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Showing papers on "Hartley transform published in 1989"


Journal ArticleDOI
TL;DR: In this article, the authors compared five real-valued orthogonal transforms in terms of learning characteristics and computational complexity, and showed that the effect of an ideal transform is to convert equal error contours that are initially hyperellipses in the parameter space into hyperspheres.
Abstract: It has been previously shown that a real-time decomposition of the incoming signal into a set of partially uncorrelated components via an orthogonal transform, and a subsequent adaptation on these individual components, leads to faster convergence rates. Here, transform domain processing is characterized by the effect of the transform on the shape of the mean-square error surface. It is shown that the effect of an ideal transform is to convert equal error contours that are initially hyperellipses in the parameter space into hyperspheres. Five specific real-valued orthogonal transforms are compared in terms of learning characteristics and computational complexity. Since the Karhunen-Loeve transform (KLT) is the ideal transform for this application, and since the KLT is defined in terms of the statistics of the input signal, it is certain that no fixed-parameter transform can deliver optimal learning characteristics for all input signals. However, the simulations suggest that transforms can be found which give much improved performance in a given situation. >

165 citations


Journal ArticleDOI
TL;DR: The authors present an electronic circuit, based on a neural (i.e. multiply connected) net to compute the discrete Fourier transform (DFT), and compare its performance to other on-chip DFT implementations.
Abstract: The authors present an electronic circuit, based on a neural (i.e. multiply connected) net to compute the discrete Fourier transform (DFT). They show both analytically and by simulation that the circuit is guaranteed to settle into the correct values within RC time constants (on the order of hundreds of nanoseconds), and they compare its performance to other on-chip DFT implementations. >

76 citations



Journal ArticleDOI
TL;DR: A fast algorithm is proposed to compute the discrete Hilbert transform via the fast Hartley transform (FHT), where the computation complexity can be greatly reduced from two complex FFTs into two real FHTs.
Abstract: A fast algorithm is proposed to compute the discrete Hilbert transform via the fast Hartley transform (FHT). Instead of the conventional fast Fourier transform (FFT) approach, the processing is carried out entirely in the real domain. Also, since many efficient FHT algorithms exist, the computation complexity can be greatly reduced from two complex FFTs into two real FHTs. >

34 citations


Journal ArticleDOI
TL;DR: A brief tutorial is given on what aliasing means and some of the conventional wisdom about aliasing is described, and why that wisdom may not be so wise is explained.
Abstract: A brief tutorial is given on what aliasing means. Plots of some relevant functions are shown. Some of the conventional wisdom about aliasing is described, and why that wisdom may not be so wise is explained. Aliasing is actually an image processing phenomenon involving the Fourier transform, convolution and the convolution theorem. >

28 citations


Journal ArticleDOI
TL;DR: The proposed transform offers a higher computational efficiency than the traditional even discrete cosine transform and yields a mean-squared error close to that of the DCT.
Abstract: A computationally efficient DCT- (discrete-cosine-transform) type orthogonal transform obtained by using a construction method developed by W. Kou and H. Zu (1986) is proposed. A recursive relation exists between a higher order and a lower order form of the kernel matrix of the transform and includes the DCT matrices as submatrices. The characteristics of the proposed transform and several fast transform algorithms are discussed. The proposed transform offers a higher computational efficiency than the traditional even discrete cosine transform and yields a mean-squared error close to that of the DCT. Since the HDCT has window spectral structures, it can be used in signal filtering and speech and image processing. >

23 citations


Journal ArticleDOI
TL;DR: In this article, a least squares IIR (infinite impulse response) algorithm, in the transformed domains, which fits each of the retained subsets of the complex transform components accurately, is presented.
Abstract: The mixed transform representation of time-varying signals uses partial sets of basis functions from the discrete Fourier transform (DFT) and the Walsh-Hadamard transform. The location, magnitude, and phase of the transform components have to be specified for proper signal reconstruction. A least-squares IIR (infinite impulse response) algorithm, in the transformed domains, which fits each of the retained subsets of the complex transform components accurately, is presented. The IIR function, while characterized by real coefficients about twice the number of the retained complex transform components, carries enough location, magnitude, and phase information for accurate signal reconstruction. To illustrate the technique's accuracy and efficiency, its application to model the DFT part of a voice speech segment is given. >

19 citations


Journal ArticleDOI
TL;DR: A new fast algorithm for computing the two-dimensional discrete Hartley transform that requires the lowest number of multiplications compared with other related algorithms is presented.
Abstract: A new fast algorithm for computing the two-dimensional discrete Hartley transform is presented. This algorithm requires the lowest number of multiplications compared with other related algorithms.

18 citations


Journal ArticleDOI
TL;DR: The use of the Hartley transform (HT) in cepstrum analysis, as a substitute for the more commonly used Fourier transform (FT), is examined.
Abstract: The use of the Hartley transform (HT) in cepstrum analysis, as a substitute for the more commonly used Fourier transform (FT), is examined. With this substitution, the input to the cepstrum must be in the real domain only. The benefits of using the HT are approximately 50% less data memory required and approximately 40% faster program execution, at no loss in accuracy. >

17 citations



Journal ArticleDOI
TL;DR: In this paper, the authors studied the wideband approximation of radar echoes, where the objects are described by their distribution D in the range velocity plane and the inverse problem of reconstructing D given the echoes of some signals is considered.
Abstract: The author studies the wideband approximation of radar echoes. The objects are described by their distribution D in the range velocity plane and he considers the inverse problem of reconstructing D given the echoes of some signals. A special choice of signals leads to an integral transform which maps D to its integrals over hyperbolae. This transform is related to the Radon transform and an inversion formula is proved.

Journal ArticleDOI
TL;DR: The Hartley transform achieves better coding performance than the Fourier transform, but is inferior to the cosine transform.
Abstract: The data compression performance of the Hartley transform on a Markov-1 signal is theoretically compared to that of the Fourier transform. Covariance distribution and residue correlation measurements have been computed for the Hartley, Fourier, and cosine transforms. The Hartley transform achieves better coding performance than the Fourier transform, but is inferior to the cosine transform. >

Journal ArticleDOI
TL;DR: A new method to evaluate the WVD of a real signal using the fast Hartley transform (FHT) is presented, compared with the existing fast Fourier transform (FFT) method in terms of computation time.
Abstract: The Wigner-Ville distribution (WVD) is of great significance in time-frequency signal analysis. In the letter we present a new method to evaluate the WVD of a real signal using the fast Hartley transform (FHT). This is compared with the existing fast Fourier transform (FFT) method in terms of computation time. The FHT method presented turns out to be much faster than the FFT method.

Journal ArticleDOI
TL;DR: The Hartley transform as discussed by the authors represents real data by real transform values, rather than complex, a feature that carries over into optical interferometry and is used for object phase determination starting from only amplitude information in the transform domain.

Journal ArticleDOI
TL;DR: The authors investigated the implementation aspects of the fast Hartley transform (FHT) in both software and hardware and describe the modifications required to convert existing fast Fourier transform programs to execute FHTs, showing the ease with which these modifications can be implemented.
Abstract: The authors investigated the implementation aspects of the fast Hartley transform (FHT) in both software and hardware. They describe the modifications required to convert existing fast Fourier transform (FFT) programs to execute FHTs, showing the ease with which these modifications can be implemented. They compare execution time and memory storage requirements of both transforms and present power spectrum calculation and convolution as illustrative examples to compare the performances of the two transform techniques. They also give a comparative survey of the performances of various microprocessors and digital signal processors in FFT and FHT computation. >

Journal ArticleDOI
TL;DR: In this paper, a series inversion of the k-plane transform is presented, and estimates for the minimum number of discretely sampled direction sets at which the kplane transform must be known in order to recover a point function up to a given degree.
Abstract: The k-plane transform encompasses both the X-ray and Radon transforms. A series inversion which operates in the unified setting of the k-plane transform is presented. The author shows that with respect to either the Jacobi or the associated Laguerre polynomial bases for square integrable point functions, the k-plane transform assumes a block-diagonal-like form. Additionally, estimates are given for the minimum number of discretely sampled direction sets at which the k-plane transform must be known in order to recover a point function up to a given degree.

Journal ArticleDOI
01 Oct 1989
TL;DR: The Hartley transform as mentioned in this paper is a real-to-real transform that can be implemented using transversal filter-type structures, and it is suitable for VLSI implementation.
Abstract: The advances in digital fabrication technology have led to a new generation of integrated circuits capable of performing fast arithmetic operations, and opened the door to the consideration of algorithms that may be implemented with simple structures. When the number of data samples in the input block is prime or a product of primes, the Hartley transform can readily be mapped to circular convolutions and then implemented using transversal filter-type structures. Such a structure is simple and regular, and hence it is suitable for VLSI implementation. The Hartley transform is real-to-real and it is the same for both forward and inverse transformations. Hence, it is simpler and may therefore be somewhat faster than the DFT implemented by the same approach.

Journal ArticleDOI
TL;DR: In this paper, a simple closed-form solution of the Fourier transform of a polygonal shape function is given. But this solution is restricted to a single polygon.
Abstract: The authors give a simple but creative closed-form solution of the Fourier transform of a polygonal shape function.

Journal ArticleDOI
O. Buneman1
TL;DR: The Johnson-Burrus method of in situ ordering for FFTs is applied to fast Hartley transforms, tying together two consecutive butterfly operations, involving eight real data, swapping some of their results.
Abstract: The Johnson-Burrus method of in situ ordering for FFTs is applied to fast Hartley transforms. It amounts to tying together two consecutive butterfly operations, involving eight real data, swapping some of their results. Pre- and postpermutation can thereby be avoided. The identification of negative indexes, significantly in Hartley transforms, is nontrivial but easily resolved. >

Proceedings ArticleDOI
30 Oct 1989
TL;DR: The authors give convolution theorems that relate the spherical transform to convolution, sampling theorem that allow the exact computation of the transform for band-limited functions, and algorithms with asymptotically improved running time for the exact computations of the harmonic expansion.
Abstract: The problem of computing the convolution of two functions on the sphere by means of a spherical transform is considered. Such convolutions are applicable to surface recognition and the location of both rotated and translated patterns in an image. The authors give convolution theorems that relate the spherical transform to convolution, sampling theorems that allow the exact computation of the transform for band-limited functions, and algorithms with asymptotically improved running time for the exact computation of the harmonic expansion. The net result is an O(n/sup 1.5/(log n)/sup 2/) algorithm for the exact computation of the convolution of two bandlimited functions sampled at n points in accordance with the sampling theorem. The techniques developed are applicable to computing other transforms, such as the Laguerre, Hermite, and Hankel transforms. >

Proceedings ArticleDOI
03 Jan 1989
TL;DR: The Fast Hartley Transform is a promising alternative to the Fast Fourier Transform when the processed data are real numbers but the slowness of the communication imposes a limitation on the speedup when a large number of processors are used.
Abstract: The Fast Hartley Transform is a promising alternative to the Fast Fourier Transform when the processed data are real numbers. The hypercube implementation of the FHT is largely dependent on the way the computation is partitioned. A partitioning algorithm is presented which generates evenly-loaded tasks on each node and demands only a regular communication topology — the Hartley graph. Mapping from the Hartley graph to the Gray graph (binary n-cube) is straightforward, since the Hartley graph has a similar structure as the Gray graph. However, the communication is not always between the nearest neighbors and thus may take some extra time. Moreover, the slowness of the communication in the presently available architectures imposes a limitation on the speedup when a large number of processors are used.

Proceedings ArticleDOI
29 May 1989
TL;DR: In this paper, the concept of self-similarity of patterns is introduced, and it is proved that the class of selfsimilar patterns is closed with respect to spectral transformation, which is the case for both the Chrestenson and the Zhang-Hartley transforms.
Abstract: The idea of mosaics (of patterns) is introduced, and this structure is analyzed in the spectral domain. Also introduced is the concept of self-similarity of patterns, and it is proved that the class of self-similar patterns is closed with respect to spectral transformation. Self-similar patterns can be studied both with the Chrestenson and the Zhang-Hartley transforms. The latter has the advantage of being a real-valued transform. >

Journal ArticleDOI
TL;DR: In this paper, the vector-radix algorithm was proposed for computing the two-dimensional discrete Hartley transform, which does not require separability and is the fastest known algorithm.
Abstract: Three methods have been described for computing the two-dimensional discrete Hartley transform. Two of these employ a separable transform, the third method, the vector-radix algorithm, does not require separability. In-place computation of the vector-radix method is described. Operation counts and execution times indicate that the vector-radix method is fastest.

Journal ArticleDOI
TL;DR: In this paper, the Radon transform of a function is defined as an integration over planes whose normals vary over the entire unit sphere, and it is shown that only the transform over a hemisphere, which can consist of disconnected parts, is required to reconstruct the original function.
Abstract: The Radon transform of a function is defined as an integration over planes whose normals vary over the entire unit sphere. The space is actually covered twice because the distance of the plane from the origin is allowed to be positive or negative. The usual inverse transform requires knowledge of the transform evaluated over the entire sphere. However, we shall show that only the transform over a hemisphere, which can consist of disconnected parts, is required to reconstruct the original function . Thus the redundancy of the double-covering is removed and only one-half of the transform is needed to recover the original function. In essence we have introduced optical coordinates. We then consider function f(x) obtained by applying the inverse Radon transform to an arbitrary function which has the same arguments as the Radon transform but is not, in general, a Radon transform. On applying the Radon transform to f(x) we find that only part of the arbitrary function, to which the inverse was applied, is reproduced. Thus the Radon transform has a left inverse but not a right inverse. However, by restricting the range of variables in the transform space, a right and left inverse can be obtained which are the same. Finally, we give Parseval’s theorem in terms of the refined Radon transform. Though we modify the older proofs for obtaining the Radon transform and its inverse, for the sake of a self-contained paper we also use new elementary proofs based on relations which we have derived between one­-dimensional and three-dimensional delta functions. We expect that our result will have consequences in tomography and other applications. We ourselves will use the result to obtain the exact fields for the scalar three-dimensional wave equation and Maxwell’s equations from fields in the wave zone, and, conversely, fields in the wave zone from the exact causal fields. In fact, the principal reason for our writing the present paper is to cast the Radon transform and its inverse in a form suitable for these applications. Though we shall prove our result for the three-dimensional case only, the proof for the general case can be inferred from our proof.

Journal ArticleDOI
TL;DR: The fast Hartley transform provides the same information as the fast Fourier transform (FFT) but with greater speed and efficiency when the input data are real.
Abstract: The fast Hartley transform provides the same information as the fast Fourier transform (FFT) but with greater speed and efficiency when the input data are real. An algorithm for taking the Hartley transform of a long sequence on a multiprocessor machine by simultaneously transforming short subsequences does not require complex arithmetic and is faster than analogous techniques which use the Fourier transform.

Proceedings ArticleDOI
23 May 1989
TL;DR: The authors describe the implementation of real and complex FFT (fast Fourier transform) algorithms on the Motorola DSP96002, a general-purpose, dual-bus IEEE standard floating-point digital signal processor that provides the basis for efficient implementation of FFTs and other fast transforms.
Abstract: The authors describe the implementation of real and complex FFT (fast Fourier transform) algorithms on the Motorola DSP96002. The DSP96002 is a general-purpose, dual-bus IEEE standard floating-point digital signal processor (DSP). At a 74-ns instruction cycle, the DSP96002 implements a 1024-point real FFT in 0.905 ms and a 1024-point complex FFT in 1.55 ms. This performance is achieved by calculating up to three floating-point results in a single instruction cycle, or 40.5 MFLOPS peak. A radix-2 FFT butterfly is executed every four cycles, an average of 33.75 IEEE MFLOPS. The instruction set and architecture of the DSP96002 provide the basis for efficient implementation of FFTs and other fast transforms, such as the discrete Walsh-Hadamard transform, discrete cosine transform, and discrete Hartley transform. >

Journal ArticleDOI
TL;DR: The Hartley transform offers a useful alternative to the Fourier transform for the conversion of a time-domain ion cyclotron resonance (ICR) signal into its corresponding frequency-domain mass spectrum, making the FHT equivalent in speed to a "real" FFT.
Abstract: The Hartley transform offers a useful alternative to the Fourier transform for the conversion of a time-domain ion cyclotron resonance (ICR) signal into its corresponding frequency-domain mass spectrum. The Hartley transform has the advantage that it eliminates the need for complex variables, when (as for linearly polarized signals) the time-domain signal can be represented by a mathematically real function. Moreover, the Hartley transform produces the same spectra (absorption mode, dispersion mode, magnitude mode) as does the Fourier transform. In particular, the discrete fast Hartley transform (FHT) produces the same spectrum at twice the speed of a complex fast Fourier transform (FFT), making the FHT equivalent in speed to a "real" FFT. Hartley and Fourier transform methods in ICR mass spectrometry are compared and demonstrated experimentally. Essentially the same advantages and computational methods should apply to the use of the Hartley transform in place of the Fourier transform in other forms of spectrometry (e.g., nuclear magnetic resonance, infrared, etc.).

Journal ArticleDOI
TL;DR: An algorithm is introduced for computing the multidimensional finite Fourier transform and offers a substantial reduction in the computational complexity.
Abstract: An algorithm is introduced for computing the multidimensional finite Fourier transform. The algorithm can be applied to data samples of any size. In most cases, it offers a substantial reduction in the computational complexity. >

Journal ArticleDOI
TL;DR: The noise performance of the binarized Hartley phase-only filter is analyzed and compared with other BPOFs as well as with the phase- only and matched filters and it can be shown that the matched filter phase is the optimum phase-Only filter.
Abstract: Binary phase-only filters (BPOFs) are a viable candidate for the replacement of matched filters in real-time image processing and pattern recognition applications The original BPOFs were binarized versions of the real or imaginary parts of the Fourier transform Recently, a filter has been proposed that is the binarized Hartley transform In this paper, the noise performance of the binarized Hartley phase-only filter is analyzed and compared with other BPOFs as well as with the phase-only and matched filters It is well known that the matched filter optimizes the signal-to-noise ratio Further, it can be shown that the matched filter phase is the optimum phase-only filter A bound on the BPOF signal-to-noise ratio relative to that of the phase-only filter is derived Finally, the analysis is applied to the generalized form of the binarized Hartley transform

Journal ArticleDOI
TL;DR: A fast algorithm for computing the Hankel transform of order one is derived by slightly modifying the algorithm developed by E.W. Hansen (1985), which enjoys computational advantages using a rapid Abel transform with shift-variant recursive filter and a fast Fourier transform.
Abstract: A fast algorithm for computing the Hankel transform of order one is derived by slightly modifying the algorithm developed by E.W. Hansen (1985). Since the algorithm uses the formal equivalency between a Hankel transform and an Abel transform followed by a Fourier transform, it enjoys computational advantages using a rapid Abel transform with shift-variant recursive filter and a fast Fourier transform. Good agreement between actual and computer transforms was obtained in the simulation with a known transform pair. >