Showing papers on "Fractional Fourier transform published in 1984"

Fast algorithms for the discrete W transform and for the discrete Fourier transform

Abstract: A systematic method of sparse matrix factorization is developed for all four versions of the discrete W transform, the discrete cosine transform, and the discrete sine transform, as well as for the discrete Fourier transform. The factorization leads to fast algorithms in which only real arithmetic is involved. A scheme for reducing multiplications and a convenient index system are introduced. This makes new algorithms more efficient than conventional algorithms for the discrete Fourier transform, the discrete cosine transform, and the discrete sine transform.

Gauss and the history of the fast fourier transform

Abstract: THE fast Fourier transform (Fm has become well known . as a very efficient algorithm for calculating the discrete Fourier Transform (Om of a sequence of N numbers. The OFT is used in many disciplines to obtain the spectrum or . frequency content of a Signal, and to facilitate the computation of discrete convolution and correlation. Indeed, published work on the FFT algorithm as a means of calculating the OFT, by J. W. Cooley and J. W. Tukey in 1965 [1], was a turning point in digital signal processing and in certain areas of numerical analysis. They showed that the OFT, which was previously thought to require N 2 arithmetic operations, could be calculated by the new FFT algorithm using only N log Noperations. This algorithm had a revolutionary effect on many digital processing methods, and remains the most Widely used method of computing Fourier transforms [2]. In their original paper, Cooley and Tukey referred only to I. J. Good's work published in 1958 [3] as having influenced their development. However, It was soon discovered there are major differences between the Cooley-Tukey FFT and the algorithm described by Good, which is now commonly referred to as the prime factor algorithm (PFA). Soon after the appearance of the CooleyTukey paper, Rudnick [4] demonstrated a similar algorithm, based on the work of Danielson and Lanczos [5] which had appeared in 1942. This discovery prompted an investigation into the history of the FFT algorithm by Cooley, Lewis, and Welch [6]. They discovered that the Oanielson-Lanczos paper referred to work by Runge published at the tu rn of the centu ry [7, 8]. The algorithm developed by Cooley and Tukey clearly had its roots in, though perhaps not a direct influence from, the early twentieth century. In a recently published history of numerical analysis [9], H. H. Goldstine attributes to Carl Friedrich Gauss, the eminent German mathematician, an algorithm similar to the FFT for the computation of the coefficients of a finite Fourier series. Gauss' treatise describing the algorithm was not published in his lifetime; it appeared only in his collected works [10] as an unpublished manuscript. The presumed year of the composition of this treatise is 1805, thereby suggesting that efficient algorithms for evaluating

Optical feature extraction via the radon transform.

Abstract: We show that by first measuring the line integrals of a two-dimensional picture f(x, y) via the Radon transform, certain feature-extraction operations useful in pattern recognition may be computed very easily; that is, the computational overhead for these feature-extraction operations is much less in the Radon space than in the direct space. In particular, we consider the following features: (1) moments of f(x, y) invariant to translation, rotation, geometric scaling, and linear contrast scaling; (2) two geometric features, polar projections and convex hull, that have similar invariance properties; (3) the Fourier power spectrum of f(x, y) integrated along radial piewedge and annular bins in the Fourier space; and (4) the Hough transform for detection of straight edges. Much of the motivation for this work lies in its implementation as an optical pattern recognition system. We show an optical-digital hardware implementation for the rapid computation of both the Radon transform and the feature extractions. By "rapid," we mean that the transform and feature-extrac-tion operations on a 512 X 512 image may be accomplished at video rates (1 /30 s per image). We point out advantages of this system over more traditional optical pattern recognition schemes that rely on the use of Fourier optics. Some experimental results are shown.

Numerical evaluation of the compound Poisson distribution: Recursion or Fast Fourier Transform?

Abstract: 1 The problem The finite vector p=(p 1,p 2, ,ps ) defines a probability distribution on the integers 1,2, ,s

Fourier Inversion of the Attenuated X-Ray Transform

Abstract: A variably attenuated x-ray transform is shown to be invertible via an integral formula for the inversion of the exponential x-ray transform.The attenuation must be known and constant in a convex set containing the unknown emitter. However the attenuation can be otherwise arbitrary.If $\mu $ denotes the attenuation constant of the exponential x-ray transform then the integral formula computes the Fourier transform of the emitter on all of $R^n $ from the values of the Fourier transform on the set $A^\mu = \{ {\sigma + i\mu \omega \in C^n |\omega \in S^{n - 1} ,\sigma \bot \omega } \}$. Of course F. Natterer [Numer. Math., 32 (1979), pp. 431–438] showed that the values of the Fourier transform of the emitter can be obtained from the Fourier transform of the exponential x-ray transform. In essence however the basic method is analytic continuation from the set $A^\mu $.A consequence of the integral formula is a uniqueness theorem for attenuated x-ray transforms of the type considered here: if the transforms ...

Short-space Fourier transform image processing

Abstract: The short-space Fourier transform (SSFT) is introduced as a means of describing discrete multi-dimensional signals of finite extent. It is an adaptation of the short-time Fourier transform developed for one-dimensional infinite-duration signals such as speech. By reflectively extending the finite signal segment, one can imagine an infinite duration signal which is "continuous." The proposed SSFT is the multidimensional generalization of the short-time Fourier transform operating upon the resulting infinite duration signal. Because boundary "discontinuities" are avoided, the proposed SSFT provides a transform representation free of extraneous spectral energy. An efficient algorithm for computing the SSET is described. SSFT image coding, an important application of the new transform method, provides localized spectral information without the undesirable phenomenon of "blocking effects."

Multi-dimensional systolic networks, for discrete fourier transform

Abstract: In this paper the problem of computing the Discrete Fourier Transform (DFT) in VLSI is considered. We describe an approach to extend the linear systolic array algorithm to the multidimensional systolic network algorithm. The proposed networks is based on the pipeline design and have regular structure. Among them the mesh-connected network matches, with a small factor, the known theoretical Ω(n2) lower bound to the (area × time2) measure of complexity in the planar VLSI.

Computation of fourier transform integrals using Chebyshev series expansions

Abstract: In this paper, we extend the Clenshaw-Curtis integration method for the computation of Fourier transform integrals. In particular, we examine the numerical stability of a recurrence relation occurring in this method.

A note on iterative fourier transform phase reconstruction from magnitude

Abstract: In this correspondence, a well-known iterative procedure for Fourier transform phase reconstruction from magnitude (usually referred to as the Gerchberg-Saxton-Fienup procedure) is revisited. The convergence of this algorithm has raised some controversy in the recent literature. In this correspondence we first point out a potential source of error in the numerical implementation of the algorithm. Then, we present a conjecture which would explain why this algorithm sometimes fails.

Image reconstruction from frequency domain data on arbitrary contours

Abstract: This paper presents a new view of the problem of image reconstruction in the spatial Fourier domain based on the generalized projections of a two-dimensional image. The algorithms that are presented are derived from the exact continuous relations between the Fourier transforms of a spatially limited image and those of its generalized projections. A Fourier domain reconstruction formula is derived that is a natural consequence of these relations giving values of the two-dimensional transform on a regular grid useful for inverse FFT. This approach is contrasted with the more ad hoc methods of Fourier domain interpolation and with the spatial domain approaches such as the backprojection and backpropagation methods.

Prime factor FFT algorithms for real-valued series

Abstract: This paper presents two techniques for computing a discrete transform of a vector of real-valued data using the Prime Factor Algorithm (PFA) with high-speed convolution. These techniques are applied to the Discrete Fourier Transform (DFT) and the Discrete Hartley Transform (DHT). The primary goals of these techniques are to eliminate unnecessary computations required when implementing a complex transform on a real-valued vector, to compute the transform in-place in the original length-N real vector, and to obtain the transform coefficients in-order. The two algorithms described require modification of the Winograd short-length transform modules to accommodate a real input. One technique replaces the modules in the Burrus-Eschenbacher PFA program with the modified real-input modules and constructs the complete transform in a final step of additions and subtractions after modules for each factor have been executed. The other technique uses these real-input DFT modules for part of the computation associated with each factor and requires complex input DFT modules for another part of the computation. These algorithms require exactly one half of the number of multiplications and slightly less than one half of the number of additions required by a complex-input PFA.

Evaluation of Fourier transform integrals using FFT with improved accuracy and its applications

Abstract: A new technique that significantly minimizes the aliasing error encountered in the conventional use of the fast Fourier transform (FFT) algorithms for the efficient evaluation of Fourier transforms of spatially limited functions (such as those that occur in the radiation pattern analysis of reflector antennas and planar near field to far field (NF-FF) transformation) is presented and illustrated through a typical example. Employing this technique and a discrete Fourier series (DFS) expansion for the integrand, a method for computing the radiation integrals of reflector antennas and planar NF-FF transformation integrals at arbitrary observation angles with optimum use of computer memory and time is also described.

Signal reconstruction from one bit of Fourier transform phase

Abstract: In this paper, we present new results on the reconstruction of signals from one bit of Fourier transform phase, defined as the sign of the real part of the Fourier transform. Specifically, we develop a new theoretical result which shows that most two-dimensional signals can in fact be reconstructed to within a scale factor from only one bit of FF phase. Experimental results showing images reconstructed from one bit of FT phase are also presented.

An algebraic approach to discrete short-time Fourier transform analysis and synthesis

Abstract: An algebraic representation of the discrete short-time Fourier transform (DSTFT) is presented for the case in which the analysis window length N equals the transform block size M . This representation allows the application of algebraic tools for determining an optimal synthesis system which minimizes the mean square error between a given modified DSTFT (which is not necessarily a valid DSTFT sequence) and the DSTFT of the synthesized signal. If no modification is applied, the result is a unity analysis-synthesis system for any given time update R of the sliding analysis window (provided that R\leqM ). It is shown that the optimal synthesis system can be implemented by the well known weighted overlap-add (WOLA) method using an optimal synthesis window. The algebraic approach enables the extension of some recent results and the relaxation of a constraint on the analysis window. The proposed approach is found also to have a potential for solving the synthesis problem for the more general case of N>M .

On the computation of discrete fourier transform using fermat number transform

Abstract: In the paper the results of a study using Fermat number transforms (FNTs) to compute discrete Fourier transforms (DFTs) are presented. Eight basic FNT modules are suggested and used as the basic sequence lengths to compute long DFTs. The number of multiplications per point is for most cases not more than one, whereas the number of shift-adds is approximately equal to the number of additions in the Winograd-Fourier-transform algorithm and the polynomial transform. Thus the present technique is very effective in computing discrete Fourier transforms.

Improvement of the high frequency content of fast Fourier transforms

Abstract: A method is described of improving the high frequency components of the spectrum derived by the fast Fourier transform. It is particularly applicable to the analysis of time series which decay with time such as in stress relaxation. The method involves the addition of components at the same frequency derived from fast Fourier transforms taken over a succession of increasing time intervals. A numerical example is given.

Symbolic network function generation via discrete Fourier transform

Abstract: A new method of symbolic network function generation is presented. The method is based upon the theory of the discrete Fourier transform and not restricted in its application to any particular type of network analysis or network configuration. It is particularly attractive when the number of symbolic variables to be handled is not large.

On window functions

Abstract: A now window function based on modified Bessel functions has been proposed. The Courier transform of the window function is obtained analytically. Computation of the Fourier transform shows a slight improvement over the Kaiser-Bessel window for suitable choice of parameters. The proposed function involves a simpler and faster computational method than that of the Kaiser-Bessel window.

Some Useful Properties of the Hilbert Transform

Abstract: The Hilbert transform is shown to be invariant under certain rational transformations of the integration variable. Examples are provided to show how this leads to new transform pairs.

Hilbert transforms using fast Fourier transforms

Abstract: Fourier transforms are unreliable near discontinuities because of the Gibbs phenomenon. Before using the fast Fourier transform technique to evaluate Hilbert transforms, it is desirable to remove any discontinuities by smoothing, as suggested by Papoulis (1962). This is especially true if one wishes to use Fourier transforms to find the Hilbert transform h(t) of a function f(t) which has an infinite discontinuity: it is then necessary to smooth f(t) as well as the Hilbert transform kernel. An alternative to smoothing f(t) is to remove a discontinuity at a point t=d by multiplying f(t) by the factor (t-d): this has the advantage of having better asymptotic behaviour. A numerical example is given and here the two methods perform about equally.