Showing papers on "Fractional Fourier transform published in 1988"
Journal Article•
TL;DR: This paper lists and interprets the numerous properties of the Zak transform and the relation with other time-frequency representations such as the Wigner distribution and the radar ambiguity function is given, and the Gabor representation problem is tackled.
310 citations
TL;DR: In this article, a unitary fast Fourier transform method for solving time dependent curve crossing problems is presented, and the procedure is described in detail and illustrated by calculations for a two curve, one dimensional example.
Abstract: We develop a unitary fast Fourier transform method for solving time dependent curve crossing problems. The procedure is described in detail and is illustrated by calculations for a two curve, one‐dimensional example. The time evolution of the wave function and mean nuclear positions and energies for each curve are shown and discussed.
137 citations
TL;DR: In this paper, two mathematical procedures of band narrowing using Fourier transforms are discussed and the methods of Fourier self-deconvolution and Fourier derivation are easier to use and to control than procedures which rely on the use of convolution functions.
113 citations
TL;DR: The circular harmonic transform (CHT) solution of the exponential Randon transform (ERT) is applied to single-photon emission computed tomography (SPECT) for uniform attenuation within a convex boundary to demonstrate that the boundary conditions are a more general property of the Radon transform and a not a property unique to rectangular coordinates.
Abstract: The circular harmonic transform (CHT) solution of the exponential Randon transform (ERT) is applied to single-photon emission computed tomography (SPECT) for uniform attenuation within a convex boundary. An important special case also considered is the linear (unattenuated) Radon transform (LRT). The solution is on the form of an orthogonal function expansion matched to projections that are in parallel-ray geometry. This property allows for efficient and accurate processing of the projections with fast Fourier transform (FFT) without interpolation or beam matching. The algorithm is optimized by the use of boundary conditions on the 2-D Fourier transform of the sinogram. These boundary conditions imply that the signal energy of the sinogram is concentrated in well-defined sectors in transform space. The angle defining the sectors depends in a direct way on the radius of the field view. These results are also obtained for fan-beam geometry and the linear Radon transform (the Fourier-Chebyshev transform of the sinogram) to demonstrate that the boundary conditions are a more general property of the Radon transform and a not a property unique to rectangular coordinates. >
107 citations
TL;DR: It is shown that the number of distinct N-point DFTs needed to calculate N*N-point two-dimensional DFT’s is equal to thenumber of linear congruences spanning the N-N grid.
Abstract: An algorithm is presented for computation of the two-dimensional discrete Fourier transform (DFT). The algorithm is based on geometric properties of the integers and exhibits symmetry and simplicity of realization. Only one-dimensional transformation of the input data is required. The transformations are independent; hence, parallel processing is feasible. It is shown that the number of distinct N-point DFTs needed to calculate N*N-point two-dimensional DFTs is equal to the number of linear congruences spanning the N*N grid. Examples for N=3, N=4, and N=10 are presented. A short APL code illustrating the algorithm is given. >
76 citations
TL;DR: Preliminary results are presented on the VLSI design and implementation of a novel algorithm for accurate high-speed Fourier analysis and synthesis, based on the number-theoretic method of Mobius inversion.
Abstract: Preliminary results are presented on the VLSI design and implementation of a novel algorithm for accurate high-speed Fourier analysis and synthesis. The arithmetic Fourier transform (AFT) is based on the number-theoretic method of Mobius inversion. Its computations proceed in parallel, and the individual operations are very simple. Except for a small number of scalings in one state of the computation, only multiplications by 0, +1, and -1 are required. If the input samples were not quantized and if ideal real-number operations were used internally, then the results would be exact. The accuracy of the computation is limited only by the input A/D (analog-to-digital) conversion process, any constraints on the word lengths of internal accumulating registers, and the implementation of the few scaling operations. Further simplifications are obtained by using delta modulation to represent the input function in digital form, so that only binary (or preferably, ternary) sequences needs to be processed in the parallel computations. The required accumulations can be replaced by up/down counters. The dynamic range of the resulting transformation can be increased by the use of adaptive delta modulation. >
63 citations
TL;DR: An algorithm for computing the Fourier transform over any finite field GF(p/sup m/) that requires only O(n(log n)/sup 2//4) additions and the same number of multiplications for an n-point transform and allows in some fields a further reduction of the number of multiplier additions.
Abstract: The Fourier transform over finite fields is mainly required in the encoding and decoding of Reed-Solomon and BCH codes. An algorithm for computing the Fourier transform over any finite field GF(p/sup m/) is introduced. It requires only O(n(log n)/sup 2//4) additions and the same number of multiplications for an n-point transform and allows in some fields a further reduction of the number of multiplications to O(n log n). Because of its highly regular structure, this algorithm can be easily implementation by VLSI technology. >
56 citations
TL;DR: An approach for realizing the N-point discrete Fourier transform (DFT) of an input sequence is presented and is combined with H.T. King's (1981) approach to construct a two-dimensional array for computing the two- dimensional DFT.
Abstract: An approach for realizing the N-point discrete Fourier transform (DFT) of an input sequence is presented. It is then combined with H.T. King's (1981) approach to construct a two-dimensional array for computing the two-dimensional DFT. This mixed model takes stream input and produces stream output. In addition, no extra I/O time delay is required before performing the row (column) transform of the two-dimensional DFT. >
55 citations
TL;DR: In this article, it was shown that the Nahm transform over flat tori is a duality transform, and that it can not be used to construct instantons over the first Brillouin zone.
Abstract: Recently P. Braam pointed out that Nahm's adaption of the ADHM procedure to the case of monopoles equally well applies to instantons over flat tori, relating them to instantons over the first Brillouin zone. We show that this construction has an inverse. Hence the Nahm transform actually is a duality transform.
54 citations
TL;DR: The HTR algorithm is outlined, and it is shown that its performance compares favorably to the popular convolution-backprojection algorithm.
Abstract: A relatively unexplored algorithm is developed for reconstructing a two-dimensional image from a finite set of its sampled projections. The algorithm, referred to as the Hankel-transform-reconstruction (HTR) algorithm, is polar-coordinate based. The algorithm expands the polar-form Fourier transform F(r, theta ) of an image into a Fourier series in theta ; calculates the appropriately ordered Hankel transform of the coefficients of this series, giving the coefficients for the Fourier series of the polar-form image f(p, phi ); resolves this series, giving a polar-form reconstruction; and interpolates this reconstruction to a rectilinear grid. The HTR algorithm is outlined, and it is shown that its performance compares favorably to the popular convolution-backprojection algorithm. >
45 citations
Journal Article•
TL;DR: In this paper, it was shown that extended zero-filling (e.g., a "zoom" transform) actually reduces the accuracy with which the spectral peak position can be determined, and that the results can be more accurate when the FFT processor operates with floating-point (preferably double-precision) rather than fixed-point arithmetic.
Abstract: A frequency-domain Lorentzian spectrum can be derived from the Fourier transform of a time-domain exponentially damped sinusoid of infinite duration. Remarkably, it has been shown that even when such a noiseless time-domain signal is truncated to zero amplitude after a finite observation period, one can determine the correct frequency of its corresponding magnitude-mode spectral peak maximum by fitting as few as three spectral data points to a magnitude-mode Lorentzian spectrum. In this paper, we show how the accuracy of such a procedure depends upon the ratio of time-domain acquisition period to exponential damping time constant, number of time-domain data points, computer word length, and number of time-domain zero-fillings. In particular, we show that extended zero-filling (e.g., a "zoom" transform) actually reduces the accuracy with which the spectral peak position can be determined. We also examine the effects of frequency-domain random noise and round-off errors in the fast Fourier transformation (FFT) of time-domain data of limited discrete data word length (e.g., 20 bit/word at single and double precision). Our main conclusions are: (1) even in the presence of noise, a three-point fit of a magnitude-mode spectrum to a magnitude-mode Lorentzian line shape can offer an accurate estimate of peak position in Fourier transform spectroscopy; (2) the results can be more accurate (by a factor of up to 10) when the FFT processor operates with floating-point (preferably double-precision) rather than fixed-point arithmetic; and (3) FFT roundoff errors can be made negligible by use of sufficiently large (> 16 K) data sets.
TL;DR: Effective methods are proposed for calculating a multidimensional discrete Fourier transform based on a new representation of it and their application to discrete number theory is proposed.
Abstract: Effective methods are proposed for calculating a multidimensional discrete Fourier transform based on a new representation of it.
TL;DR: In this paper, the exact expression for the discrete Fourier transform of a sum of exponentially damped sinusoids is derived, and its applicability for describing the general DFT spectrum is demonstrated.
TL;DR: A method for computing the short-time Fourier transform of a discrete-time signal by convolving it with one of a family of infinite-duration windows h(nT)=(nT)ke−αnT, k integer which greatly alleviates the problem of spectral leakage which is inherent in the more traditional DFT method.
TL;DR: The authors propose an orthogonal, unitary transformation called the modified Hermite transformation (MHT) and its extension, which is called the modular modified Hermites transformation (MMHT), which is efficient computationally and comparable to the DCT for AR(1) source models with positive correlation coefficients.
Abstract: The authors propose an orthogonal, unitary transformation called the modified Hermite transformation (MHT) and its extension, which is called the modular modified Hermite transformation (MMHT). The MHT algorithm, which is an efficient algorithm, is explained and explored. The MHT is compared to the discrete cosine transform (DCT) for various AR(1) input signal source models using the performance criterion of gain over PCM, denoted by /sup N/G/sub TC/. The MHT algorithm requires only 2N real multiplications or divisions for a transformation of a signal block of N samples. It is also used for the inverse transformation, IMHT, and makes this new transform attractive. It is efficient computationally and comparable to the DCT for AR(1) source models with positive correlation coefficients, it is somewhat better than the DCT for negative correlation coefficients. >
Patent•
18 Oct 1988TL;DR: In this paper, the conversion process of a picture signal is performed in block unit, not in frame unit, and the products between the orthogonal transform matrix and matrices for realizing picture resolution conversion, image manipulation processes such as expansion, compression and rotation, and various kinds of linear filtering are provided in a coefficient memory as new transform matrices.
Abstract: An image signal processor in which the conversion process of a picture signal is performed in block unit, not in frame unit, and in order to raise the speed of the process, besides an orthogonal transform matrix, the products between the orthogonal transform matrix and matrices for realizing picture resolution conversion, image manipulation processes such as expansion, compression and rotation, and various kinds of linear filtering are provided in a coefficient memory as new transform matrices, and these transform matrices are properly changed-over in accordance with the content of the pertinent transform, thereby to perform an orthogonal transform or an inverse orthogonal transform.
TL;DR: In this paper, the authors used a desktop computer to imitate a pendulum having the physical characteristics of a finite damping constant and drift in the zero potential energy position to measure the period of a torsion pendulum with precision.
Abstract: Measuring the period of torsion pendulums with precision has long been a formidable challenge in gravitation experiments, particularly those measuring the Newtonian gravitational constant G. An alternative method to fitting the position signal of the pendulum to a sine wave is the use of the power spectrum generated by the fast Fourier transform (FFT) as the source of information from which the period of oscillation can be determined. There are, however, known limitations to the use of a FFT to measure the period of a physical oscillator with precision. These limitations include two effects due to the finiteness of the duration of the sinusoidal data record and one effect due to the uncertainty of the starting phase of the oscillator relative to the window imposed by this duration. We have done a phenomenological study of the FFT using a desktop computer to imitate a precision oscillator having the physical characteristics of a finite damping constant and drift in the zero potential‐energy position. Also,...
01 Dec 1988
TL;DR: It is shown that by using an index mapping scheme, the multidimensional discrete Hartley transform can be changed into convolutions that can be calculated very efficiently via the Fermat number transform.
Abstract: It is shown that by using an index mapping scheme, the multidimensional discrete Hartley transform can be changed into convolutions that can be calculated very efficiently via the Fermat number transform. Compared with existing algorithms, the number of multiplications is reduced by a factor of 8 to 20, at the expense of a slight increase in the number of shift and add operations, that are assumed to be simpler than multiplications.
03 Aug 1988
TL;DR: In this article, the relationship between linear (short-time Fourier transform, wavelet transform) and bilinear (Wigner-Ville distribution, affine Wigner distribution) approaches is investigated.
Abstract: General results are presented for time-frequency and time-scale methods. Attention is given to both linear (short-time Fourier transform, wavelet transform) and bilinear (Wigner-Ville distribution, affine Wigner distribution) approaches, with emphasis put on their relationships. Also considered is the relationship of the methods examined to such approaches as constant-Q analysis and ambiguity functions. >
01 Jan 1988
TL;DR: In this article, the multiplicative complexity of the discrete Fourier transform (DFT) was analyzed and the complexity of DFT for any positive integer was shown. But the complexity was not shown for any integer.
Abstract: In this chapter the multiplicative complexity of the discrete Fourier transform (DFT) is analyzed. The next several sections define the DFT and then show how the complexity of the DFT is determined when the number of inputs is prime, a power of an odd prime, a power of two, and finally for any positive integer.
TL;DR: It is shown that for an analysis filter length that does not exceed a given value, the optimal synthesis scheme is independent of the duration of the given MDSTT and is an extension of the weighted overlap add (WOLA) synthesis method.
Abstract: The discrete short-time transform (DSTT) is a generalization of the discrete short-time Fourier transform (DSTFT). The necessary and sufficient conditions on the analysis filter, under which perfect reconstruction of the input signal is possible (when the DSTT is not modified), are presented. The class of linear modifications for which the original input can be reconstructed when the modification is applied is characterized. The synthesis of an optimal (in the minimum-mean-square-error sense) signal from a modified DSTT (MDSTT) of finite duration is presented. It is shown that for an analysis filter length that does not exceed a given value, the optimal synthesis scheme is independent of the duration of the given MDSTT and is an extension of the weighted overlap add (WOLA) synthesis method. For longer analysis filters, the optimal synthesis scheme becomes quite cumbersome, and therefore, a steady-state solution (as the duration of the MDSTT approaches infinity) is presented for this case. It is shown that this solution can be approximated with arbitrarily small reconstruction error. >
TL;DR: It is shown that the use of equal spacings in the logarithmic time and frequency domains provides a very efficient transform algorithm that is applicable for the analysis of systems with moderate dynamic behavior over several frequency decades.
Abstract: Discrete Fourier transforms are derived which allow the use of nonequally spaced time-domain samples. It is shown that the use of equal spacings in the logarithmic time and frequency domains provides a very efficient transform algorithm. The applicability of this algorithm for the analysis of systems with moderate dynamic behavior over several frequency decades is demonstrated by examples. An error analysis is given. >
TL;DR: Based on a multichannel incoherent optical correlator, a new simple scheme was proposed for performing a complex discrete Fourier transform as mentioned in this paper, where a complex value is represented by using three nonnegative reals, and every real is encoded with the area of a rectangular aperture.
TL;DR: In this paper, an orthonormal basis for the x-ray transform in Rn is described in terms of spherical harmonics and Jacobi polynomials, and the inversion of the X-ray transformation is done by using the projection slice theorem.
Abstract: An inversion for the x-ray transform in Rn is given for square integrable functions supported in the unit ball E. An orthonormal basis for L2(E) is described in terms of spherical harmonics and Jacobi polynomials and the inversion is done in terms of the generalised Fourier series expansion with respect to that basis. The result is derived by using the projection slice theorem to show that the X-ray transform is block diagonal with respect to the above basis for L2(E) and the transform of that basis along a fixed direction omega 0.
TL;DR: In this article, the effects of a discrete Fourier transform (DFT) filter on a time series of ocean currents and sea levels were investigated and it was shown that ringing in the retransformed time series can be minimized by careful choice of filter bandwidth and the amount of tapering of the sides of the filter.
Abstract: Many different filters exist to remove unwanted signals from time series of ocean currents and sea levels, and in this paper we examine two: a tidal harmonic “filter” and a discrete Fourier transform (DFT) filter. DFT filters are particularly easy to use if working in the frequency domain, but as with all filters it is necessary to understand the effects of a DFT filter before it can be used with complete confidence. A number of sample time series are used, some artificial and some real, to test DFT filters. It is shown that “ringing” in the retransformed time series can be minimized by careful choice of the filter bandwidth and the amount of tapering of the sides of the filter.
11 Apr 1988
TL;DR: The fast real Fourier transform (FRFT) algorithms discussed are the radix-2 decimation-in-time (DIT), theRadix-4 DIT, the split-radix DIF, the prime factor, and the Winograd FRFT algorithm.
Abstract: Fast algorithms for the computation of the real discrete Fourier transform (RDFT) are discussed. Implementations based on the RDFT are always efficient, whereas the implementations based on the DFT are efficient only when signals to be processed are complex. The fast real Fourier transform (FRFT) algorithms discussed are the radix-2 decimation-in-time (DIT), the radix-4 DIT, the split-radix DIT, the split-radix DIF, the prime factor, and the Winograd FRFT algorithm. >
TL;DR: In this article, a functional program for the Fast Fourier Transform for multiplication of polynomials is presented for the Parallel Reduction Machine (PRM) project, which is used to verify experimentally two claims by functional programmers: functional programming is good for writing structured software; better so than the imperative von Neumann languages.
Abstract: This paper is written as a contribution to the Parallel Reduction Machine Project. Its
purpose is to present a functional program for a well-known application of the fundamental
algorithmic method Fast Fourier Transform for multiplication of polynomials. This in
order to verify experimentally two claims by functional programmers [BvL]:
(i) functional programming is good for writing structured software; better so than the
so-called imperative von Neumann-languages.
(ii) functional programming allows for a parallel evaluation of subexpressions,
provided a proper implementation.
TL;DR: In this paper, the Fourier spectrum of a multiply periodic system appears complex due to resonances at integer combinations of the fundamental frequencies, and it is shown how to "reorder" the peaks in the spectrum to yield a simpler structure which reveals important aspects of the time series which are not clearly seen in the ordinary Fourier transform.
IBM1
TL;DR: In this article, the authors obtained explicit formulas for the probability distributions of such bursts and for the errors that the clipping induce in the decoder, which can help the FTDM code designer to decide on an appropriate average power constraint.
Abstract: Inherent in the method of Fourier transform division multiplexing (FTDM) is the possibility that the FTDM encoder will yield spurious power bursts, which can affect the linearity of the channel. A common way for dealing with such bursts is to clip the signal at some predetermined peak power level. The authors obtain explicit formulas for the probability distributions of such bursts and for the errors that the clipping induce in the decoder. The formulas can help the FTDM code designer to decide on an appropriate average power constraint. >