The original paper is in English. Non-English content has been machine-translated and may contain typographical errors or mistranslations. ex. Some numerals are expressed as "XNUMX".
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The original paper is in English. Non-English content has been machine-translated and may contain typographical errors or mistranslations. Copyrights notice
No processamento digital de sinais, o teorema da amostragem afirma que qualquer função com valor real f pode ser reconstruída a partir de uma sequência de valores de f que são amostrados discretamente com uma frequência pelo menos duas vezes maior que a frequência máxima do espectro de f. Este teorema também pode ser aplicado a funções em domínio finito. Então, a faixa de frequências de f pode ser expresso com mais detalhes usando um conjunto limitado em vez da frequência máxima. Uma função cuja faixa de frequências está confinada a um conjunto limitado é chamada de função limitada por banda. E um teorema de amostragem para funções com banda limitada no domínio booleano foi obtido. Aqui, é importante obter um teorema de amostragem para funções com banda limitada não apenas no domínio booleano (GF(2)n domínio), mas também sobre GF(q)n domínio, onde q é uma potência primordial e GF(q) é o campo de ordem de Galois q. Por exemplo, em projetos experimentais, embora o modelo possa ser expresso como uma combinação linear das funções da base de Fourier e os níveis de cada fator possam ser representados por GF(q), o número de níveis geralmente assume um valor maior que dois. No entanto, o teorema da amostragem para funções com banda limitada ao longo de GF(q)n domínio não foi obtido. Por outro lado, os pontos de amostragem estão intimamente relacionados com as palavras-código de um código linear. No entanto, a relação entre a matriz de verificação de paridade de um código linear e quaisquer vetores de erro distintos não foi obtida, embora seja necessária para a compreensão do significado do teorema de amostragem para funções limitadas em banda. Neste artigo, generalizamos o teorema de amostragem para funções com banda limitada sobre o domínio booleano para um teorema de amostragem para funções com banda limitada sobre GF(q)n domínio. Apresentamos também um teorema para a relação entre a matriz de verificação de paridade de um código linear e quaisquer vetores de erro distintos. Por fim, esclarecemos a relação entre o teorema da amostragem para funções sobre GF(q)n domínio e códigos lineares.
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Yoshifumi UKITA, Tomohiko SAITO, Toshiyasu MATSUSHIMA, Shigeichi HIRASAWA, "A Note on a Sampling Theorem for Functions over GF(q)n Domain" in IEICE TRANSACTIONS on Fundamentals,
vol. E93-A, no. 6, pp. 1024-1031, June 2010, doi: 10.1587/transfun.E93.A.1024.
Abstract: In digital signal processing, the sampling theorem states that any real valued function f can be reconstructed from a sequence of values of f that are discretely sampled with a frequency at least twice as high as the maximum frequency of the spectrum of f. This theorem can also be applied to functions over finite domain. Then, the range of frequencies of f can be expressed in more detail by using a bounded set instead of the maximum frequency. A function whose range of frequencies is confined to a bounded set is referred to as bandlimited function. And a sampling theorem for bandlimited functions over Boolean domain has been obtained. Here, it is important to obtain a sampling theorem for bandlimited functions not only over Boolean domain (GF(2)n domain) but also over GF(q)n domain, where q is a prime power and GF(q) is Galois field of order q. For example, in experimental designs, although the model can be expressed as a linear combination of the Fourier basis functions and the levels of each factor can be represented by GF(q), the number of levels often take a value greater than two. However, the sampling theorem for bandlimited functions over GF(q)n domain has not been obtained. On the other hand, the sampling points are closely related to the codewords of a linear code. However, the relation between the parity check matrix of a linear code and any distinct error vectors has not been obtained, although it is necessary for understanding the meaning of the sampling theorem for bandlimited functions. In this paper, we generalize the sampling theorem for bandlimited functions over Boolean domain to a sampling theorem for bandlimited functions over GF(q)n domain. We also present a theorem for the relation between the parity check matrix of a linear code and any distinct error vectors. Lastly, we clarify the relation between the sampling theorem for functions over GF(q)n domain and linear codes.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E93.A.1024/_p
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@ARTICLE{e93-a_6_1024,
author={Yoshifumi UKITA, Tomohiko SAITO, Toshiyasu MATSUSHIMA, Shigeichi HIRASAWA, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={A Note on a Sampling Theorem for Functions over GF(q)n Domain},
year={2010},
volume={E93-A},
number={6},
pages={1024-1031},
abstract={In digital signal processing, the sampling theorem states that any real valued function f can be reconstructed from a sequence of values of f that are discretely sampled with a frequency at least twice as high as the maximum frequency of the spectrum of f. This theorem can also be applied to functions over finite domain. Then, the range of frequencies of f can be expressed in more detail by using a bounded set instead of the maximum frequency. A function whose range of frequencies is confined to a bounded set is referred to as bandlimited function. And a sampling theorem for bandlimited functions over Boolean domain has been obtained. Here, it is important to obtain a sampling theorem for bandlimited functions not only over Boolean domain (GF(2)n domain) but also over GF(q)n domain, where q is a prime power and GF(q) is Galois field of order q. For example, in experimental designs, although the model can be expressed as a linear combination of the Fourier basis functions and the levels of each factor can be represented by GF(q), the number of levels often take a value greater than two. However, the sampling theorem for bandlimited functions over GF(q)n domain has not been obtained. On the other hand, the sampling points are closely related to the codewords of a linear code. However, the relation between the parity check matrix of a linear code and any distinct error vectors has not been obtained, although it is necessary for understanding the meaning of the sampling theorem for bandlimited functions. In this paper, we generalize the sampling theorem for bandlimited functions over Boolean domain to a sampling theorem for bandlimited functions over GF(q)n domain. We also present a theorem for the relation between the parity check matrix of a linear code and any distinct error vectors. Lastly, we clarify the relation between the sampling theorem for functions over GF(q)n domain and linear codes.},
keywords={},
doi={10.1587/transfun.E93.A.1024},
ISSN={1745-1337},
month={June},}
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TY - JOUR
TI - A Note on a Sampling Theorem for Functions over GF(q)n Domain
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1024
EP - 1031
AU - Yoshifumi UKITA
AU - Tomohiko SAITO
AU - Toshiyasu MATSUSHIMA
AU - Shigeichi HIRASAWA
PY - 2010
DO - 10.1587/transfun.E93.A.1024
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E93-A
IS - 6
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - June 2010
AB - In digital signal processing, the sampling theorem states that any real valued function f can be reconstructed from a sequence of values of f that are discretely sampled with a frequency at least twice as high as the maximum frequency of the spectrum of f. This theorem can also be applied to functions over finite domain. Then, the range of frequencies of f can be expressed in more detail by using a bounded set instead of the maximum frequency. A function whose range of frequencies is confined to a bounded set is referred to as bandlimited function. And a sampling theorem for bandlimited functions over Boolean domain has been obtained. Here, it is important to obtain a sampling theorem for bandlimited functions not only over Boolean domain (GF(2)n domain) but also over GF(q)n domain, where q is a prime power and GF(q) is Galois field of order q. For example, in experimental designs, although the model can be expressed as a linear combination of the Fourier basis functions and the levels of each factor can be represented by GF(q), the number of levels often take a value greater than two. However, the sampling theorem for bandlimited functions over GF(q)n domain has not been obtained. On the other hand, the sampling points are closely related to the codewords of a linear code. However, the relation between the parity check matrix of a linear code and any distinct error vectors has not been obtained, although it is necessary for understanding the meaning of the sampling theorem for bandlimited functions. In this paper, we generalize the sampling theorem for bandlimited functions over Boolean domain to a sampling theorem for bandlimited functions over GF(q)n domain. We also present a theorem for the relation between the parity check matrix of a linear code and any distinct error vectors. Lastly, we clarify the relation between the sampling theorem for functions over GF(q)n domain and linear codes.
ER -