ε-arithmetics for real vectors and linear processing of real vector-valued signals
ε-arithmetics for real vectors and linear processing of real vector-valued signals
- 信息与智能学报（英文） 2023年1卷第1期
- 作者机构：
1. Department of Electrical and Computer Engineering, University of Delaware,DE,USA,19716
- 作者简介：
- 基金信息：
DOI：10.1016/j.jiixd.2022.08.001
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Xiang-Gen Xia. ε-arithmetics for real vectors and linear processing of real vector-valued signals[J]. 信息与智能学报（英文）, 2023,1(1):2-10.
ε-arithmetics for real vectors and linear processing of real vector-valued signals[J]. Journal of Information and Intelligence, 2023,1(1):2-10.
Xiang-Gen Xia. ε-arithmetics for real vectors and linear processing of real vector-valued signals[J]. 信息与智能学报（英文）, 2023,1(1):2-10. DOI： 10.1016/j.jiixd.2022.08.001.
ε-arithmetics for real vectors and linear processing of real vector-valued signals[J]. Journal of Information and Intelligence, 2023,1(1):2-10. DOI： 10.1016/j.jiixd.2022.08.001.
摘要
In this paper, we introduce a new concept, namely ,ε,-arithmetics, for real vectors of any fixed dimension. The basic idea is to use vectors of rational values (called rational vectors) to approximate vectors of real values of the same dimension within ,ε, range. For rational vectors of a fixed dimension ,m, they can form a field that is an ,m,th order extension Q(,α,) of the rational field Q where ,α, has its minimal polynomial of degree ,m, over Q. Then, the arithmetics, such as addition, subtraction, multiplication, and division, of real vectors can be defined by using that of their approximated rational vectors within ,ε, range. We also define complex conjugate of a real vector and then inner product and convolutions of two real vectors and two real vector sequences (signals) of finite length. With these newly defined concepts for real vectors, linear processing, such as linear filtering, ARMA modeling, and least squares fitting, can be implemented to real vector-valued signals with real vector-valued coefficients, which will broaden the existing linear processing to scalar-valued signals.
Abstract
In this paper, we introduce a new concept, namely ,ε,-arithmetics, for real vectors of any fixed dimension. The basic idea is to use vectors of rational values (called rational vectors) to approximate vectors of real values of the same dimension within ,ε, range. For rational vectors of a fixed dimension ,m, they can form a field that is an ,m,th order extension Q(,α,) of the rational field Q where ,α, has its minimal polynomial of degree ,m, over Q. Then, the arithmetics, such as addition, subtraction, multiplication, and division, of real vectors can be defined by using that of their approximated rational vectors within ,ε, range. We also define complex conjugate of a real vector and then inner product and convolutions of two real vectors and two real vector sequences (signals) of finite length. With these newly defined concepts for real vectors, linear processing, such as linear filtering, ARMA modeling, and least squares fitting, can be implemented to real vector-valued signals with real vector-valued coefficients, which will broaden the existing linear processing to scalar-valued signals.
关键词
Keywords
ArithmeticsRational vectorsReal vectorsRational fieldAlgebraic number fieldField extensionAlgebraic numberInner productLinear processing of real vector-valued signals
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