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Showing papers on "Hartley transform published in 1997"


Journal ArticleDOI
TL;DR: An improved DFRFT is proposed that provides transforms similar to those of the continuous fractional Fourier transform and also retains the rotation properties.
Abstract: The fractional Fourier transform is a useful mathematical operation that generalizes the well-known continuous Fourier transform. Several discrete fractional Fourier transforms (DFRFT's) have been developed, but their results do not match those of the continuous case. We propose a new DFRFT. This improved DFRFT provides transforms similar to those of the continuous fractional Fourier transform and also retains the rotation properties.

185 citations


Journal ArticleDOI
TL;DR: The purpose of this paper is to introduce extensions of the FT¿s convolution theorem, dealing with the FRFT of a product and of a convolution of two functions.
Abstract: The fractional Fourier transform (FRFT) is a generalization of the classical Fourier transform (FT). It has recently found applications in several areas, including signal processing and optics. Many properties of this transform are already known, but an extension of the FT?s convolution theorem is still missing. The purpose of this paper is to introduce extensions of this theorem, dealing with the FRFT of a product and of a convolution of two functions.

150 citations


Journal ArticleDOI
TL;DR: The wavelet transform, which has had a growing importance in signal and image processing, has been generalized by association with both the wavelettransform and the fractional Fourier transform.
Abstract: The wavelet transform, which has had a growing importance in signal and image processing, has been generalized by association with both the wavelet transform and the fractional Fourier transform. Possible implementations of the new transformation are in image compression, image transmission, transient signal processing, etc. Computer simulations demonstrate the abilities of the novel transform. Optical implementation of this transform is briefly discussed.

128 citations


Journal ArticleDOI
TL;DR: In this article, a new technique based on double affine Hecke algebras was applied to the Harish-Chandra theory of spherical zonal functions, and the formulas for the Fourier transforms of the multiplications by the coordinates were obtained as well as a simple proof of the inversion theorem using the Opdam transform.
Abstract: We apply a new technique based on double affine Hecke algebras to the Harish-Chandra theory of spherical zonal functions. The formulas for the Fourier transforms of the multiplications by the coordinates are obtained as well as a simple proof of the Harish-Chandra inversion theorem using the Opdam transform.

67 citations


Journal ArticleDOI
TL;DR: An extension of the Fresnel transform to first-order optical systems that can be represented by an ABCD matrix is analyzed in this article, which is recognized to belong to the class of linear canonical transforms.
Abstract: An extension of the Fresnel transform to first-order optical systems that can be represented by an ABCD matrix is analyzed. We present and discuss a definition of the generalized transform, which is recognized to belong to the class of linear canonical transforms. A general mathematical characterization is performed by listing a number of meaningful theorems that hold for this operation and can be exploited for simplyfying the analysis of optical systems. The relevance to physics of this transform and of the theorems is stressed. Finally, a comprehensive number of possible decompositions of the generalized transform in terms of elementary optical transforms is discussed to obtain further insight into this operation.

67 citations


Journal ArticleDOI
TL;DR: The proposed unified systolic arrays exhibit advantages in terms of the number of PE's and latency and can be employed for computation of the inverse DCT/DST/DHT (IDCT/IDST/IDHT).
Abstract: We propose unified systolic arrays for computation of the one-dimensional (1-D) and two-dimensional (2-D) discrete cosine transform/discrete sine transform/discrete Hartley transform (DCT/DST/DHT). By decomposing the transforms into even- and odd-numbered frequency samples, the proposed architecture computes the 1-D DCT/DST/DHT. Compared to the conventional methods, the proposed systolic arrays exhibit advantages in terms of the number of PE's and latency. We generalize the proposed structure for computation of the 2-D DCT/DST/DHT. The unified systolic arrays can be employed for computation of the inverse DCT/DST/DHT (IDCT/IDST/IDHT).

61 citations


Journal ArticleDOI
TL;DR: A new linear integral transform is defined, which is called the exponential chirp transform, which provides frequency domain image processing for space-variant image formats, while preserving the major aspects of the shift-invariant properties of the usual Fourier transform.
Abstract: Space-variant (or foveating) vision architectures are of importance in both machine and biological vision. In this paper, we focus on a particular space-variant map, the log-polar map, which approximates the primate visual map, and which has been applied in machine vision by a number of investigators during the past two decades. Associated with the log-polar map, we define a new linear integral transform, which we call the exponential chirp transform. This transform provides frequency domain image processing for space-variant image formats, while preserving the major aspects of the shift-invariant properties of the usual Fourier transform. We then show that a log-polar coordinate transform in frequency provides a fast exponential chirp transform. This provides size and rotation, in addition to shift, invariant properties in the transformed space. Finally, we demonstrate the use of the fast exponential chirp algorithm on a database of images in a template matching task, and also demonstrate its uses for spatial filtering.

51 citations




Journal ArticleDOI
TL;DR: This work shows how to implement the fractional Hilbert transform for two-dimensional inputs, which is now suitable for image processing.
Abstract: The classical Hilbert transform can be implemented optically as a spatial-filtering process, whereby half the Fourier spectrum is π-phase shifted. Recently the Hilbert transform was generalized. The generalized version, called the fractional Hilbert transform, is quite easy to implement optically if the input is one dimensional. Here we show how to implement the fractional Hilbert transform for two-dimensional inputs. Hence the new transform is now suitable for image processing.

47 citations


Journal ArticleDOI
TL;DR: In this article, the Fourier transform for compactly supported smooth functions on NA, when NA is a symmetric space of rank one, is defined and the corresponding inversion formula and Plancherel Theorem are obtained.
Abstract: Given a group N of Heisenberg type, we consider a one-dimensional solvable extension NA of N, equipped with the natural left-invariant Riemannian metric, which makes NA a harmonic (not necessarily symmetric) manifold. We define a Fourier transform for compactly supported smooth functions on NA, which, when NA is a symmetric space of rank one, reduces to the Helgason Fourier transform. The corresponding inversion formula and Plancherel Theorem are obtained. For radial functions, the Fourier transform reduces to the spherical transform considered by E. Damek and F. Ricci.

Proceedings ArticleDOI
09 Jun 1997
TL;DR: A novel unified systolic architecture which can efficiently implement various discrete trigonometric transforms (DXT) including the discrete Fourier transform (DFT), the discrete Hartley transform, the discrete cosine transform, and the discrete sine transform is described.
Abstract: In this paper, a novel unified systolic architecture which can efficiently implement various discrete trigonometric transforms (DXT) including the discrete Fourier transform (DFT), the discrete Hartley transform (DHT), the discrete cosine transform (DCT), and the discrete sine transform (DST) is described. Based on Clenshaw's recurrence formula, a set of efficient recurrences for computing the DXT is developed first. Then, the inherent symmetry of the trigonometric functions is further exploited to render a hardware-efficient, systolic structure. For the computation of any N-point DXT of interest, the proposed structure requires only about N/2 multipliers and N adders, thus providing substantial hardware savings compared with previous works. Furthermore, the new scheme can be easily adapted to compute any type of DXT with only minor modification. The complete I/O buffers have been addressed as well which allows for a continuous flow of successive blocks of input data and transformed results in natural order.

Journal ArticleDOI
TL;DR: A split-radix algorithm that can flexibly compute the discrete Hartley transforms of various sequence lengths is presented and it shows that the length-3*2/sup m/ DHTs need a smaller number of multiplications than thelength-2/Sup m/DHTs, but they both require about the same computational complexity.
Abstract: This paper presents a split-radix algorithm that can flexibly compute the discrete Hartley transforms of various sequence lengths. Comparisons with previously reported algorithms are made in terms of the required number of additions and multiplications. It shows that the length-3*2/sup m/ DHTs need a smaller number of multiplications than the length-2/sup m/ DHTs. However, they both require about the same computational complexity in terms of the total number of additions and multiplications. Optimized computation of length-12, -16 and -24 DFTs are also provided.

Journal ArticleDOI
TL;DR: This method is based on the cosine Fourier transform between the angle and order domains of the Chebyshev operator and is applicable to matrices of any functions of the Hamiltonian operator.


Journal ArticleDOI
01 Oct 1997
TL;DR: The authors propose one-dimensional and two-dimensional systolic architectures for the discrete Hilbert transform that have the features of massive parallelism, high pipelining, regular data flow, modular nature and local interconnections.
Abstract: A new fast parallel array algorithm to compute the discrete Hilbert transform for radix-2 length sequences is proposed. Unlike the existing fast methods which use transforms such as the fast Fourier transform, the proposed algorithm does not require the help of any fast transforms. This array algorithm offers a suitable expression for developing a VLSI systolic array for the discrete Hilbert transform. The authors propose one-dimensional and two-dimensional systolic architectures for the discrete Hilbert transform. The proposed architectures have the features of massive parallelism, high pipelining, regular data flow, modular nature and local interconnections. These arrays offer high speed computation of the discrete Hilbert transform for real-time signal processing applications.

Journal ArticleDOI
TL;DR: In this paper, Shih et al. proposed a new fractional Fourier transform with four periodic eigenvalues with respect to the order of Hermite-Gaussian functions.

Journal ArticleDOI
TL;DR: The FRGT provides analyses of signals in both the real space and the FRFT frequency domain simultaneously, and has an additional freedom, compared with the conventional GT, i.e., the transform order.
Abstract: A fractional Gabor transform (FRGT) is proposed. This new transform is a generalization of the conventional Gabor transform (GT) based on the Fourier transform to the windowed fractional Fourier transform (FRFT). The FRGT provides analyses of signals in both the real space and the FRFT frequency domain simultaneously. The space-FRFT frequency pattern can be rotated as the fractional order changes. The FRGT has an additional freedom, compared with the conventional GT, i.e., the transform order. The FRGT may offer a useful tool for guiding optimal filter design in the FRFT domain in signal processing.

Journal ArticleDOI
TL;DR: The fractional Fourier transform of an object can be observed in the free-space Fresnel diffraction pattern of the object.
Abstract: The fractional Fourier transform of an object can be observed in the free-space Fresnel diffraction pattern of the object.

Journal ArticleDOI
TL;DR: A new parallel approach for computing the running DHT and DWT's is proposed by establishing the relationship between the running discrete Hartley transform, the discrete W transforms, and the LMS algorithm.
Abstract: The computation of block-based discrete orthogonal transforms using the adaptive least-mean-square (LMS) algorithm has been studied in literature. The authors extend this work by establishing the relationship between the running discrete Hartley transform (DHT), the discrete W transforms (DWT's), and the LMS algorithm. As a result a new parallel approach for computing the running DHT and DWT's is proposed.

Journal ArticleDOI
TL;DR: This work combines two concepts: the joint transform correlator and the fractional Fourier transform and it is shown that this combination is almost as convenient experimentally as the classical joint Transform correlator.
Abstract: We combine two concepts: the joint transform correlator and the fractional Fourier transform. This combination is almost as convenient experimentally as the classical joint transform correlator. As a processor it is as versatile as the standard fractional Fourier correlator.

Posted Content
TL;DR: The Replica Fourier Transform introduced previously is related to the standard definition of Fourier transforms over a group and its use is illustrated by block-diagonalizing the eigenvalue equation of a four-replica Parisi matrix as discussed by the authors.
Abstract: The Replica Fourier Transform introduced previously is related to the standard definition of Fourier transforms over a group. Its use is illustrated by block-diagonalizing the eigenvalue equation of a four-replica Parisi matrix.

Journal ArticleDOI
TL;DR: In this paper, a new frame-of-reference based on the use of Hartley's transform and a three-phase thyristor controlled reactor (TCR) harmonic model are presented.
Abstract: The main objectives of this paper are to present a new frame-of-reference based on the use of Hartley's transform and to present a three-phase thyristor controlled reactor (TCR) harmonic model which uses Hartley's domain. Solutions using the new frame-of-reference are between two to four times faster than solutions using an established frame-of-reference based on Fourier's transform because Hartley's transform makes use of the real plane as opposed to the complex plane. Harmonic switching vectors in Hartley's domain have been developed for maximum computer efficiency. Their use, combined with discrete convolution operations, provide cleaner and faster operations than those afforded by the fast Hartley transform. The TCR model is completely general and caters for any kind of plant imbalances, e.g. uneven firing angles and inductances. Network imbalances are accounted for via the excitation voltage. The new frame-of-reference accommodates any number of buses, phases, harmonics and cross-couplings between harmonics. It provides a reliable and efficient means for the iterative solution of power systems harmonic problems through a Newton-Raphson method which exhibits quadratic convergence.

Journal ArticleDOI
TL;DR: The scaled fractional Fourier transform is suggested and is implemented optically by one lens for different values of phi and output scale and relates the FRT with the general lens transform-the optical diffraction between two asymmetrically positioned planes before and after a lens.
Abstract: The scaled fractional Fourier transform is suggested and is implemented optically by one lens for different values of phi and output scale. In addition, physically it relates the FRT with the general lens transform-the optical diffraction between two asymmetrically positioned planes before and after a lens. (C) 1997 Optical Society of America.

Proceedings ArticleDOI
10 Dec 1997
TL;DR: In this paper, a frequency weighted least squares (FWLS) formulation is given for identifying the parameters of Hammerstein-type nonlinear continuous-time systems (1930) based on input and noise contaminated output data observed over a finite time interval.
Abstract: A frequency weighted least squares (FWLS) formulation is given for identifying the parameters of Hammerstein-type nonlinear continuous-time systems (1930) based on input and noise contaminated output data observed over a finite time interval. The Hartley modulating functions (HMF) method (1942) starts from a priori knowledge of the Hammerstein system structure with unknown parameters. The approach converts the nonlinear differential equation describing the nonlinear system into a Hartley spectrum equation and circumvents the need to estimate unknown initial conditions through the use of certain modulation properties. The unknown system parameters can then be estimated in the frequency domain by a FWLS-algorithm. A root mean square normalized error criterion is applied to measure the bias of the estimate for different values of the mode number and order of the Hartley transformation as well as for different levels of the noise-to-signal ratio in order to investigate some computational considerations associated with the methodology. The illustrative Monte Carlo simulation studies suggest that this method lies in the potential of being able to accurately estimate the parameters of a nonlinear continuous-time Hammerstein system in the presence of significant output measurement disturbances.

Journal ArticleDOI
TL;DR: In this article, a sampling theorem associated with boundary-value problems involving a one-dimensional system of Dirac operators is derived, which is a special case of the sampling theorem for the Hartley transform of a bandlimited function.

Journal ArticleDOI
TL;DR: The method is based on an orthogonal transform based on discrete Legendre polynomials that provides several advantages for compressing ECG signals when compared with conventional Fourier or cosine transforms.

Proceedings ArticleDOI
09 Sep 1997
TL;DR: A uniform algebraic framework for computing hybrid spectral transforms in an efficient manner is given, based on properties of the Kronecker product, which leads naturally to an algorithm for computing such transforms efficiently.
Abstract: We give a uniform algebraic framework for computing hybrid spectral transforms in an efficient manner. Based on properties of the Kronecker product, we derive a set of recursive equations, which leads naturally to an algorithm for computing such transforms efficiently. As a result, many applications of transforms like the Walsh transform and the Reed-Muller transform, which were previously impossible because of memory constraints, have now become feasible. The same set of recursive equations also gives a new way of explaining hybrid transform diagrams, an efficient data-structure for integer valued Boolean functions.

Proceedings ArticleDOI
12 Apr 1997
TL;DR: The results of this work will serve as a framework for creating an object-oriented, poly-functional FFT implementation which will automatically choose the most efficient algorithm given user-specified constraints.
Abstract: A large number of fast Fourier transform (FFT) algorithms have been developed over the years. Among these, the most promising are the radix-2, radix-4, split-radix, fast Hartley transform (FHT), quick Fourier transform (QFT), and the decimation-in-time-frequency (DITF) algorithms. We present a rigorous analysis of these algorithms that includes the number of mathematical operations, computational time, memory requirements, and object code size. The results of this work will serve as a framework for creating an object-oriented, poly-functional FFT implementation which will automatically choose the most efficient algorithm given user-specified constraints.