<< Chapter < Page | Chapter >> Page > |
This module will explain norms, a mathematical concept that provides a notion of the size of a vector. Specifically, the general definition of a norm will be discussed and discrete time signal norms will be presented.
The norm of a vector is a real number that represents the "size" of the vector.
In $\mathbb{R}^{2}$ , we can define a norm to be a vectors geometric length.
$x=\left(\begin{array}{c}{x}_{0}\\ {x}_{1}\end{array}\right)$ , norm $(x)=\sqrt{{x}_{0}^{2}+{x}_{1}^{2}}$
Mathematically, a norm $(\xb7)$ is just a function (taking a vector and returning a real number) that satisfies three rules.
To be a norm, $(\xb7)$ must satisfy:
A vector space with a well defined norm is called a normed vector space or normed linear space .
$\mathbb{R}^{n}$ (or $\mathbb{C}^{n}$ ), $x=\left(\begin{array}{c}{x}_{0}\\ {x}_{1}\\ \dots \\ {x}_{n-1}\end{array}\right)$ , $(, x)=\sum_{i=0}^{n-1} \left|{x}_{i}\right|$ , $\mathbb{R}^{n}$ with this norm is called ${\ell}^{1}\left(\right[0,n-1\left]\right)$ .
$\mathbb{R}^{n}$ (or $\mathbb{C}^{n}$ ), with norm $(, x)=\sum_{i=0}^{n-1} \left|{x}_{i}\right|^{2}^{\left(\frac{1}{2}\right)}$ , $\mathbb{R}^{n}$ is called ${\ell}^{2}\left(\right[0,n-1\left]\right)$ (the usual "Euclidean"norm).
$\mathbb{R}^{n}$ (or
$\mathbb{C}^{n}$ ,
with norm
$()$∞
We can define similar norms for spaces of sequences and functions.
Discrete time signals = sequences of numbers $$x(n)=\{\dots , {x}_{-2}, {x}_{-1}, {x}_{0}, {x}_{1}, {x}_{2}, \dots \}$$
For continuous time functions:
Notification Switch
Would you like to follow the 'Signals and systems' conversation and receive update notifications?