ε-arithmetics for real vectors and linear processing of real vector-valued signals
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 dimensio...
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| Format: | Article |
| Language: | English |
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KeAi Communications Co., Ltd.
2023-05-01
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| Series: | Journal of Information and Intelligence |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S2949715922000014 |
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| author | Xiang-Gen Xia |
| author_facet | Xiang-Gen Xia |
| author_sort | Xiang-Gen Xia |
| collection | DOAJ |
| description | 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 mth 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. |
| format | Article |
| id | doaj-art-1f21cabd417a438f8879116893dbdfe3 |
| institution | Kabale University |
| issn | 2949-7159 |
| language | English |
| publishDate | 2023-05-01 |
| publisher | KeAi Communications Co., Ltd. |
| record_format | Article |
| series | Journal of Information and Intelligence |
| spelling | doaj-art-1f21cabd417a438f8879116893dbdfe32025-08-20T03:32:49ZengKeAi Communications Co., Ltd.Journal of Information and Intelligence2949-71592023-05-011121010.1016/j.jiixd.2022.08.001ε-arithmetics for real vectors and linear processing of real vector-valued signalsXiang-Gen Xia0Department of Electrical and Computer Engineering, University of Delaware, DE 19716, USAIn 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 mth 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.http://www.sciencedirect.com/science/article/pii/S2949715922000014ArithmeticsRational vectorsReal vectorsRational fieldAlgebraic number fieldField extension |
| spellingShingle | Xiang-Gen Xia ε-arithmetics for real vectors and linear processing of real vector-valued signals Journal of Information and Intelligence Arithmetics Rational vectors Real vectors Rational field Algebraic number field Field extension |
| title | ε-arithmetics for real vectors and linear processing of real vector-valued signals |
| title_full | ε-arithmetics for real vectors and linear processing of real vector-valued signals |
| title_fullStr | ε-arithmetics for real vectors and linear processing of real vector-valued signals |
| title_full_unstemmed | ε-arithmetics for real vectors and linear processing of real vector-valued signals |
| title_short | ε-arithmetics for real vectors and linear processing of real vector-valued signals |
| title_sort | ε arithmetics for real vectors and linear processing of real vector valued signals |
| topic | Arithmetics Rational vectors Real vectors Rational field Algebraic number field Field extension |
| url | http://www.sciencedirect.com/science/article/pii/S2949715922000014 |
| work_keys_str_mv | AT xianggenxia earithmeticsforrealvectorsandlinearprocessingofrealvectorvaluedsignals |