ECE633F09_HW4solutions

ECE633F09_HW4solutions - ECE633 Signals and Systems I Fall...

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ECE633 Signals and Systems I, Fall 2009 – Homework 4 Solutions 1 From: Linear Systems and Signals, 2 nd ed . B. P. Lathi, Oxford University Press, 2005. 2.2-1 An LTIC system is specified by the equation ( ) ( ) ( )() 2 56 1 DD y tDx t ++ =+ (a) Find the characteristic polynomial, characteristic equation, characteristic roots, and characteristic modes of this system. (b) Find () 0 yt , the zero-input component of the response ( ) for 0 t , if the initial conditions are 0 02 y = and ( ) 0 01 y = − ± . (a) Characteristic polynomial: 2 λλ Characteristic equation: ( )( ) 2 2 30 λ + = Characteristic roots: 2 =− and 3 = − Characteristic modes: 2 t e and 3 t e (b) 23 012 tt yt c e c e −− and 2 c e c e ± ( ) ( ) 20 2 1 2 yc e c e c c =+= + = ( ) ( ) ( ) 2 1 2 1 1 1 3 2 3 1 2 3 2 6 e c e c c c c c =− =− = − ± 1 5 c = and 2 3 c 0 53 e e 2.2-7 Repeat problem 2.2-1 for ( )( ) ( ) ( ) 2 15 6 Dy t D x t + = with 0 y = , 0 y ± and 0 05 y = ±± . (a) Characteristic polynomial: ( ) 2 6 + Characteristic equation: ( )( )( ) 2 6 1 2 3 0 + = + + + = Characteristic roots: 1 , 2 and 3 = − Characteristic modes: t e , 2 t e and 3 t e
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This note was uploaded on 02/11/2010 for the course ECE 633 taught by Professor Staff during the Fall '09 term at University of Texas.

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ECE633F09_HW4solutions - ECE633 Signals and Systems I Fall...

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