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Norms

By Michael Haag, Justin Romberg

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.

Example 1

In R2 2 , we can define a norm to be a vectors geometric length.

Figure 1

x= x 0 x 1 T x x 0 x 1 , norm x= x 0 2+ x 1 2 x 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.

To be a norm, · · must satisfy:

  1. the norm of every vector is positive x,xS:x>0 x x S x 0
  2. scaling a vector scales the norm by the same amount αx=|α|x α x α x for all vectors x x and scalars α α
  3. Triangle Property: x+yx+y x y x y for all vectors x x, y 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

Example 2

Rn n (or Cn n ), x= x 0 x 1 x n - 1 x x 0 x 1 x n - 1 , x1=i=0n1| x i | 1 x i 0 n 1 x i , Rn n with this norm is called 1 ( [ 0 , n - 1 ] ) 1 ( [ 0 , n - 1 ] ) .

Figure 2: Collection of all xR2 x 2 with x1=1 1 x 1
Example 3

Rn n (or Cn n ), with norm x2=i=0n1| x i |212 2 x i 0 n 1 x i 2 1 2 , Rn n is called 2 ( [ 0 , n - 1 ] ) 2 ( [ 0 , n - 1 ] ) (the usual "Euclidean"norm).

Figure 3: Collection of all xR2 x 2 with x2=1 2 x 1
Example 4

Rn n (or Cn n , with norm x=maxii| x i | x i x i is called ( [ 0 , n - 1 ] ) ( [ 0 , n - 1 ] )

Figure 4: xR2 x 2 with x=1 x 1

Spaces of Sequences and Functions

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

Discrete time signals = sequences of numbers xn= x -2 x -1 x 0 x 1 x 2 x n x -2 x -1 x 0 x 1 x 2

  • xn1=i=|xi| 1 x n i x i , xn 1 ( ) (x1<) x n 1 ( ) 1 x
  • xn2=i=|xi|212 2 x n i x i 2 1 2 , xn 2 ( ) (x2<) x n 2 ( ) 2 x
  • xnp=i=|xi|p1p p x n i x i p 1 p , xn p ( ) (xp<) x n p ( ) p x
  • xn= sup i | x [ i ] | x n sup i | x [ i ] | , xn ( ) (x<) x n ( ) x

For continuous time functions:

  • ftp=|ft|pdt1p p f t t f t p 1 p , ft L p ( ) (ftp<) f t L p ( ) p f t
  • (On the interval) ftp=0T|ft|pdt1p p f t t 0 T f t p 1 p , ft L p ( [ 0 , T ] ) (ftp<) f t L p ( [ 0 , T ] ) p f t