ε-arithmetics for real vectors and linear processing of real vector-valued signals
Date
2022-09-19
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Journal of Information and Intelligence
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 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.
Description
This article was originally published in Journal of Information and Intelligence. The version of record is available at: https://doi.org/10.1016/j.jiixd.2022.08.001
Keywords
arithmetics, rational vectors, real vectors, rational field, algebraic number field, field extension, inner product, linear processing of real vector-valued signals
Citation
Xia, Xiang-Gen. “ε-Arithmetics for Real Vectors and Linear Processing of Real Vector-Valued Signals.” Journal of Information and Intelligence, September 2022. https://doi.org/10.1016/j.jiixd.2022.08.001.