Showing papers on "Fractional Fourier transform published in 1986"

Phase-retrieval stagnation problems and solutions

Abstract: The iterative Fourier-transform algorithm has been demonstrated to be a practical method for reconstructing an object from the modulus of its Fourier transform (i.e., solving the problem of recovering phase from a single intensity measurement). In some circumstances the algorithm may stagnate. New methods are described that allow the algorithm to overcome three different modes of stagnation: those characterized by (1) twin images, (2) stripes, and (3) truncation of the image by the support constraint. Curious properties of Fourier transforms of images are also described: the zero reversal for the striped images and the relationship between the zero lines of the real and imaginary parts of the Fourier transform. A detailed description of the reconstruction method is given to aid those employing the iterative transform algorithm.

The Hartley transform

Abstract: The author describes the fast algorithm he discovered for spectral analysis and indeed any purpose to which Fourier Transforms and the Fast Fourier Transform are normally applied.

Fringe-pattern analysis using a 2-D Fourier transform

Abstract: A refinement of the Fourier transform fringe-pattern analysis technique which uses a 2-D Fourier transform is described. The 2-D transform permits better separation of the desired information components from unwanted components than a 1-D transform. The accuracy of the technique when applied to real data recorded by a system with a nonlinear response function is investigated. This leads to simple techniques for optimizing an interferogram for analysis by these Fourier transform methods and to an estimate of the error in the retrieved fringe shifts. This estimate is tested on simulated data and found to be reliable.

Modeling the noise influence on the Fourier coefficients after a discrete Fourier transform

Abstract: An analysis is made to study the influence of time-domain noise on the results of a discrete Fourier transform (DFT). It is proven that the resulting frequency-domain noise can be modeled using a Gaussian distribution with a covariance matrix which is nearly diagonal, imposing very weak assumptions on the noise in the time domain.

Fast two-dimensional Hartley transform

Abstract: The fast Hartley transform algorithm introduced in 1984 offers an alternative to the fast Fourier transform, with the advantages of not requiring complex arithmetic or a sign change of i to distinguish inverse transformation from direct. A two-dimensional extension is described that speeds up Fourier transformation of real digital images.

Novel Properties Of The Fourier Decomposition Of The Sinogram

Abstract: The double Fourier decomposition of the sinogram is obtained by first taking the Fourier transform of each parallel-ray projection and then calculating the coefficients of a Fourier series with respect to angle for each frequency component of the transformed projections. The values of these coefficients may be plotted on a two-dimensional map whose coordinates are spatial frequency w (continuous) and angular harmonic number n (discrete). For |w| large, the Fourier coefficients on the line n=kw of slope k through the origin of the coefficient space are found to depend strongly on the contributions to the projection data that, for each view, come from a certain distance to the detector plane, where the distance is a linear function of k. The values of these coefficients depend only weakly on contributions from other distances from the detector. The theoretical basis of this property is presented in this paper and a potential application to emission computerized tomography is discussed.

Fast computation of discrete cosine transform through fast Hartley transform

Abstract: A relationship between the discrete cosine transform (DCT) and the discrete Hartley transform (DHT) is derived. It leads to a new fast and numerically stable algorithm for the DCT.

Fast algorithms for the multidimensional discrete Fourier transform

Abstract: In this paper, the prime factor algorithm for the evaluation of a one-dimensional discrete Fourier transform is generalized to the evaluation of multidimensional discrete Fourier transforms defined on arbitrary periodic sampling lattices. It is shown that such an algorithm is equivalent in computational complexity to the evaluation of a rectangular discrete Fourier transform. As a sidelight to the derivation of the algorithm, a Chinese remainder theorem is derived for integer lattices.

Conversion of frequency-domain data to the time domain

Abstract: The advantages and disadvantages of three different algorithms for transforming frequency-domain data to the time domain are reviewed The algorithms are a direct computation of the Fourier series, the fast Fourier transform, and the chirp-z transform It is concluded that the fast Fourier transform still has the advantage of speed, but the chirp-z transform offers some additional flexibility that makes it more useful in many applications

Calculation of geometric moments using Fourier plane intensities.

Abstract: A new system for calculating the geometric moments of an input image is presented. The system is based on a mathematical derivation that relates the geometric moments of the input image to the intensity of the Fourier transform of the image. Since optical systems are very efficient at obtaining Fourier transforms, an optical system is used to provide the transform of the input image in this design. An array of detectors is then used to sample the Fourier plane, and a digital postprocessor performs a differentiation process on these Fourier magnitude samples to obtain a vector of values which are combined in a predetermined fashion to provide the geometric moments of the original input function.

Solution to some electromagnetic problems using fast Fourier transform with conjugate gradient method

Abstract: Using the fast Fourier transform (FFT) to compute the convolution integrals that appear in the conjugate-gradient method (CGM), an efficient numerical procedure to solve electromagnetic problems is obtained. In comparison with the method of moments (MM), the proposed FFT-CGM avoids the storage of large matrices and reduces the computer time by orders of magnitude.

Doubly pipelined Cordic array for digital signal processing algorithms

Abstract: In this paper, we present a doubly pipelined VLSI Cordic array processor for digital signal processing computations The basic notion of doubly pipelined CORDIC computation will be introduced first Then, some potential applications to digital signal processing problems will be discussed Specifically, we shall demonstrate how a doubly pipelined CORDIC processor array can be applied to compute discrete Fourier transform and Fast Fourier transform, to implement Lattice filters, to solve Toeplitz systems as well as matrix QR factorizations It is shown that by adopting a secondary pipelining, about one third hardware can be saved, and sometimes the throughput of the entire CORDIC processor array may be doubled

On the spatial Fourier transforms of the Pascal-Sierpinski gaskets

Abstract: It has been shown here that the Fourier transform, which can be obtained experimentally, of an evolving fractal can be manipulated to yield the fractal dimension. Although only the Pascal-Sierpinski gaskets have been considered here, because of their generality, it is expected that this technique can be utilized to identify other planar fractals as well.

Signals and Systems: Models and Behaviour

Abstract: Part 1 Introduction to signals and systems: signal classification signal processing systems linearity and time-invariance signal types and definitions signal symmetry and orthogonality signal sampling. Part 2 Time-domain models: discrete-time systems unit-sample response and convolution convolution for continuous systems. Part 3 Frequency-domain models: the frequency-domain approach the Fourier transform Fourier transforms of signals input-output relationships symmetry properties the inverse Fourier transform. Part 4 Laplace transforms: the Laplace integral Laplace model of signals properties of Laplace transforms the system transform function pole-zero models. Part 5 Z-transforms: the Z-transform the transfer function system response pole-zero models frequency response of a discrete-time system. Chapter 6 Periodic signals: strictly periodic signals the Fourier exponential series Fourier series and the Fourier integral input-output relationships band limited signals. Part 7 Fourier analysis of discrete-time signals and systems: the Laplace and Z-transforms the Fourier transform and the DTFT further properties of the DTFT signal sampling and aliasing frequency resolution the discrete Fourier transform. Appendices: A short table of Laplace transform pairs some Laplace transform properties some Z-transform pairs.

Optical coherence calculations with the split-step fast Fourier transform method.

Abstract: Björn Hermansson is with Swedish Telecommunications Administration, Technology Department, S-123 86 Farsta, Sweden; D. Yevick is with Pennsylvania State University, Department of Electrical Engineering, University Park, Pennsylvania 16802, and A. T. Friberg is with Helsinki University of Technology, Department of Technical Physics, SF-02150 Espoo 15, Finland. Received 4 February 1986. 0003-6935/86/162645-02$02.00/0. © 1986 Optical Society of America. Although a large number of problems involving the paraxial propagation of spatially coherent electric fields have recently been analyzed with the aid of the split-step fast Fourier transform (SSFFT) method, the technique has not yet been applied to incoherent or partially coherent light beams. An obvious reason for this omission is that to describe incoherent light propagation through an inhomogeneous optical medium with the SSFFT, many individual realizations of the noncoherent electric field must be propagated, resulting in an unreasonable expenditure of computer time. A resolution of this difficulty, recently proposed by two of us (Yevick and Hermansson), is to reformulate the SSFFT in terms of Green's function matrices as follows. First, an equidistant set of N transverse grid points χ1 = XL, X2 = xL + Δ χ , . . . ,ΧΝ = xL + (N 1)Δχ is specified along a line or plane at z = 0 perpendicular to the optical axis. For a given transverse grid point Xj, we consider the electric field distribution Ei(xj) = δij, where δij, is the Kronecker delta function which is one on the given grid point and zero on the remaining points. After propagating this electric field a distance Z through an inho­ mogeneous optical medium with the aid of the SSFFT, we obtain an output electric field vector Ep(xq). The SSFFT method involves repeated application of a three-step procedure according to which the electric field is first fast Fourier transformed and propagated a distance Δz/ 2 in a homogeneous medium with a refractive index equal to some representative index n0 of the optical medium. Next, the field is inverse Fourier transformed and multiplied by a phase term obtained by exponentiating i n 0 zπ/λ times the average of [n(x,y,z)/n0 1] over the interval zi < z < zi + Δz. Finally the field is propagated again in the homogeneous medium a distance z/2. Here λ denotes the vacuum wave­ length of the monochromatic incoming light beam. Special­ izing for simplicity to a 2-D system, we may summarize this procedure in the following formula, valid to order (Δz):

Fourier transform magnitudes are unique pattern recognition templates

Abstract: Fourier transform magnitudes are commonly used in the generation of templates in pattern recognition applications. We report on recent advances in Fourier phase retrieval which are relevant to pattern recognition. We emphasise in particular that the intrinsic form of a finite, positive image is, in general, uniquely related to the magnitude of its Fourier transform. We state conditions under which the Fourier phase can be reconstructed from samples of the Fourier magnitude, and describe a method of achieving this. Computational examples of restoration of Fourier phase (and hence, by Fourier transformation, the intrinsic form of the image) from samples of the Fourier magnitude are also presented.

Accuracy of angular spectral analysis with an anamorphic Fourier transformer

Abstract: Angular measurements in symmetrical and nonsymmetrical Fourier spectra are compared. The coefficient of angular magnification of a spectrum and the effective angular extent of a scanning wedge filter are introduced. Better accuracy of angular spectral analysis with an anamorphic Fourier transformer is explained and experimentally proved.

The Laplace Transform and the z-transform

Abstract: The Laplace Transform and the z-transform are closely related to the Fourier Transform, and to our work in the two preceding chapters. There are several good reasons for covering these additional transforms in a book on electronic signals and systems. The Laplace Transform is somewhat more general in scope than the Fourier Transform, and is widely used by engineers for describing continuous circuits and systems, including automatic control systems. It is particularly valuable for analysing signal flow through causal LTI systems with nonzero initial conditions. The Laplace Transform also overcomes some of the convergence problems associated with the continuous-time Fourier Transform, and can handle a broader class of signal waveforms. The z-transform, on the other hand, is especially suitable for dealing with discrete signals and systems. It offers a more compact and convenient notation than the discrete-time Fourier Transform. The Laplace Transform and the z-transform should be viewed as complementary to the Fourier Transform, rather than essentially different. Since we have already covered much of the relevant conceptual framework, we will concentrate on those features of the transforms which shed additional light on electronic systems.

The performance of a hybrid videoconferencing coder using displacement estimation in the transform domain

Abstract: In this paper we extend the transform domain oriented estimation algorithm introduced in [1] in which the calculation of the displacement vector was obtained from the transform domain coefficients. The performance of the algorithm is verified within a hybrid coding configuration. In this paper only transform domain block matching algorithms are considered. The block-match procedure makes use of the displacement matrix H defined in [1]. A matrix decomposition method is described in order to show that a practical implementation is very well possible. The properties of the translation invariant matrices are explained by using the ordered Walsh Hadamard transform as an example. The procedure however enables the use any other orthogonal transform. An important issue with respect to the hardware complexity of this motion compensated hybrid coder is the use of only one transform. The performance of the proposed new algorithm is shown and a video tape containing a very critical videoconferencing scene (i.e. split screen and a hard switch to full screen with heavy motion) will be presented. The sequences are coded at a bitrate of 384 kbit/s and 64 kbit/s respectively. Results of the compensation in the pixel domain and in the transform domain are compared and it is shown that the latter approach results in an improved performance of the estimation procedure.

Analysis of Electrical Networks

Abstract: The Laplace Transform The Laplace Transform in Circuit Analysis External Characterization of Networks Interconnections of Networks Network Topology Network Functions and Responses Convolution and Superposition Frequency Response and Graphical Methods State Variables I: Formulation State Variables II: Solution Stability Periodic Waveforms and the Fourier Series The Fourier Transform Time Varying Networks Appendixes Index.

Optimal algorithm for computing the two-dimensional discrete fourier transform.

Abstract: Since for each t ∈ [1, N/2], we have W t+N/2 = −W , one can also represent component (1) at the point (p, s) by the corresponding N/2-dimensional vector F̄ ′ p,s = (f ′ p,s,1, f ′ p,s,2, ..., f ′ p,s,N/2), whose components are calculated from the components of the corresponding initial vector F̄p,s by formula f ′ p,s,t = fp,s,t − fp,s,t+N/2, t = 1 ÷ N/2. (5) We call such representation of each component Fp,s of the spectrum in the form of the corresponding N/2-dimensional vector F̄ ′ p,s the paired vector representation, to distinct it from the original vector representation F̄p,s, and the constructed tensor of the 3rd order (f ′ p,s,t; p, s, = 1 ÷ N, t = 1 ÷ N/2 to be the paired tensor of the Fourier-spectrum. As for the original tensor representation of the spectrum of the signal, when for any p, s and k the following formula was valid [1]