SP_V_P02_4p - Introduction to Signal Processing 167...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
Introduction to Signal Processing 167 Copyright © 2005-2009 – Hayder Radha H. System Response To External Input: The Zero-State Response Consider an n-th order system [ ][ 0 ] [ 1 ] [1 2 ] [2 ] ][ 1 ] [2 ][ 2 ] [][ ] m x nx n x n xnxn xm n m δ δδ =−∞ =+ + + +− ++− + + =− " " Introduction to Signal Processing 168 Copyright © 2005-2009 – Hayder Radha [] x ny n Input - Output Introduction to Signal Processing 169 Copyright © 2005-2009 – Hayder Radha x n nh n Unit impulse input – Unit impulse response output Introduction to Signal Processing 170 Copyright © 2005-2009 – Hayder Radha x n n nm h nm −⇒ Time invariance of LTID systems
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Introduction to Signal Processing 171 Copyright © 2005-2009 – Hayder Radha [] x ny n nh n δ nm h nm −⇒ [][ ] ] x mn m x m h n m Linearity of LTID systems Introduction to Signal Processing 172 Copyright © 2005-2009 – Hayder Radha x n n ] ] x m x m h n m [ ] [ ] mm xn yn x m x m h n m ∞∞ =−∞ =−∞ ∑∑ ±²²³²²´ Linearity of LTID systems Introduction to Signal Processing 173 Copyright © 2005-2009 – Hayder Radha [ ] [ ] x m x m h n m =−∞ =−∞ Left hand side is (decomposed) input: ] m x m =−∞ Right hand side is output: [ ][ ] m xmhn m =−∞ =− Introduction to Signal Processing 174 Copyright © 2005-2009 – Hayder Radha The output expression on the right side is the convolution sum, [] [] ] m x n x m h =−∞ ∗= . [] [] [] xn hn =∗
Background image of page 2
Introduction to Signal Processing 175 Copyright © 2005-2009 – Hayder Radha Properties of the Convolution Sum The Commutative Property 12 21 [] [] [] x nx n x n ∗=∗ The Distributive Property 3 1 23 [ ] ( [ ] [ ]) ( [ ] [ ]) [ ] x n x n x n x n ∗+=+∗ The Associative Property 3 1 [ ] ( [ ] [ ]) ( [ ] [ ]) [ ] x n x n x n x n ∗∗=∗∗ Introduction to Signal Processing 176 Copyright © 2005-2009 – Hayder Radha The Shifting Property If, x n c n ∗= Then, [ ] x nm xnp c nmp −∗ −= The Convolution with an Impulse [] [] [] x nn x n δ Introduction to Signal Processing 177 Copyright © 2005-2009 – Hayder Radha Width W of a signal is one less than then number of its elements L . The Width Property Consider two signals 1 x n and 2 x n with finite widths and lengths. Then the width of the convolution sum of the two signals equals the sum of their individual widths. Introduction to Signal Processing 178 Copyright © 2005-2009 – Hayder Radha If, 11 No.of elementsof [ ] x nL = and 22 No.of elementsof [ ] x = And, 1 Widthof [ ] 1 x nWL == and 2 Widthof [ ] 1 x nW L , Then the width of 1 2 1 2 No.of elementsof [ ] [ ] 1 x nWW LL + = + .
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Introduction to Signal Processing 179 Copyright © 2005-2009 – Hayder Radha Causality and Zero-State Response For a system with causal input and causal unit impulse response the limits of the convolution sum can be simplified. 0 [] [ ][ ] n m yn xmhn m = =− Introduction to Signal Processing 180 Copyright © 2005-2009 – Hayder Radha Example 3.13 Determine [] [] [] cn xn gn =∗ for [] ( 0 . 8 ) [] n x nu n = and 0 . 3 n gn un = .
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 06/08/2009 for the course ECE 366 taught by Professor Staff during the Spring '08 term at Michigan State University.

Page1 / 19

SP_V_P02_4p - Introduction to Signal Processing 167...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online