A Closed Set of Normal Orthogonal Functions

Walsh, J. L.

doi:10.2307/2387224

Cited by 832 publications

(197 citation statements)

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“…In this section, we will briefly discuss some of those applications in error-control coding Peterson (1961), and to define Walsh functions Walsh (1923) that can be used for signal modelling Harmuth (1960) and transformation of information in image compression and image encoding Jain (1989). We will also consider the use of Hadamard matrices to design spreading sequences for application in code division multiple access (CDMA) direct sequence (DS) spread spectrum (SS) systems Lam and Tantaratana (1994).…”

Section: Applications Of Hadamard Matrices In Telecommunications and mentioning

confidence: 99%

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On some applications of Hadamard matrices

Seberry

Wysocki

2005

Metrika

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Modern communications systems are heavily reliant on statistical techniques to recover information in the presence of noise and interference. One of the mathematical structures used to achieve this goal is Hadamard matrices. They are used in many different ways and some examples are given. This paper concentrates on code division multiple access systems where Hadamard matrices are used for user separation. Two older techniques from design and analysis of experiments which rely on similar processes are also included. We give a short bibliography (from the thousands produced by a google search) of applications of Hadamard matrices appearing since the paper of Hedayat and Wallis in 1978 and some applications in telecommunications. Disciplines Physical Sciences and Mathematics Publication DetailsThis article was originally published as Seberry, J, Wysocki, Bj and Wysocki, TA, On some applications of Hadamard matrices, Metrika, 62, (2005) Abstract Modern communications systems are heavily reliant on statistical techniques to recover information in the presence of noise and interference. One of the mathematical structures used to achieve this goal is Hadamard matrices. They are used in many different ways and some examples are given. This paper concentrates on code division multiple access systems where Hadamard matrices are used for user separation. Two older techniques from design and analysis of experiments which rely on similar processes are also included. We give a short bibliography (from the thousands produced by a google search) of applications of Hadamard matrices appearing since the paper of Hedayat and Wallis in 1978 and some applications in telecommunications.

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Section: Applications Of Hadamard Matrices In Telecommunications and mentioning

confidence: 99%

“…The Walsh functions constitute a complete set of orthogonal functions Walsh (1923), which means that every function defined on the interval [0, 1] and having finite norm can be expanded into the generalized Fourier series [26] using Walsh functions.…”

Section: W Al(0 T) W Al(7 T) W Al(3 T) W Al(4 T) W Al(1 T) W Al(mentioning

confidence: 99%

On some applications of Hadamard matrices

Seberry

Wysocki

2005

Metrika

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“…The WHT is an orthogonal transform whose basis functions consist of a set of rectangular discontinuous waveforms that take the value -1 and 1 [10]. The main advantage over other orthogonal transforms is its low computational complexity since a small number of additions and subtractions are required.…”

Section: The Proposed Algorithmmentioning

confidence: 99%

Complexity adaptation in H.264/AVC video coder for static cameras

Kim

O’Connor

2009

2009 Picture Coding Symposium

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H.264/AVC uses variable block size motion estimation (VBS-ME) to improve coding gain. However, its complexity is significant and fixed regardless of the required quality or of the scene characteristics. In this paper, we propose an adaptive complexity algorithm based on using the Walsh Hadamard Transform (WHT). VBS automatic partition and skip mode detection algorithms also are proposed. Experimental results show that 70% -5% of the computation of H.264/AVC is required to achieve the same PSNR.

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“…Again we confine ourselves to base 2 (for more information see [3,24,25,30]). In the following let N 0 denote the set of non-negative integers.…”

Section: Walsh Functionsmentioning

confidence: 99%

On the mean square weighted L₂discrepancy of randomized digital (t,m,s)-nets over Z₂

Dick

Pillichshammer²

2005

Acta Arith.

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On the mean square weighted L 2 discrepancy of randomized digital (t, m, s)-nets over Z 2 by Josef Dick (Sydney) and Friedrich Pillichshammer (Linz)1. Introduction. We study distribution properties of point sets in the s-dimensional unit cube [0, 1) s . There are various measures for the equidistribution of such point sets (see for example [7,10,11,16,19]). The one we consider here is based on the following function. For a set P N = {x 0 , . . . , x N −1 } of points in the s-dimensional unit cube [0, 1) s the discrepancy function is defined asThe discrepancy function measures the difference of the portion of points in an axis parallel box containing the origin and the volume of this box. Hence it is a measure of the irregularity of distribution of a point set in [0, 1) s . There are of course other functions which serve a comparable purpose, though this function has drawn a great deal of attention as various connections with applications have been pointed out, notably in numerical integration of functions (see for example [19,29]). Further, we can use different norms of the discrepancy function, again yielding different quality measures. Amongst those norms especially the L 2 norm and the L ∞ norm are of considerable interest and have been studied extensively (see for example [19,29]). In the following we introduce some notation and define the weighted L 2 discrepancy of a point set, which will be the focus of this paper.Let D = {1, . . . , s}. For u ⊆ D let γ u be a non-negative real number, |u| the cardinality of u and for a vector x ∈ [0, 1) s let x u denote the vector from 2000 Mathematics Subject Classification: 11K38, 11K06.

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A Closed Set of Normal Orthogonal Functions

Cited by 832 publications

References 1 publication

On some applications of Hadamard matrices

On some applications of Hadamard matrices

Complexity adaptation in H.264/AVC video coder for static cameras

On the mean square weighted L₂discrepancy of randomized digital (t,m,s)-nets over Z₂

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A Closed Set of Normal Orthogonal Functions

Cited by 832 publications

References 1 publication

On some applications of Hadamard matrices

On some applications of Hadamard matrices

Complexity adaptation in H.264/AVC video coder for static cameras

On the mean square weighted L2discrepancy of randomized digital (t,m,s)-nets over Z2

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Product

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On the mean square weighted L₂discrepancy of randomized digital (t,m,s)-nets over Z₂