15.2 Norms

Introduction

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.

Norms

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 $(·)$ is just a function (taking a vector and returning a real number) that satisfies three rules.

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To be a norm, $(·)$ must satisfy:

1. the norm of every vector is positive $\forall x, x\in S\colon (x)> 0$
2. scaling a vector scales the norm by the same amount $(\alpha x)=\left|\alpha \right|(x)$ for all vectors $x$ and scalars $\alpha$
3. Triangle Property: $(x+y)\le (x)+(y)$ for all vectors $x$ , $y$ . "The "size" of the sum of two vectors is less than or equal to the sum of their sizes"

A vector space with a well defined norm is called a normed vector space or normed linear space .

Examples

$\mathbb{R}^n$ (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 ${\ell }^1\left(\right[0,n-1\left]\right)$ .

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$\mathbb{R}^n$ (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).

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$\mathbb{R}^n$ (or $\mathbb{C}^n$ , with norm $()$∞ x i x i is called ${\ell }^{\infty }\left(\right[0,n-1\left]\right)$

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Spaces of sequences and functions

We can define similar norms for spaces of sequences and functions.

Discrete time signals = sequences of numbers $$x(n)=\{\dots , x_{-2}, x_{-1}, x_0, x_1, x_2, \dots \}$$

• $(, x(n))=\sum_{i=()}$∞ ∞ x i , $x(n)\in {\ell }^1\left(\mathbb{Z}\right)\implies ((, x))$∞
• $(, x(n))=\sum_{i=()} ^{}$∞ ∞ x i 2 1 2 , $x(n)\in {\ell }^2\left(\mathbb{Z}\right)\implies ((, x))$∞
• $(, x(n))=\sum_{i=()} ^{}$∞ ∞ x i p 1 p , $x(n)\in {\ell }^p\left(\mathbb{Z}\right)\implies ((, x))$∞
• $()$∞ x n sup i | x [ i ] | , $x(n)\in {\ell }^{\infty }\left(\mathbb{Z}\right)\implies (())$∞ x ∞

For continuous time functions:

• $(, f(t))=\int_{()} \,d t^{}$∞ ∞ f t p 1 p , $f(t)\in L^p\left(ℝ\right)\implies ((, f(t)))$∞
• (On the interval) $(, f(t))=\int_0^T \left|f(t)\right|^p\,d t^{\left(\frac{1}{p}\right)}$ , $f(t)\in L^p\left(\right[0,T\left]\right)\implies ((, f(t)))$∞