Showing papers on "Hartley transform published in 1985"
TL;DR: A complete set of fast algorithms for computing the discrete Hartley transform is developed, including decimation-in-frequency, radix-4, split radix, prime factor, and Winograd transform algorithms.
Abstract: The discrete Hartley transform (DHT) is a real-valued transform closely related to the DFT of a real-valued sequence. Bracewell has recently demonstrated a radix-2 decimation-in-time fast Hartley transform (FHT) algorithm. In this paper a complete set of fast algorithms for computing the DHT is developed, including decimation-in-frequency, radix-4, split radix, prime factor, and Winograd transform algorithms. The philosophies of all common FFT algorithms are shown to be equally applicable to the computation of the DHT, and the FHT algorithms closely resemble their FFT counterparts. The operation counts for the FHT algorithms are determined and compared to the counts for corresponding real-valued FFT algorithms. The FHT algorithms are shown to always require the same number of multiplications, the same storage, and a few more additions than the real-valued FFT algorithms. Even though computation of the FHT takes more operations, in some situations the inherently real-valued nature of the discrete Hartley transform may justify this extra cost.
275 citations
263 citations
TL;DR: In this paper, it was shown that the Fourier transform belongs to Lq(r, da) for a certain natural measure on the su.rface of a circular cone in R3.
Abstract: Let r be the su.rface of a circular cone in R3. We show that if 1 < p < 4/3, 1/q = 3(1-1/p) and f E LP(R3), then the Fourier transform of f belongs to Lq(r, da) for a certain natural measure a on r. Following P. Tomas we also establish bounds for restrictions of Fourier transforms to conic annuli at the endpoint p = 4/3, with logarithmic growth of the bound as the thickness of the annulus tends to zero.
77 citations
TL;DR: In this paper, the authors present a Hankel transform algorithm using a fast (linear time) Abel transform, followed by an FFT, which maps an axisymmetric two-dimensional function into a line integral projection, and a one-dimensional Fourier transform.
Abstract: The Hankel, or Fourier-Bessel, transform is an important computational tool for optics, acoustics, and geophysics. It may be computed by a combination of an Abel transform, Which maps an axisymmetric two-dimensional function into a line integral projection, and a one-dimensional Fourier transform. This paper presents a Hankel transform algorithm using a fast (linear time) Abel transform, followed by an FFT.
67 citations
TL;DR: La transformation de Hartley est reelle et offre un accroissement significatif dans la vitesse de calcul des spectres des images and dans le calcul des operations de filtrage des images.
Abstract: La transformation de Hartley est reelle et offre un accroissement significatif dans la vitesse de calcul des spectres des images et dans le calcul des operations de filtrage des images. Deux configurations optiques possibles ont ete trouvees pour representer la transformation de Hartley d'un objet
61 citations
TL;DR: The discrete Hartley transform as mentioned in this paper is a new tool for the analysis, design and implementation of digital signal processing algorithms and systems, which is strictly symmetric concerning the transformation and its inverse.
Abstract: The discrete Hartley transform is a new tool for the analysis, design and implementation of digital signal processing algorithms and systems. It is strictly symmetrical concerning the transformation and its inverse. A new fast Hartley transform algorithm has been developed. Applied to real signals, it is faster than a real fast Fourier transform, especially in the case of the inverse transformation. The speed of operation for a fast convolution can thus be increased.
46 citations
TL;DR: In this article, the nilpotent Fourier transform on spaces of distributions is used to prove the equivalence between *-products on g* for nil-potent g.
Abstract: We study the nilpotent Fourier transform on spaces of distributions. We use it to prove the equivalence between *-products on g* for nilpotent g.
33 citations
TL;DR: In this article, the analytical and numerical properties of the Fourier transform of a two-center product of exponentially declining functions (exponential-type functions, ETFs) are derived with the help of Fourier convolution theorem and Feynman's identity.
29 citations
01 Dec 1985
TL;DR: In this paper, it was shown that the DFT of a real sequence, formed via the Fast Hartley Transform, can be computed at most only 2 times faster than using a complex Fast Fourier Transform.
Abstract: It is shown that the DFT of a real sequence, formed via the Fast Hartley Transform, can be computed at most only 2 times faster than by using a complex Fast Fourier Transform. However, more sophisticated FFT algorithms exist which give the same speedup factor. A simple FHT subroutine is presented to illustrate the similarity of the FHT and FFT butterflies in their simplest forms.
27 citations
01 Dec 1985
TL;DR: An in-place version of the Fast Hartley Transform is presented, similar to that presented in the above paper, but with a new timing diagram for the computation of the DHT.
Abstract: In the above paper, Bracewell presents a new timing diagram for the computation of the DHT. We present here an in-place version of the Fast Hartley Transform.
20 citations
TL;DR: A new method to generate optical Hartley transform for 2D real images is proposed, based on polarization encoding of the coherent optical beam, and different coherent optical image processing techniques are discussed.
Patent•
19 Mar 1985TL;DR: The Discrete Bracewell Transform (DBT) as discussed by the authors is a special purpose computer and method of computation for performing an N-length real-number discrete transform, which is performed without need for employing real and imaginary parts, and in efficient embodiments, is executed efficiently and in less time than the Discrete Fourier Transform (DFT).
Abstract: A special purpose computer (35) and method of computation for performing an N-length real-number discrete transform. For a real-valued function f(tau) where tau has the values 0,1,....,(N-1), the Discrete Bracewell Transform (DBT) H (v) is as in (I), where, v = 0,1,....,N-1; cas = cos + sin. The DBT is performed without need for employing real and imaginary parts, and in efficient embodiments, is executed efficiently and in less time than the Discrete Fourier Transform (DFT). The process steps for the original transform and the inverse retransformation are the same.
TL;DR: A hybrid system has been constructed to perform the complex Fourier transform of real 2-D data based on the Radon transform, which is performed with SAW filters via the chirp transform algorithm.
Abstract: A hybrid system has been constructed to perform the complex Fourier transform of real 2-D data The system is based on the Radon transform; ie, operations are performed on 1-D projections of the data The projections are derived optically from transmissive or reflective objects, and the complex Fourier transform is performed with SAW filters via the chirp transform algorithm The real and imaginary parts of the 2-D transform are produced in two bipolar output channels
26 Apr 1985
TL;DR: A new multidimensional Hartley transform is defined and a vector-radix algorithm for fast computation of the transform is developed that is shown to be faster (in terms of multiplication and addition count) compared to other related algorithms.
Abstract: A new multidimensional Hartley Transform is defined and a vector-radix algorithm for fast computation of the transform is developed. The algorithm is shown to be faster (in terms of multiplication and addition count) compared to other related algorithms.
TL;DR: In this article, the authors proposed a method for phase retrieval from the observed modulus at the Fourier transform plane of an object in two dimensions, which consists of the logarithmic Hilbert transform in one dimension.
Abstract: This paper proposes a method for solving the phase retrieval problem from the observed modulus at the Fourier transform plane of an object in two dimensions. This method consists of the logarithmic Hilbert transform in one dimension, based on the reduction by the sampling theorem of the two-dimensional (2-D) Fourier transform of the object to the one-dimensional (1-D) Fourier transform of an effective object function. The usefulness of the method is shown in computer simulation studies of the phase retrieval from the 2-D modulus at the Fourier transform plane, for the 2-D real and positive objects. The zero information in the complex lower half-plane must be obtained from another observation for the phase evaluation using the logarithmic Hilbert transform.
TL;DR: In this paper, the authors presented a solution of the initial value problem for the Bloch equation of NMR in the case of T 1 = infinity and to relate it to a non-linear version of the Fourier transform that is termed the bloch transform.
Abstract: The author presents a solution of the initial-value problem for the Bloch equation of NMR in the case of T1= infinity and to relate it to a non-linear version of the Fourier transform that is termed the Bloch transform. The invertibility of this map is an open problem.
TL;DR: This work study how the finite Fourier transform relates to certain polynomial identities involving weight enumerator polynomials of linear codes, which include the generalized MacWilliams identities and theorems originally due to R. Gleason concerningPolynomial algebras containing weight enumerators of linear code.
Abstract: The role of the finite Fourier transform in the theory of error correcting codes has been explored in a recent text by Richard Blahut. In this work we study how the finite Fourier transform relates to certain polynomial identities involving weight enumerator polynomials of linear codes. These include the generalized MacWilliams identities and theorems originally due to R. Gleason concerning polynomial algebras containing weight enumerator polynomials. The Heisenberg group model of the finite Fourier transform provides certain algebras of classical theta functions which will be applied to reprove Gleason's results.
11 Jul 1985
TL;DR: The DHT coding system incorporated with a human visual system model is studied and this system offers about the same subjective image quality as a straight-forward DCT coding system.
Abstract: The discrete Hartley transform (DHT) and its fast algorithm were introduced recently. One of the advantages of the DHT is that the forward and inverse transforms are of the same form except for a normalization constant. Therefore, the forward and the inverse transform can be implemented by the same subroutine or hardware when the normalization constant is properly taken care of. In this paper, the applications of the DHT to image compression are studied. The distribution of the DHT coefficients is tested using the Kolmogorov-Smirnov goodness-of-fit test. The compression efficiency of DHT coding is found to be about the same as discrete Fourier transform (DFT) coding. The DHT coding system incorporated with a human visual system model is also studied and this system offers about the same subjective image quality as a straight-forward DCT coding system.
TL;DR: In this paper, a second continuous Jacobi transform is defined for suitable functions f(x) and properties of the transform are studied, and an extension of Campbell's sampling theorem is presented.
Abstract: This paper continues the work started in [1]; a second continuous Jacobi transform is defined for suitable functions f(x). Properties of the transform are studied. In particular, the first continuous Jacobi transform in [1] and the second continuous Jacobi transform are shown to be inverse to each other. The paper concludes with an extension of Campbell's sampling theorem [2].
TL;DR: The purpose of this article is to make more explicit the interplay between ring-theoretic structures and the algorithms for the finite Fourier transform and to use this new insight to construct new algorithms for evaluating the finite fourier transform on the groups Z.
TL;DR: The generalized structure of the orthogonal transformation relation by using the vector space model has been presented in this paper, which can be used for potential extension or modification of the existing orthogonality transformations.
Abstract: The introduction of an orthogonal transformation pair is generally begun with the definition, followed by the proofs of the orthogonality and the associated Parseval's relation shown as one of the properties of the transform pair. This procedure has to be repeated for various transform relations. In this article, we present the generalized structure of the orthogonal transformation relation by using the vector space model. This method enables us to visualize the similarities as well as the differences among the orthogonal transformations used in signal processing. In addition to the examples, transform pairs including Fourier transform, Fourier series, discrete-time Fourier transform (DFI), Hankel transform, Hilbert transform, the sampling theorem, Legendre, Laguerre, Hermite, and Chebychev decomposition methods are tabulated and shown as special cases of the generalized model. This approach can be used for potential extension or modification of the existing orthogonal transformations. It can be also applied to the design of special-purpose orthogonal mapping techniques.
01 Apr 1985
TL;DR: A new Hankel transform algorithm designed for uniformly sampled data is presented, which is the most elementary of several algorithms derived from the Tchebycheff and Fourier transforms.
Abstract: A new Hankel transform algorithm designed for uniformly sampled data is presented. Although data of this type occur frequently, previous algorithms require interpolations and/or numerical evaluations of Bessel functions. These difficulties can be avoided by using a Tchebycheff transform followed by a Fourier transform. The basic structure and performance of any Hankel transform algorithm derived from this two-step process depends on the combined results from the numerical methods used to compute the Tchebycheff and Fourier transforms. The algorithm presented here is the most elementary of several algorithms derived from this procedure. Examples are presented and errors associated with the results are discussed.
19 Dec 1985
TL;DR: The recursive nature of the FHT algorithm derived in this paper enables us to generate the next higher-order FHT from two identical lower- order FHTs, which offers flexibility in programming different sizes of transforms, while the orderly structure of its signal flow graphs indicates an ease of implementation in VSLI.
Abstract: The Fast Hartley TransformH. S. HouElectronics and Optics Division, The Aerospace Corporation2350 E. El Segundo Blvd., El Segundo, California 90245AbstractThe Fast Hartley Transform (FHT) is similar to the Cooley -Tukey Fast Fourier Transform(FFT) but performs much faster because it requires only real arithmetic computationscompared to the complex arithmetic computations required by the FFT. Through use of theFHT, Discrete Cosine Transforms (DCT) and Discrete Fourier Transforms (DFT) can be obtained.The recursive nature of the FHT algorithm derived in this paper enables us to generate thenext higher -order FHT from two identical lower -order FHTs. In practice, this recursiverelationship offers flexibility in programming different sizes of transforms, while theorderly structure of its signal flow graphs indicates an ease of implementation in VSLI.IntroductionRecently, Bracewe111,2 introduced the Discrete Hartley Transform (DHT) as a new member ofthe transform family. The DHT uses the real variable cos(2,rkn /N) + sin(2nkn /N) as thetransform kernel, while the Discrete Fourier Transform (DFT) uses the complex exponential,Exp(i2n kn /N), as the transform kernel. Thus, the DHT is intuitively simpler and hence,faster than the Fast Fourier Transform (FFT) since the multiplication of two complex varia-bles is equivalent to four real multiplications, and a complex addition is two ie.§.l addi-
01 Jan 1985
TL;DR: A bit-level systolic array system is proposed for the Winograd Fourier Transform Algorithm and it is demonstrated how long transforms can be implemented with components designed to perform short length transforms.
Abstract: A bit-level systolic array system is proposed for the Winograd Fourier Transform Algorithm. The design uses bit-serial arithmetic and, in common with other systolic arrays, features nearest neighbour interconnections, regularity and high throughput. The short interconnections in this method contrast favourably with the long interconnections between butterflies required in the FFT. The structure is well suited to VLSI implementations. It is demonstrated how long transforms can be implemented with components designed to perform short length transforms. These components build into longer transforms preserving the regularity and structure of the short length transform design.
TL;DR: In this article, the Padic transform bases and lengths can be easily determined by the Hensel code weight sums defined over a finite ring Pr. The maximum transform length is found to be Pr−−1(P−1) instead of(p−1), it has much wider choice and is longer than before in terms of transform size.
Abstract: The P-adic transform bases and lengths can be easily determined by the Hensel code weight sums defined over a finite ring Pr. The maximum transform length is found to be Pr−−1(P−1) instead of(P−1), it has much wider choice and is longer than before in terms of transform size.
01 Oct 1985
TL;DR: Originally presented as author's thesis (M.S. -- Massachusetts Institute of Technology), 1985.
Abstract: Originally presented as author's thesis (M.S. -- Massachusetts Institute of Technology), 1985.
06 Nov 1985
TL;DR: The prime factor Fourier transform system arestructed based on a new designed parallel processing array processor called Vector Engine and compared with some well known FFT algorithms, the performance analysis is discussed.
Abstract: A new method for implementing prime factor discrete Fourier transforms on array processors is presented. 'The prime factor Fourier transform system are mnstructed based on a new designed parallel processing array processor called Vector Engine. Basic architecture for short length Prime number Fourier transform implementations are discussed. By applying the short length architecture, the implemelltation of long length prime factor Fourier tansforms are discussed and designed. Compared with some well known FFT algorithms, the performance analysis of this system is also discussed.
01 Jan 1985
TL;DR: The discrete representation of linear transformations discussed in the pre-ceeding chapter requires efficient algorithmic implementation and the various aspects of this problem form the subject of this chapter.
Abstract: The discrete representation of linear transformations discussed in the pre-ceeding chapter requires efficient algorithmic implementation. The various aspects of this problem form the subject of this chapter.