$\epsilon$-Arithmetics for Real Vectors and Linear Processing of Real Vector-Valued Signals with Real Vector-Valued Coefficients

TL;DR

This paper introduces $$-arithmetics, a method for approximating real vectors with rational vectors within a specified error, enabling advanced linear processing of vector-valued signals.

Contribution

The paper proposes a novel $$-arithmetics framework for real vectors, allowing linear processing of vector-valued signals with rational vector approximations.

Findings

Defined $$-arithmetics for real vectors using rational vector approximations.

Extended linear processing techniques to vector-valued signals.

Demonstrated applications in filtering, ARMA modeling, and least squares fitting.

Abstract

In this paper, we introduce a new concept, namely $\epsilon$-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 $\epsilon$ range. For rational vectors of a fixed dimension $m$, they can form a field that is an $m$th order extension $\mathbf{Q}(\alpha)$ of the rational field $\mathbf{Q}$ where $\alpha$ has its minimal polynomial of degree $m$ over $\mathbf{Q}$. 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 $\epsilon$ 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 filtering, ARMA modeling, and least squares fitting, with real vector-valued coefficients can be implemented to real vector-valued signals, which will broaden the existing linear processing to scalar-valued signals.