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lecture9

# lecture9 - EE313 Linear Systems and Signals Spring 2009...

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EE313 Linear Systems and Signals Spring 2009 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 occurs when 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 s [ n ] - ½ y s [ n -1] = n 2
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lecture9 - EE313 Linear Systems and Signals Spring 2009...

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