# lecture9 - EE313 Linear Systems and Signals Fall 2010...

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EE313 Linear Systems and Signals Fall 2010 Initial conversion of content to PowerPoint by Dr. Wade C. Schwartzkopf Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin Difference Equations

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9 - 2 Iterative Solutions Example: y [ n ] - ½ y [ n -1] = x [ n ] How many initial conditions do we need? For x [ n ] = n 2 u [ n ] and y [-1] = 16 , causal system, y [ n ] = ½ y [ n -1] + x [ n ] Compute answer iteratively: y [0] , y [1] , … y [0] = ½ y [-1] + x [0] = ½ (16) + 0 = 8 y [1] = ½ (8) + (1) 2 = 5 y [2] = 6.5 y [3] = 12.25 y [4] = 22.125
9 - 3 Stability Is the system bounded-input bounded-output (BIBO) stable? y [ n ] - ½ y [ n -1] = x [ n ] Impulse response is the output y [ n ] for input x [ n ] = δ [ n ] Zero-state response y s [ n ] to input δ [ n ] is 1, 0.5, 0.25, …, for n = 0, 1, 2, … System appears to be BIBO stable. x [ n ] = n 2 u [ n ] is unbounded in amplitude as n goes to infinity

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9 - 4 Zero-Input, Zero-State Solutions Example: y [ n ] - ½ y [ n -1] = x [ n ] with y 0 [-1] = 16 Zero-input solution y 0 [ n ] - ½ y 0 [ n -1] = 0 y 0 [ n ] = ½ y 0 [ n -1] y 0 [ n ] = 8 (½) n u [ n ] 8, 4, 2, 1, ½, … [ ] = - = n m m n s m n y 0 2 2 1 Zero-state solution y
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## This note was uploaded on 02/16/2011 for the course ECE 313 taught by Professor Evans during the Spring '11 term at University of Texas at Austin.

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lecture9 - EE313 Linear Systems and Signals Fall 2010...

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