# Connexions

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# Derived copy of Infinite Length DT Signals

Module by: carlos rodriguez-solano. E-mail the author

Based on: Infinite Length DT Signals by Richard Baraniuk

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

What should happened 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 )?

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