Infinite Length DT Signals 1.1 2004/07/01 13:31:09 GMT-5 2004/07/29 15:25:27.856 GMT-5 Richard G. Baraniuk richb@rice.edu Richard G. Baraniuk richb@rice.edu Harika Basana ilsai@rice.edu Malan Shiralkar mshiralkar05@hotmail.com Infinite Length DT Signals infinite length DT signals fall 2003 elec 301 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 . That is, vector x N becomes infinitely long. x ... x 2 x 1 x 0 x 1 x 2 ... We can still keep all notions of vectors, vector spaces, inner products, norms, l p spaces...

General ∞-length inner product x y n ∞ ∞ y n x n

lp norm p x n x n p 1 p 1 p x x n n

lp(Z) spaces These are vector spaces comprising all ∞-length vectors with finite l p norm... l p ( ) x p x Why is this a vector space? What is the dimension of l p ( ) ? Not every ∞-length vector x belongs to an l p ( ). x n 1 , n

1 x 2 x x What are the conditions on x to be in an l p ( )?