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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...?

Figure 1
Figure 1 (fig2.png)

We can view this as letting N→∞ N . That is, vector x∈RN x N becomes infinitely long.

x=...x⁢−2x⁢−1x⁢0x⁢1x⁢2... 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 =−∞∞y⁢n¯⁢x⁢n x y n ∞ ∞ y n x n
(2)

lp norm

∥x∥p=∑ n =−∞∞|x⁢n|p1p 1≤p<∞ p x n x n p 1 p 1 p
(3)

∥x∥∞=max|x⁢n| −∞<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 ∥x∥p<∞ 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

x⁢n=1 x n 1 , −∞<n<∞ n

Figure 2
Figure 2 (fig1.png)
∥x∥1= 1 x

∥x∥2= 2 x

∥x∥∞= x

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

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