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Showing papers on "Discrete-time Fourier transform published in 1986"


Book
01 Jan 1986
TL;DR: 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.
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.

437 citations


Journal ArticleDOI
TL;DR: A refinement of the Fourier transform fringe-pattern analysis technique which uses a 2-D Fouriertransform permits better separation of the desired information components from unwanted components than a 1-D 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.

363 citations


Journal ArticleDOI
TL;DR: In this paper, the influence of time-domain noise on the results of a discrete Fourier transform (DFT) was studied and it was shown that the resulting frequency domain noise can be modeled using a Gaussian distribution with a covariance matrix which is nearly diagonal.
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.

150 citations


Journal ArticleDOI
01 Sep 1986
TL;DR: The fast Hartley transform algorithm as discussed by the authors 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.
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.

128 citations


Proceedings ArticleDOI
01 Jan 1986
TL;DR: 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 as discussed by the authors.
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.

125 citations


Journal ArticleDOI
TL;DR: It is shown that such an algorithm is equivalent in computational complexity to the evaluation of a rectangular discrete Fourier transform, and a Chinese remainder theorem is derived for integer lattices.
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.

64 citations


Journal ArticleDOI
01 Jan 1986
TL;DR: 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.
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

39 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that the most natural extension of the DHT to two dimensions fails to be separate in two dimensions, and is therefore inefficient, and an alternative separable form is considered, corresponding convolution theorem is derived.
Abstract: Bracewell has proposed the Discrete Hartley Transform (DHT) as a substitute for the Discrete Fourier Transform (DFT), particularly as a means of convolution. Here, it is shown that the most natural extension of the DHT to two dimensions fails to be separate in the two dimensions, and is therefore inefficient. An alternative separable form is considered, corresponding convolution theorem is derived. That the DHT is unlikely to provide faster convolution than the DFT is also discussed.

35 citations


Journal ArticleDOI
TL;DR: An optical system is used to provide the transform of the input image in this design 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 provided the geometric moments of the original input function.
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.

35 citations



Journal ArticleDOI
TL;DR: 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 as discussed by the authors.
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.

Proceedings ArticleDOI
07 Apr 1986
TL;DR: A considerable increase in accuracy can be obtained with only a small penalty in execution time, by applying an alternating form of rounding rather than truncation to the discrete Fourier transform calculation.
Abstract: The calculation of the discrete Fourier transform using a fast Fourier transform (FFT) algorithm with fixed-point arithmetic is considered. The input data is scaled to prevent overflow and to maintain accuracy. The implementation uses 16-bit fixed-point representation for the data and provides for double precision accumulation of sums and products. Algorithm variants as well as different rounding options are compared. Execution times for implementations based on a single chip signal processor are given. These show that a considerable increase in accuracy can be obtained with only a small penalty in execution time, by applying an alternating form of rounding rather than truncation.

Journal ArticleDOI
TL;DR: In this article, it was shown that the Fourier transform of an evolving fractal can be manipulated to yield the fractal dimension, which can be used to identify planar fractals as well.
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.

Journal ArticleDOI
TL;DR: In this article, the Fourier transform mechanical spectroscopy (FTMS) was used for determining the dynamic mechanical behavior of viscoelastic materials. But the results of using the new method on Nylon over the frequency range 5*10-4 to 20 Hz and temperatures from 25 to 110 degrees C were presented.
Abstract: A powerful new experimental method of determining the dynamic mechanical behaviour of viscoelastic materials is described. The method, Fourier transform mechanical spectroscopy (FTMS), consists of superposing and deconvoluting discrete Fourier transforms obtained from a single stress relaxation experiment using a novel sampling scheme. The theory of the sampling scheme and the superposition of the transforms is described elsewhere. In this paper the authors present the results of using the new method on Nylon over the frequency range 5*10-4 to 20 Hz and temperatures from 25 to 110 degrees C. There is good agreement between the values of storage and loss moduli, obtained by means of the transform method, and those obtained by conventional means. An activation energy of 143 kcal mol-1 is obtained for the relaxation in this temperature range. The new method has considerable potential for the rapid determination of the mechanical properties of novel polymers and composites.

Book
01 Apr 1986
TL;DR: Fourier analysis of discrete-time signals and systems: the Laplace and Z-transforms the Fourier transform and theDTFT further properties of the DTFT signal sampling and aliasing frequency resolution the discrete Fouriertransform.
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.

Journal ArticleDOI
TL;DR: In this paper, the generalized sampling theorem for band-limited functions is shown to admit generalization in the following sense: a bandlimited Fourier transform can be arbitrarily extended on the frequency axis.
Abstract: The classical sampling theorem for band-limited functions is shown to admit generalization in the following sense: a bandlimited Fourier transform can be arbitrarily extended on the frequency axis. If this extension is performed in such a fashion that the result is an almost periodic function, the sampling theorem can be shown to hold for sampling at nonuniformly spaced instants in time. This result is proved, and the generalized sampling theorem is shown to provide a simple, constructive proof of the multichannel sampling theorem. A new sampled data Fourier transform is introduced which retains information about the sampling instants.

Journal ArticleDOI
TL;DR: 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.
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):


Journal ArticleDOI
TL;DR: It is emphasised in particular that the intrinsic form of a finite, positive image is, in general, uniquely related to the magnitude of its Fourier transform.
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.

Proceedings ArticleDOI
01 Apr 1986
TL;DR: This paper develops an approach to deriving special FFT and convolution algorithms, considering the architecture of special-purpose DSP micros with a hardware multiplier and an adjacent accumulator, and uses Winograd's ideas for the implementation.
Abstract: This paper develops an approach to deriving special FFT and convolution algorithms, considering the architecture of special-purpose DSP micros with a hardware multiplier and an adjacent accumulator. Because of the build-in accumulator, it is possible to combine multiplications and additions into one operation, and hence the optimality criterion is redefined to minimize the total number of additions, multiplications and combined multiply-additions, which results in new interesting algorithms. The structure of the algorithms depends strongly on the nature of N - the sequence length. Where most conventional fast algorithms decompose convolutions completely, it turns out that because the adjacent accumulator can hides some of the additions, they should only be partly decomposed. The decomposition is done by index mapping and the implementation uses Winograd's ideas. Discrete Fourier transforms are also only partly decomposed, in contrast to most algorithms. The DFTs are converted into convolutions, and the convolutions are implemented as block processing. Comparisons show up to a 35 % improvement in execution speed.

Journal ArticleDOI
TL;DR: Angular measurements in symmetrical and nonsymmetrical Fourier spectra are compared and better accuracy of angular spectral analysis with an anamorphic Fourier transformer is explained and experimentally proved.
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.

Journal ArticleDOI
TL;DR: In this article, a new approximate method for the analysis of delay time-invariant and lime-varying systems via Fourier series is presented, which utilizes the shift and product operational matrices of the Fourier-series for linear systems.
Abstract: The paper introduces a new approximate method for the analysis of delay time-invariant and lime-varying systems via Fourier series. The present approach utilizes the shift and product operational matrices of the Fourier series for the analysis of linear systems via the Fourier operational matrix of integration. The method developed is applied to the analysis of a number of illustrative model systems.

Book
17 Jan 1986
TL;DR: 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 as discussed by the authors
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.

01 Jan 1986
TL;DR: A paired tensor representation of each component Fp,s of the spectrum of the signal in the form of the corresponding N/2-dimensional vector F̄ ′ p,s the paired vector representation is called.
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]

Proceedings ArticleDOI
Pierre Duhamel1, Martin Vetterli
01 Apr 1986
TL;DR: Two approaches using Fourier or Hartley transforms are first compared, showing that the recently proposed FFT algorithms for real data present a lower arithmetic complexity than the corresponding DHT-based approach.
Abstract: Recently, new fast transforms (such as the discrete Hartley transform in particular) have been proposed which are best suited for the computation of cyclic convolution of real sequences. Two approaches using Fourier or Hartley transforms are first compared, showing that the recently proposed FFT algorithms for real data present a lower arithmetic complexity than the corresponding DHT-based approach. Improvements are made to both types of algorithms, leading to different trade offs between arithmetic and structural complexity. We also present a new Hartley Transform algorithm with lower arithmetic complexity than any previously published one.

Patent
03 Jun 1986
TL;DR: In this article, a fast Fourier transform (FFT) data address pre-scrambler technique and cuit for selectively generating prescrambled bit reversed, data address sequences needed to perform radix 2, radix 4, and mixed radix-2/4 FFT transforms are presented.
Abstract: A fast Fourier transform (FFT) data address pre-scrambler technique and cuit for selectively generating pre-scrambled bit reversed, data address sequences needed to perform radix-2, radix-4 and mixed radix-2/4 fast Fourier transforms.

Proceedings ArticleDOI
01 Apr 1986
TL;DR: In this article, the authors investigate the usefulness of support-limited extrapolation as a tool for improving the estimate of the magnitude of a finite extent complex-valued signal from samples of its Fourier transform that are offset from the origin.
Abstract: We investigate the usefulness of support-limited extrapolation as a tool for improving the estimate of the magnitude of a finite extent complex-valued signal from samples of its Fourier transform that are offset from the origin. The motivating application is spotlight mode synthetic aperture radar where the complex-valued signal to be recovered is microwave reflectivity and the measured Fourier samples of this reflectivity constitute the signal to be extrapolated. Both analytical and simulation results are presented supporting an earlier conjecture that extrapolation will be of little benefit unless the width of the known Fourier data is less than or equal to the bandwidth of just the magnitude of the reflectivity.

Journal ArticleDOI
TL;DR: In this paper, the discrete frequency Fourier transform (DFFT) is shown to be a useful transform in its own right for spatial domain image reconstruction, filling a gap in the theory and aids the designer in understanding problems which have inherently sampled frequency domains.
Abstract: In certain signal processing applications it may be required to reconstruct a spatial domain image form samples of its Fourier transform. For problems such as this it may be useful to use the dual of the well-known discrete time Fourier transform (DTFT) for purposes of analysis and design. In this paper, this dual concept, called the discrete frequency Fourier transform (DFFT), is shown to be a useful transform in its own right. In addition to being useful for certain physical problems, the DFFT fills a gap in the theory and aids the designer in understanding problems which have inherently sampled frequency domains.

Journal ArticleDOI
TL;DR: A single-lens imaging system is described that provides spatial frequency filtering to an extended incoherent light source and the modified image is derived from the filtered diffraction pattern by Fourier synthesis.
Abstract: A single-lens imaging system is described that provides spatial frequency filtering to an extended incoherent light source. Two ways to calculate the modified image are considered: First, the image is derived from the filtered diffraction pattern by Fourier synthesis. Second, the convolution of the object function with a Fourier transform of the filter function has to be evaluated with regard to a phase function that corresponds to the light source position.

Proceedings ArticleDOI
01 Apr 1986
TL;DR: This paper describes a new method to locate sound sources using many sensors by inverse Fourier transforming the estimated spectra, and the wave form of a specified sound source can be obtained.
Abstract: This paper describes a new method to locate sound sources using many sensors. Spectra of all the sensor outputs are calculated by discrete Fourier transform. In the proposed method, following parameters are assumed; (1) the number of sound sources, (2) a position of each sound source, and (3) the spectrum of the O-th sensor output due to each sound source. The spectra of all the sensor outputs are estimated using the parameters under the assumption of free field. Then, introduced is an error function which is the mean-square value of the difference between the calculated spectra and the estimated ones. By changing the values of the parameters, the minimum value of the error function is searched. When the error function takes the minimum value, the parameters represent the estimates; the number of sound sources, and the positions and spectra of the sound sources. By inverse Fourier transforming the estimated spectra, the wave form of a specified sound source can be obtained.