# Connexions

You are here: Home » Content » Infinite Length DT Signals

### Recently Viewed

This feature requires Javascript to be enabled.

# Infinite Length DT Signals

Module by: Richard Baraniuk. E-mail the author

Summary: In this section, you will learn about Infinite Length DT Signals.

What happens as we let signals become longer and longer...?

We can view this as letting N N . That is, vector xRN x N becomes infinitely long.

x=...x2x1x0x1x2... x ... x 2 x 1 x 0 x 1 x 2 ...
(1)

## Note:

We can still keep all notions of vectors, vector spaces, inner products, norms, l p l p spaces...

## General ∞-length inner product

x,y= n =yn¯xn x y n y n x n
(2)

## lp norm

xp= n =|xn|p1p 1p< p x n x n p 1 p 1 p
(3)

x=max|xn| <n< x x n n
(4)

## lp(Z) spaces

These are vector spaces comprising all ∞-length vectors with finite l p l p norm...

l p (Z)= x xp< l p ( ) x p x
(5)

### Exercise 1

Why is this a vector space?

### Exercise 2

What is the dimension of l p (Z) l p ( ) ?

### note:

Not every ∞-length vector x x belongs to an l p l p ( Z ).

### Exercise 3

xn=1 x n 1 , <n< n

x1= 1 x

x2= 2 x

x= x

What are the conditions on x x to be in an l p l p ( Z )?

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks