jean Baptiste Jo-seph Fourier, suggested the usage of the Fourier theorem. Thereafter, the Fourier trans-form (FT) has been applied widely in many scientific disciplines, and has played i…     

On Factorizing the Symbolic U-resultant — Application of the ddet operator —

This paper presents a new, practical, efficient algorithm for factorizing the U-resultant of a system of algebraic equations with finitely many solutions. Given a matrix whose determinant is the U-resultant, our algorithm obtains the true linear factors with exact multiplicities, directly from the matrix without expanding the multivariate determinant. The main focuses are laid upon the use of a new operator, which enables us to treat only matrices of univariate polynomial elements, and also upon the exact treatment of multiplicities, which is made possible by the symbolic representation of solutions in simple algebraic extensions of Q. The algorithm is probabilistic in the sense that there exists no deterministic method to give an appropriate parameterization. The efficiency of our algorithm will be demonstrated with some empirical problems.     

Evaluation of a linear system impulse response and its Fourier transform†

Impulse response of a linear time invariant system is partitioned by dividing the time axis into equal intervals of time. Then the impulse response is expressed as a sum of these partitioned portions. Each individual portion is approximated by a finite sum of orthogonalized sinusoids satisfying integral squared error criteria. Four different sets are given for this purpose. If the time reversed functions from these sets are applied to the system then the sampled values of the system response at the partitioning instants directly yield the system coefficients as required for the least integral squared error. Knowing these coefficients the best approximation to the impulse response can be constructed as illustrated by the examples considered. Sampled values of the Fourier transform of system impulse response are obtained as a by-product.     

Theory and algorithms for two-dimensional warped discrete Fourier transform

In this paper, the two-dimensional Warped Discrete Fourier Transform （2-D WDFT） is developed based on the concept of the 1-D WDFT. An exact computation algorithm is developed for 2-D WDFT based on matrix factorizing with special structure. A fast algorithm is then proposed to reduce greatly the computational complexity of the inverse 2-D WDFT. Finally, numerical examples are given to show the efficiency of the proposed approach.     

On the numerical solution of differential equations of Lane–Emden type

In this paper, a numerical method which produces an approximate polynomial solution is presented for solving Lane–Emden equations as singular initial value problems. Firstly, we use an integral operator (Yousefi (2006) [4]) and convert Lane–Emden equations into integral equations. Then, we convert the acquired integral equation into a power series. Finally, transforming the power series into Padé series form, we obtain an approximate polynomial of arbitrary order for solving Lane–Emden equations. The advantages of using the proposed method are presented. Then, an efficient error estimation for the proposed method is also introduced and finally some experiments and their numerical solutions are given; and comparing between the numerical results obtained from the other methods, we show the high accuracy and efficiency of the proposed method.     

On the fast and fully discretized solution¶of integral and pseudo-differential equations¶on smooth curves

For the fast numerical solution of a fully discrete variant of the trigonometric Galerkin equations associated with periodic integral equations, we consider approximations with small residuals and provide order-optimal estimates for the associated error. The CGNR method is considered as a method with a simple iteration scheme where these approximations can be obtained by a total number of ℳ(N log N ) arithmetical operations, with N denoting the dimension of the space of trigonometric trial polynomials associated with the Galerkin method. Noise in the model of the problem as well as in the right-hand side is admitted. Received: August 1999 / Revised version: July 2000     

On variable-step methods for the numerical solution of Schrödinger equation and related problems

In this paper we present a review for the construction of variable-step methods for the numerical integration of the Schr?dinger equation. Phase-lag and stability are investigated. The methods are variable-step because of a simple natural error control mechanism. Numerical results obtained for coupled differential equations arising from the Schr?dinger equation and for the wave equation show the validity of the approach presented.     

Some local Poincaré inequalities for the composition of the sharp maximal operator and the Green’s operator

In this paper, we establish the local Poincaré-type inequalities for the composition of the sharp maximal operator and the Green’s operator with an Orlicz norm.     

On finite difference methods for the solution of the Schrödinger equation

A review for the numerical methods used for the solution of the Schrödinger equation is presented.     

On the transient solution of the unforced duffing equation with large damping†

The refined elliptic-function method previously developed (Soudack and Barkham 1970) is applied to the unforced Duffing equation with large damping. The theoretical development for this case, and an example, are presented. The Kryloff-Bogoliuboff solution is shown for comparison with the present result.

The advantage of this approach, well illustrated by the example, is in its ability to predict the solution phase. The approach is therefore superior to quasi-linear methods, such as the Kryloff-Bogoliuboff method, which rapidly fails to give a good approximation due to phase errors.     

Evaluation of Fourier transform of impulse response of linear time invariant systems†

Evaluation of the Fourior transform of the system impulse response is an important aspect of the design of control systems. A method suggested hero requires only one cycle of sine or cosine wave to be applied as an input to the system. It is proved that the sum of the sampled values of the output response, at the sampling interval equal to the period of the input wave, directly yields the sign and cosine transforms respectively. The procedure is generalized to any number of complete cycles of input wave, as well as to n/2 cycles whore n is any odd positive integer.     

On a collocation B-spline method for the solution of the Navier–Stokes equations

A novel B-spline collocation method for the solution of the incompressible Navier–Stokes equations is presented. The discretization employs B-splines of maximum continuity, yielding schemes with high-resolution power. The Navier–Stokes equations are solved by using a fractional step method, where the projection step is considered as a Div–Grad problem, so that no pressure boundary conditions need to be prescribed. Pressure oscillations are prevented by introducing compatible B-spline bases for the velocity and pressure, yielding efficient schemes of arbitrary order of accuracy. The method is applied to two-dimensional benchmark flows, and mass lumping techniques for cost-effective computation of unsteady problems are discussed.     

On the stability of collocation methods for the two-dimensional Burgers equation — The Fourier case

We investigate Fourier collocation approximations of the evolutionary twodimensional Burgers equation. The numerical schemes are not required to be semi-conservative. We obtain stability estimates in theH ¹() norm that are uniform in time. Our results show that collocation techniques do not yield instability, at least if the resolution is fine enough.     

On kalandiya’ method for the numerical solution of singular integral equations

For the numerical solution of one-dimensional singular integral equations with Cauchy type kernels, one can use an appropriate quadrature rule and an appropriate set of collocation points for the reduction of this equation to a system of linear equations. In this short paper, we use as collocation points the nodes of the quadrature rule and we rederive, in a more direct manner, Kalandiya’ method for the numerical solution of the aforementioned class of equations, which was originally based on a trigonometric interpolation formula. Furthermore, we test this method in numerical applications. Finally, a discussion on the accuracy of the same method is made.