Showing papers on "Hadamard transform published in 1973"
390 citations
TL;DR: The Hadamard product of two matrices multiplied together elementwise is a rather neglected concept in matrix theory, and has found only brief and scattered application in statistical analysis.
316 citations
01 Mar 1973
TL;DR: In this paper, an initial bounding-value problem for the Korteweg-deVries equation is considered for x, t ≥ 0, where x is a uniform open channel with a wavemaker at one end.
Abstract: An initial-boundary-value problem for the equationis considered for x, t ≥ 0. This system is a model for long water waves of small but finite amplitude, generated in a uniform open channel by a wavemaker at one end. It is shown that, in contrast to an alternative, more familiar model using the Korteweg–deVries equation, the solution of (a) has good mathematical properties: in particular, the problem is well set in Hadamard's classical sense that solutions corresponding to given initial data exist, are unique, and depend continuously on the specified data.
109 citations
TL;DR: For complex Hadamard matrices, Turyn as discussed by the authors showed that symmetric conference matrices cannot exist for any order larger than 1+a2 +b2 where a, b are integers.
Abstract: R. J. Turyn introduced complex Hadamard matrices and showed that if there is a complex Hadamard matrix of order c and a real Hadamard matrix of order h> > 1, then there is a real Hadamard matrix of order he. Previously, complex Hadamard matrices were only known for a few small orders and the orders for which symmetric conference matrices were known. These latter are known only to exist for orders which can be written as 1+a2 +b2 where a, b are integers. We give many constructions for new infinite classes of complex Hadamard matrices and show that they exist for orders 306,650, 870,1406,2450 and 3782: for the orders 650, 870, 2450 and 3782, a symmetric conference matrix cannot exist.
68 citations
TL;DR: The new design philosophy of a three-stage structure is believed to offer at least a suboptimal search strategy for recognizing printed Chinese characters with a dictionary of 7000–8000 characters.
55 citations
TL;DR: High-speed algorithms to compute the discrete Hadamard and Walsh transforms of speech waveforms have been developed and a digital processor, which allows real-time analysis of speech to be conducted on the system, is described.
Abstract: High-speed algorithms to compute the discrete Hadamard and Walsh transforms of speech waveforms have been developed. Intelligible speech has been reconstructed from dominant Hadamard or Walsh coefficients on a medium sized computer in a non-real-time mode. Degradation of some phonemes was noted at low bit rates of reconstruction, but the reconstruction could be improved by varying the position of the sampling window. A digital processor, which allows real-time analysis of speech to be conducted on the system, is described.
37 citations
TL;DR: In this paper, a non-recursive transform equation for the Fourier transform of a discrete Walsh function with a rectangular pulse was developed, which is a function of the bits of the Gray code number for the order of the Walsh function.
Abstract: Convolution of a discrete Walsh function with a rectangular pulse simplifies the derivation of an expression for the Fourier transform of a Walsh function. The nonrecursive transform equation that is developed is a function of the bits of the Gray code number for the order of the Walsh function.
33 citations
29 citations
Patent•
02 Oct 1973
TL;DR: In this paper, a method and apparatus for transforming the analog waveform of a signal into its Hadamard characterization by performing a matrix multiplication using the hadamard matrix is presented.
Abstract: A method and apparatus for transforming the analog waveform of a signal into its Hadamard characterization by performing a matrix multiplication using the Hadamard matrix and for analyzing the resulting Hadamard characterization of the signal for identification purposes. A parallel adder system employing recirculating shift registers utilizes the unique properties of the Hadamard matrix so as to reduce the matrix multiplication required in the transformation to a minimal number of simple addition and subtraction operations.
28 citations
TL;DR: The related combinatorial designs are the main concern of this paper; some results dealing with the existence and construction of modular Hadamard matrices will be included in a later paper.
26 citations
TL;DR: The advantages and drawbacks of Hadamard transform spectroscopy are examined in detail and existing theory, as reported in the literature, is not consistent with the claims made for the technique.
Abstract: Recent articles have claimed a significant S/N advantage of Hadamard transform spectroscopy over Fourier transform spectroscopy. The scanty published data does not support this assertion, and the possibility that the claim is valid in theory is examined. Existing theory, as reported in the literature, is not consistent with the claims made for the technique. The advantages and drawbacks of Hadamard transform spectroscopy are examined in detail.
TL;DR: A hybrid special-purpose computer based on a recursive algorithm for discrete Walsh transform computations is described, which uses feedback in time to reduce the required number of summing junctions to N, the number of data points to be transformed.
Abstract: A hybrid special-purpose computer based on a recursive algorithm for discrete Walsh transform computations is described. The device uses feedback in time to reduce the required number of summing junctions to N, the number of data points to be transformed. Computations are completed in 700 ?s when N = 256.
TL;DR: The motivation for this study is to show that while the WHT is inherently associated with the notion of dyadic time shifts, it does have analogous properties with respect to cyclic time shifts.
Abstract: It is shown that the complete set of circular shift invariants called the Q-spectrum, of the Walsh-Hadamard transform (WHT) of a periodic sequence is related to the cyclic autocorrelation of the given sequence through the Hadamard matrices. It is also shown that the modified WHT (MWHT) of the cyclic autocorrelation yields the Q-spectrum within some scale factors. This is analogous to the discrete Fourier transform (DFT) case, i.e., the DFT of the autocorrelation of a sequence yields the shift invariant power spectrum. The Q-spectrum can be computed efficiently using the MWHT rather than the WHT. A physical interpretation for the Q-spectrum is also provided. The motivation for this study is to show that while the WHT is inherently associated with the notion of dyadic time shifts, it does have analogous properties with respect to cyclic time shifts.
TL;DR: A very simple scheme of the transform is found, where digitized videophone signals are transformed to Hadamard components all digitally just by n digital adders and some shift registers for 2nth-order transform, extendable to two-dimensional transform with ease.
Abstract: Various image coding schemes have been studied for digital transmission of videophone signals. The Hadamard transform, which is now studied for the transmission of pictures such as those from satellites, has been considered too complicated for public use, though the characteristics such as the ratio of bit-rate reduction are more desirable than those of differential pulse-code modulation (DPCM). We have found a very simple scheme of the transform, where digitized videophone signals are transformed to Hadamard components all digitally just by n digital adders and some shift registers for 2nth-order transform. For example, three adders are necessary for eighth-order transform. It is extendable to two-dimensional transform with ease. We have made an experimental model running in real time. Experiments and theoretical calculation have shown that 3 bits/sample are required for good picture quality in the case of two-dimensional (4 \times 2) th transform and 0.5 bits more for one-dimensional eighthorder transform.
TL;DR: It is shown that if four suitable matrices of order m exist then there are Hadamard matricesof order 28m, 36m, and 44m and that if n is the order of a conference matrix there is an Hadamards matrix of order 4mn.
01 Jan 1973
TL;DR: The Hadamard difference set as mentioned in this paper is a subset of a group G of order v such that every nonidentity element of G can be expressed as a product d 1 d 2 -1 of elements of D in exactly λ ways.
Abstract: This chapter discusses Hadamard difference sets. The difference set is a subset D of size k of a group G of order v such that every nonidentity element of G can be expressed as a product d 1 d 2 -1 of elements of D in exactly λ ways. A difference set D is said to be cyclic, Abelian, non-Abelian, if the group G has the corresponding property. A difference set with λ= 1 is sometimes called planar or simple. If G is an abelian group written in additive notation, the defining condition is that every nonzero element of G can be written as a difference of elements of D in exactly λ ways.
TL;DR: A modified factorization procedure is developed that is likely as economical in storage requirements and in the number of computational operations as the conventional fast Hadamard transform.
Abstract: For the Hadamard matrix, a modified factorization procedure is developed that is likely as economical in storage requirements and in the number of computational operations as the conventional fast Hadamard transform. Using this specific factoring method, the procedure for obtaining the fast Hadamard transform may be interpreted as operations on an H diagram. The H diagram was originally derived by Marihugh and Anderson [1] to provide a graphical representation for logic functions.
01 May 1973
TL;DR: In this article, a sampling theorem for sequency-bandlimited waveforms is formulated and the requisite data-reconstruction filter discussed based upon some fundamental relations of the transform theory using Walsh functions.
Abstract: Based upon some fundamental relations of the transform theory using Walsh functions two alternative expressions for the Walsh transform of a sampled time function are derived. A sampling theorem for sequency-bandlimited waveforms is then formulated and the requisite data-reconstruction filter discussed.
TL;DR: A computational algorithm is developed to arrange the transform in an increasing order of sequencies and the unscrambling is achieved by exchanges of components and requires no additional storage.
Abstract: The fast computational algorithm based on matrix Kronecker products yields the Hadamard transform in a scrambled order of sequencies. A computational algorithm is developed to arrange the transform in an increasing order of sequencies. The unscrambling is achieved by exchanges of components and requires no additional storage.
TL;DR: In this article, the authors investigated the possibilities for the number of positive terms in the permanent expansion of an n ¾ n ~- 1-matrix and obtained partial results.
Abstract: odd. The impossibility is easy to establish either by the use of Hadamard's determinant theorem (see, for example, [2] page 418) or directly. This problem suggests the more general question of investigating what possibilities exist for the signs of the n! terms in the permanent expansion of an n n ~- 1-matrix. We ask a simpler question than this; namely, what are the possibilities for the number of positive terms ? Some partial results are obtained in this paper. 1.2. Unless otherwise stated, the (square) matrices considered throughout have elements 4- 1 ; and we shall generally use a ,grid' such as
TL;DR: A "natural" choice of the short-time functions does not lead to a Walsh-Fourier transform pair (dyadic Wiener-Khintchine theorem), and thus a second kind of short- time dacf and short- Time Walsh energy spectrum are defined as the Walsh-fourier transforms of the first kind.
Abstract: A short-time dyadic autocorrelation function (dacf) and a short-time Walsh energy spectrum of the first kind are defined in the Walsh-Fourier domain. The "natural" choice of the short-time functions does not lead to a Walsh-Fourier transform pair (dyadic Wiener-Khintchine theorem), and thus a second kind of short-time dacf and short-time Walsh energy spectrum are defined as the Walsh-Fourier transforms of the first kind. This leads to a meaningful and convenient Walsh transform pair between the first short-time Walsh energy spectrum and the second short-time dacf. The measurement procedures for both kinds of functions are discussed, and the mean values of these short-time functions are shown to be related to the corresponding long-time functions.
Journal Article•
01 Jan 1973
TL;DR: The existence of Szekeres difference sets, X and Y, of size 2f with y E Y = -y E Y, where q = 4f + 1 is a prime power, q = 5 (mod 8) and q = p2 + 4, is demonstrated in this paper.
Abstract: The existence of Szekeres difference sets, X and Y, of size 2f with y E Y = -y E Y, where q = 4f + 1 is a prime power, q = 5 (mod 8) and q = p2 + 4, is demonstrated This gives amicable Hadamard matrices of order 2(q + 1), and if 2q is also the order of a symmetric conference matrix, a regular symmetric Hadamard matrix of order 4q2 with constant diagonal Disciplines Physical Sciences and Mathematics Publication Details Jennifer Seberry Wallis, A note on amicable Hadamard matrices, Utilitas Mathematica, 3, (1973), 119-125 This journal article is available at Research Online: http://rouoweduau/infopapers/948 A NOTE ON AMICABLE HADAMARD MATRICES
Patent•
04 Jun 1973
TL;DR: In this article, a Walsh transform computer is used to transform input data, such as, a stream of sampled signals, a digital signal, or the output of a photo-diode array, according to a Kronecker-matrix rule.
Abstract: A Walsh transform computer to transform input data, such as, a stream of sampled signals, a digital signal, or the output of a photo-diode array, according to a Kronecker-matrix rule.
TL;DR: This correspondence treats the derivation of natural moments from their corresponding Walsh characteristic function via the dyadic derivative operator via the Dyadic derivativeoperator.
Abstract: This correspondence treats the derivation of natural moments from their corresponding Walsh characteristic function via the dyadic derivative operator. The derivation of a result concerning Walsh transforms of dyadic derivatives of functions is also considered. However, some established ideas such as Walsh transform, dyadic stationarity, and dyadic correlation are introduced first.
TL;DR: The paper presents and compares quantitatively various compression techniques based on the Shannon-Fano, ‘run-length’ and Hadamard transformation methods of source encoding, and the compression ratios obtained when applying the techniques to actual satellite data are given.
Abstract: This paper reports on an investigation into the application of data compression techniques as a means of reducing the ‘on-ground’ data storage requirements that are associated with many space research programmes. The paper presents and compares quantitatively various compression techniques based on the Shannon-Fano, ‘run-length’ and Hadamard transformation methods of source encoding. The compression ratios obtained when applying the techniques to actual satellite data are given and some new basic theory relating to ‘run-length’ encoding is presented.