Showing papers on "Hadamard transform published in 1970"
TL;DR: In this paper, strongly regular graphs derived from combinatorial designs are derived from the use of adjacency matrices that have elements 0 on the diagonal, −1 or + 1 elsewhere accordingly as the corresponding vertices are adjacent or nonadjacent, respectively.
Abstract: Publisher Summary This chapter explores strongly regular graphs derived from combinatorial designs. It presents graphs that are undirected, without loops, and without multiple edges. It discusses the use of adjacency matrices that have elements 0 on the diagonal, −1 or + 1 elsewhere accordingly as the corresponding vertices are adjacent or nonadjacent, respectively. The chapter presents a fiber-type construction method for graphs, which is applied to block designs with λ = 1 and yields strongly regular graphs. It also discusses block designs in which the number of points in the intersection of any pair of blocks attains only two values. The methods are applied to the construction of symmetric Hadamard matrices with constant diagonal. These matrices are related to special strong graphs. Symmetric Hadamard matrices with constant diagonal are involved in various current investigations.
183 citations
TL;DR: It is shown how the Kronecker product can be mathematically defined and efficiently implemented using a matrix factorization method and a generalized spectral analysis is suggested, and a variety of examples are presented displaying various properties of the decompositions possible.
Abstract: A technique is presented to implement a class of orthogonal transformations on the order of pN log p N operations. The technique is due to Good [1] and implements a fast Fourier transform, fast Hadamard transform, and a variety of other orthogonal decompositions. It is shown how the Kronecker product can be mathematically defined and efficiently implemented using a matrix factorization method. A generalized spectral analysis is suggested, and a variety of examples are presented displaying various properties of the decompositions possible. Finally, an eigenvalue presentation is provided as a possible means of characterizing some of the transforms with similar parameters.
123 citations
TL;DR: It is pointed out the applicability of optical Hadamard-transform coding to detector- noise-limited image scanning and approximate signal-to-noise ratio gains over conventional point-by-point scanning for several whole-image multiplex scanning schemes.
Abstract: We point out the applicability of optical Hadamard-transform coding to detector-noise-limited image scanning and give approximate signal-to-noise ratio gains over conventional point-by-point scanning for several whole-image multiplex scanning schemes.
55 citations
TL;DR: In this paper, the authors provide an overview of symmetric Hadamard matrices of order 36, showing that the Steiner triple systems of order 15 can be switched into strongly regular graphs.
Abstract: Publisher Summary This chapter provides an overview of symmetric Hadamard matrix of order 36. A square matrix H is a Hadamard matrix if its elements are + 1 and if it is orthogonal. If in addition, H is symmetric, has a constant diagonal I, and has order 36, then H2 = 36I, H = HT, H = A + I, (A−5I) (A+7I) = 0.The chapter explains a method to obtain symmetric Hadamard matrices of order 36 from the 12 Latin squares of order 6, and from the 80 Steiner triple systems of order 15 . The graphs obtained in this way are shown to be nonequivalent with respect to switching, with one exceptional pair. The method consists of counting void and complete sub-graphs, and is capable of wider application. The chapter also explains how graphs of the Steiner type can be switched into strongly regular graphs. Among these graphs, there are 23 that can be made regular with valency 21, whereas all 80 can be made regular with valency I5. This follows from a computer search by which each of the 80 Steiner triple systems appears to contain a 3-factor, that is, a subsystem of 15 triples in which every symbol occurs exactly three times.
41 citations
Patent•
30 Apr 1970TL;DR: In this article, the difference signal is distributed by transformation prior to coding, e.g., by quantizing, and transmission, preferably by means of the Fourier, Hadamard, or other unitary matrix transforms, to "'''scramble'' them relatively homogeneously in the domain of the transformed variable''.
Abstract: Immunity to transmission errors and worthwhile bandwidth reduction are achieved by distributing the difference signal, developed in differentially coding an image signal, over a spatially large interval or area, and transmitting the coded distributed signal instead of the image or differential signal. Line or frame image difference signals are, accordingly, dispersed by transformation prior to coding, e.g., by quantizing, and transmission, preferably by means of the Fourier, Hadamard, or other unitary matrix transforms, to ''''scramble'''' them relatively homogeneously in the domain of the transformed variable. Each transmitted image element thus represents a weighted sum of many or all of the elements of the corresponding line or frame. Simpler coding and other economies are achieved, particularly for relatively slowly varying sequences of images.
32 citations
TL;DR: This correspondence describes a method of computing the Hadamard transform and the R-transform that produces the transform components in the order of the increasing sequency of the Walsh functions represented by the rows of a symmetric hadamard matrix.
Abstract: This correspondence describes a method of computing the Hadamard transform and the R-transform. This method produces the transform components in the order of the increasing sequency of the Walsh functions represented by the rows of a symmetric Hadamard matrix for which the sequency of each row is larger than the sequency of the preceding row.
32 citations
TL;DR: The theory of the finite Walsh transform is presented as a background to current interest in the fast Walsh transform algorithm.
Abstract: The theory of the finite Walsh transform is presented as a background to current interest in the fast Walsh transform algorithm.
22 citations
TL;DR: The matrix form of the Walsh functions as defined in the above-mentioned short note can be generated by the modulo-2 product of two generating matrices: the natural binary code, and the transpose of the bit-reversedform of the first.
Abstract: The matrix form of the Walsh functions as defined in the above-mentioned short note [1] can be generated by the modulo-2 product of two generating matrices: the natural binary code, and the transpose of the bit-reversed form of the first. As a result, the coefficients of the Walsh transform occur in bit-reversed order. By simply reordering the Walsh functions themselves to correspond to generation by the product of two such code matrices, neither or both in bit-reversed form, the Walsh coefficients occur in their natural order.
15 citations
TL;DR: Pedagogical aspects pertaining to the algorithms that are used to compute the discrete Fourier and the Hadamard transforms are considered to introduce the student to this class of algorithms before proceding with more rigorous developments.
Abstract: SOme pedagogical aspects pertaining to the algorithms that are used to compute the discrete Fourier and the Hadamard transforms are considered. Elementary matrix partitioning techniques are used to illustrate the manner in which these algorithms work and how they are related. It is felt that this approach can be used to good advantage to introduce the student to this class of algorithms before proceding with more rigorous developments.
14 citations
TL;DR: In this article, the properties of BIFORE (Hadamard) transforms are compared with those of discrete Fourier transform (d.f.t.) and compare their properties to those of the Hadamard transform.
Abstract: Elementary properties of discrete Fourier transforms (d.f.t.) have appeared in recent literature. Corresponding properties of BIFORE (Hadamard) transforms are summarised and compared with those of the d.f.t.
12 citations
Patent•
24 Aug 1970
TL;DR: A clock driven generator which produces desired Walsh functions in response to binary number input command signals is described in this paper, where the input commands are connected to various circuitry and thereby produce output signals which are individually a particular bit of a desired Walsh function.
Abstract: A clock driven generator which produces desired Walsh functions in response to binary number input command signals. The input command signals are connected to be operated on by various circuitry and thereby produce output signals which are individually a particular bit of a desired Walsh function. The Walsh function is obtained by scanning the output signals.
TL;DR: The problem of determining Ihe distributions of all possible outputs of the proposed system for other classes of inputs is shown to be equivalent to the unsolved problem of finding the weight enumerators for the cosets of the first order Reed-Muller codes.
Abstract: M. R. Schroeder recently proposed a scheme for compression of motion picture data by taking the difference of two successive frames and then smearing.1 The smearing is accomplished by a Hadamard matrix. If the Hadamard matrix is of a certain particularly well-understood type, then we show that if the input differential picture consists of a small odd number of large pulses of identical magnitudes (but arbitrary signs), then the output will consist of three components: (i) Large pulses of equal magnitude and the correct signs, matching each of the input pulses. (ii) One additional “stray” large pulse, of magnitude equal to the others, but located at a point where the input was zero. (iii) Scatlered pulses of amplitude low relative to the pulses of types i and ii, but so numerous that they consume (π − 2)/π of the total energy of the output differential picture. We give an explicit formula for the amplitude of each of these pulses. The problem of determining Ihe distributions of all possible outputs of the proposed system for other classes of inputs is shown to be equivalent to the unsolved problem of finding the weight enumerators for the cosets of the first order Reed-Muller codes.
TL;DR: In this paper, an explicit formula for the maximum cardinality of such an orthogonal set when the ring involved is Zm, the integers modulo m, is given.
TL;DR: A note on Hadamard products can be found in this paper, where the authors discuss the properties of hadamards and their applications in the field of algebraic geometry, e.g.
Abstract: (1970). A Note on Hadamard Products. The American Mathematical Monthly: Vol. 77, No. 10, pp. 1087-1087.
TL;DR: Hadamard elementary solutions for the tri-axially symmetric potential equation in space of three dimensions and for the bi-axial symmetric possible equation in two dimensions were found in this article.
Abstract: Hadamard elementary solutions are found for the tri-axially symmetric potential equation in space of three dimensions and for the bi-axially symmetric potential equation in space of two dimensions. The elementary solutions involve hypergeometric functions of several variables.
TL;DR: The nature of the Hadamard algorithm is analyzed, and a simple method is outlined for constructing the hadamard and Predvoditelev algorithms.
Abstract: The nature of the Hadamard algorithm is analyzed, and a simple method is outlined for constructing the Hadamard and Predvoditelev algorithms. Generalized conditions which hold at interfaces in transport problems are found.
01 Jan 1970
TL;DR: Wallis et al. as discussed by the authors presented configurations and Hadamard matrices for the first time in 1970, and the results showed that configurations and hadamard matrix matrices are equivalent to configurations and configurations.
Abstract: v, k, lambda) Configirations and Hadamard matrics Disciplines Physical Sciences and Mathematics Publication Details Jennifer Seberry Wallis, (v, k, lambda) configurations and Hadamard matrices, Journal of the Australian Mathematical Society, 11, (1970), 297-309. This journal article is available at Research Online: http://ro.uow.edu.au/infopapers/933 (v, k, ).) CONFIGURATIONS AND HADAMARD MATRICES
TL;DR: Pseudo-random binary sequences are shown to be perfectly analogous to cyclic difference sets and this is used to develop a programme to identify all such sequences.
Abstract: Pseudo-random binary sequences are shown to be perfectly analogous to cyclic difference sets and this is used to develop a programme to identify all such sequences. Residue, recursive and Hadamard sequences are shown to be describable by the properties of the respective difference set and a classification system is proposed.