$ε$ -实向量的算法和具有实向量系数的实向量值信号的线性处理
$ε$-Arithmetics for Real Vectors and Linear Processing of Real Vector-Valued Signals with Real Vector-Valued Coefficients
摘要：
本文对任意固定维数的实向量引入了一个新的概念，即$ε$ -算术。其基本思想是使用有理值的向量（称为有理向量）来近似$ε$范围内相同维数的实值向量。对于固定维数$m$的有理向量，它们可以形成一个域，该域是有理域$\mathbf{Q}$的$m$阶扩展$\mathbf{Q}(α)$，其中$α$在$\mathbf{Q}$上具有其次数为$m$的最小多项式。然后，利用实向量在$ε$范围内的近似有理向量的算术定义实向量的加、减、乘、除等算术。我们还定义了复共轭的实向量，然后内积和卷积的两个实向量和两个实向量序列（信号）的有限长度。利用这些新定义的实向量概念，可以对实向量值信号实施具有实向量值系数的线性处理，诸如滤波、ARMA建模和最小二乘拟合，这将扩展现有的标量值信号的线性处理。
英文摘要：
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 $\mathbf{Q}(α)$ of the rational field $\mathbf{Q}$ where $α$ 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 $ε$ 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 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.
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