Showing papers on "Fractional Fourier transform published in 1980"

The Fractional Order Fourier Transform and its Application to Quantum Mechanics

Abstract: We introduce the concept of Fourier transforms of fractional order, the ordinary Fourier transform being a transform of order 1. The integral representation of this transform can be used to construct a table of fractional order Fourier transforms. A generalized operational calculus is developed, paralleling the familiar one for the ordinary transform. Its application provides a convenient technique for solving certain classes of ordinary and partial differential equations which arise in quantum mechanics from classical quadratic hamiltonians. The method of solution is first illustrated by its application to the free and to the forced quantum mechanical harmonic oscillator. The corresponding Green's functions are obtained in closed form. The new technique is then extended to three-dimensional problems and applied to the quantum mechanical description of the motion of electrons in a constant magnetic field. The stationary states, energy levels and the evolution of an initial wave packet are obtained by a systematic application of the rules of the generalized operational calculus. Finally, the method is applied to the second order partial differential equation with time-dependent coefficients describing the quantum mechanical dynamics of electrons in a time-varying magnetic field.

Fourier Transform and Its Applications

Abstract: This paper analyses Fourier transform used for spectral analysis of periodical signals and emphasizes some of its properties. It is demonstrated that the spectrum is strongly depended of signal duration that is very important for very short signals which have a very rich spectrum, even for totally harmonic signals. Surprisingly is taken the conclusion that spectral function of harmonic signals with infinite duration is identically with Dirac function and more of this no matter of duration, it respects Heisenberg fourth uncertainty equation. In comparison with Fourier series, the spectrum which is emphasized by Fourier transform doesn’t have maximum amplitudes for signals frequencies but only if the signal lasting a lot of time, in the other situations these maximum values are strongly de-phased while the signal time decreasing. That is why one can consider that Fourier series is useful especially for interpolation of nonharmonic periodical functions using harmonic functions and less for spectral analysis. Key-Words — signals, Fourier transform, continuous spectrum properties, Quantum Physics, Fourier series, discrete spectrum

Time-frequency representation of digital signals and systems based on short-time Fourier analysis

Abstract: This paper develops a representation for discrete-time signals and systems based on short-time Fourier analysis. The short-time Fourier transform and the time-varying frequency response are reviewed as representations for signals and linear time-varying systems. The problems of representing a signal by its short-time Fourier transform and synthesizing a signal from its transform are considered. A new synthesis equation is introduced that is sufficiently general to describe apparently different synthesis methods reported in the literature. It is shown that a class of linear-filtering problems can be represented as the product of the time-varying frequency response of the filter multiplied by the short-time Fourier transform of the input signal. The representation of a signal by samples of its short-time Fourier transform is applied to the linear filtering problem. This representation is of practical significance because there exists a computationally efficient algorithm for implementing such systems. Finally, the methods of fast convolution age considered as special cases of this representation.

A weighted overlap-add method of short-time Fourier analysis/Synthesis

Abstract: In this correspondence we present a new structure and a simplified interpretation of short-time Fourier synthesis using synthesis windows. We show that this approach can be interpreted as a modification of the overlap-add method where we inverse the Fourier transform and window by the synthesis window prior to overlap-adding. This simplified interpretation results in a more efficient structure for short-time synthesis when a synthesis window is desired. In addition, we show how this structure can be used for analysis/synthesis applications which require different analysis and synthesis rates, such as time compression or expansion.

Recursive discrete Fourier transformation

Abstract: This paper presents new discrete Fourier transform methods which are recursive, expressible in state variable form, and which involve real number computations. The algorithms are especially useful for running Fourier transformation and for general and multirate sampling. Numerical examples are given which illustrate the ability of these spectral observers to operate at sampling rates other than the Nyquist rate, to perform one-step-per-sample updating, and to converge to the spectrum in the presence of severe numerical truncation error.

An integral transform related to quantization

Abstract: We study in some detail the correspondence between a function f on phase space and the matrix elements. (Qf)(a,b) of its quantized Qf between the coherent states ‖ a〉 and ‖ b〉. It is an integral transform: Qf(a,b) = F{a,b ‖ v}f(v) dv, which resembles in many ways the integral transform of Bargmann. We obtain the matrix elements of Qf between harmonic oscillator states as the Fourier coefficients of f with respect to an explicit orthonormal system.

Computation of the Hankel transform using projections

Abstract: In this paper two new algorithms for computing an nth‐order Hankel transform are proposed. The algorithms are based on characterizing a circularly symmetric function and its two‐dimensional Fourier transform by a radial section and interpreting the Hankel transform as the relationship between the radial section in the two domains. By utilizing the property that the projection of a two‐dimensional function in one domain transforms to a radial section in the two‐dimensional Fourier transform or inverse Fourier transform domain, several efficient procedures for computing the Hankel transform exploiting the one‐dimensional FFT algorithm are suggested.

On an extension problem, generalized fourier analysis, and an entropy formula

Abstract: Let h(t) be an n × n matrix valued function on the interval |t| ⩽ τ with summable entries. Let ĥ denote the Fourier transform of h and let e denote the n × n identity matrix. Necessary and sufficient conditions for the existence of an extension u of h to the full line such that e-u admits either a left or a right canonical factorization and the inverse transform of (e-u)−1-e vanishes for |t| ⩾ τ are presented and discussed. The connections between these extensions and a generalized Fourier transform are then explored in detail with the help of the theory of triangular factorization. It is then shown that if an allied finite Wiener-Hopf operator based on h is positive, then h admits exactly one extension of the type alluded to above. This extension is then characterized in terms of an entropy integral.

High-resolution narrow-band spectra by FFT pruning

Abstract: A new method of computing high-resolution narrow-band spectra faster than the chirp z transform (CZT) and direct computation of discrete Fourier transform (DFT) is presented. This is achieved by a generalization of Markel's pruning algorithm and in combination with Skinner's pruning algorithm for the decimation-in-time FFT formulation. However, for very high resolutions it is shown that the CZT is selectively superior to the new method.

Ambiguity of the phase-reconstruction problem

Abstract: Is it possible to determine a function with a finite support from the modulus of its Fourier transform? This problem, the so-called phase problem, is studied theoretically and numerically. It is shown theoretically that, at least for a wide class of functions, such determination is not possible. The theory developed in this Letter is essentially two dimensional. Examples are given and studied numerically.

Fourier and laplace transforms

Abstract: This chapter discusses Fourier and Laplace transforms. The inversion of the Laplace transform is accomplished for analytic functions f (p) of order O (p -k ) with k > 1 by means of the inversion integral. The inversion of the Fourier transform is accomplished by means of the inversion integral. The Fourier sine and cosine transforms of the function f(x), denoted by F s (ξ) and F c (ξ), respectively, are defined by the integrals. The functions f(x) and F s ( ξ ) are called a Fourier sine transform pair, and the functions f(x) and F c ( ξ ) a Fourier cosine transform pair, and knowledge of either F s ( ξ ) or F c ( ξ ) enables f(x) to be recovered. The inversions of the Fourier sine and Fourier cosine transform are accomplished by means of the inversion integral.

Fast Fourier Transform for Step-Like Functions: The Synthesis of Three Apparently Different Methods

Abstract: In 1965 Cooley and Tukey published an algorithm for rapid calculation of the discrete Fourier transform (DFT), a particularly convenient calculating technique, which can well be applied to impulse-like functions whose beginning and end lie at the same level. Independently, various propositions were made to overcome the truncation error which arises, if a step-like function, i.e. one whose end level differs from its starting level, is treated in the same way. It was argued that they behave differently under the influence of noise, band-limited violation, and other experimental inconveniences. The aim of this paper is to show that the three widely and satisfactorily used techniques of Samulon, Nicolson, and Gans, which originate from apparently different ideas, are exactly the same. An extended DFT and fast Fourier transform (FFT) formula is deduced which is adapted as well to impulse-like as to step-like functions.

Numerical computation of the fourier transform using Laguerre functions and the Fast Fourier Transform

Abstract: In this paper we propose a numerical technique for the computation of Fourier transforms. It uses a bilateral expansion of the unknown transformed function with respect to Laguarre functions. The expansion coefficients are obtained via trigonometric interpolation and may be computed very efficiently by means of the Fast Fourier Transform. The convergence of the algorithm is analyzed and numerical results are presented which confirm that it works well.

On the spectral decomposition of stationary time series using walsh

Abstract: The paper looks at the asymptotic properties of the finite Walsh-Fourier transform applied to a discrete-time stationary time series, and shows that in many ways we have analogous results to those obtained when using the finite trigonometric Fourier transform.

A stepwise approach to computing the multidimensional fast Fourier transform of large arrays

Abstract: We consider the problem of performing a two-dimensional fast Fourier transform (FFT) on a very large matrix in limited core memory. We propose a decomposition of the Cooley-Tukey algorithm to allow efficient utilization of core memory and mass storage. The number of input/output operations is greatly reduced, with no increase in the computational burden. The method is suitable for nonsquare matrices and arrays of three or more dimensions.

Phase-only signal reconstruction

Abstract: In this paper, we develop a set of conditions under which a sequence is uniquely specified by the phase or samples of the phase of its Fourier transform. These conditions are applicable to mixed-phase one-dimensional and multi-dimensional sequences. Under the specified conditions, we also present several algorithms which may be used to reconstruct a sequence from its phase.

Signal duration and the Fourier transform

Abstract: Using the variance measure of duration, it is shown that the duration of a signal is composed of two terms. The first term is the duration of the zero-phase equivalent signal, and the second term is the variance of the phase derivative of the Fourier transformed signal.

Apparatus for computing two-dimensional discrete Fourier transforms

Abstract: An apparatus for computing the two-dimensional discrete Fourier transform (DFT) of an image comprised of N×N samples. The samples within each row are respectively multiplied by W-n.sbsp.1, n1 =0, 1, . . . , N-1 and stored in a memory 17. A device 20 derives therefrom N polynomials of N terms by means of a polynomial transform. The terms of each of these polynomials are multiplied by Wn.sbsp.1 and a device 28 computes the one-dimensional DFT thereof, thereby providing the N2 terms of the transform of said image.

On vectorizing the fast fourier transform

Abstract: A variant of the Cooley-Tukey algorithm due to Stockham is derived and vectorized and is shown to be on a par with the Pease algorithm. The Stockham algorithm is then proposed for the entire computation of the two-dimensional fast Fourier transform on a vector computer.

On Fourier Transforms Over Extensions of Finite Rings

Abstract: A divisibility criterion is given for the existence of a Fourier transform over algebraic extensions of the ring of integers mod M which is generally easy to apply. We also present examples of how such transforms may be used to compute two-dimensional convolutions.

Power spectra from approximate transforms

Abstract: Power spectra may be computed via a discrete fourier transform with severely rounded-off trigonometric terms, and recently the ultimate round-off – that is to rectangular waves – has been proposed.1 A comparison is made between this and a 3 level round-off, +1, −1 and 0.