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Infinite Length DT Signals

Authors
Publication Date
Keywords
  • Infinite Length Dt Signals
  • Infinite
  • Length
  • Dt
  • Signals
  • Fall 2003
  • Elec 301
Disciplines
  • Mathematics

Abstract

Connexions module: m12345 1 Infinite Length DT Signals ∗ Richard Baraniuk This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License † Abstract In this section, you will learn about Infinite Length DT Signals. What happens as we let signals become longer and longer...? Figure 1 We can view this as letting N →∞. That is, vector x ∈ RN becomes infinitely long. x =  ... x [−2] x [−1] x [0] x [1] x [2] ...  (1) ∗ Version 1.2: Jul 30, 2004 10:47 am GMT-5 † http://creativecommons.org/licenses/by/1.0 http://cnx.org/content/m12345/1.2/ Connexions module: m12345 2 note: We can still keep all notions of vectors, vector spaces, inner products, norms, lp spaces... 1 General ∞-length inner product < x, y >= ∞∑ n=−∞ y [n]x [n] (2) 2 lp norm ‖ x ‖p = ( ∞∑ n=−∞ (|x [n] |)p ) 1 p 1 ≤ p <∞ (3) ‖ x ‖∞ = max {|x [n] |} −∞ < n <∞ (4) 3 lp(Z) spaces These are vector spaces comprising all ∞-length vectors with finite lp norm... lp (Z) = { x | ‖ x ‖p <∞ } (5) Exercise 1 Why is this a vector space? Exercise 2 What is the dimension of lp (Z)? note: Not every ∞-length vector x belongs to an lp (Z). Exercise 3 x [n] = 1, −∞ < n <∞ Figure 2 ‖ x ‖1 = ‖ x ‖2 = ‖ x ‖∞ = What are the conditions on x to be in an lp (Z )? http://cnx.org/content/m12345/1.2/

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