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 5 6 1 D D y t D x t + + = + (a) Find the characteristic polynomial, characteristic equation, characteristic roots, and characteristic modes of this system. (b) Find ( ) 0 y t , the zero-input component of the response ( ) y t for 0 t , if the initial conditions are ( ) 0 0 2 y = and ( ) 0 0 1 y = − ± . (a) Characteristic polynomial: 2 5 6 λ λ + + Characteristic equation: ( )( ) 2 5 6 2 3 0 λ λ λ λ + + = + + = Characteristic roots: 2 λ = − and 3 λ = − Characteristic modes: 2 t e and 3 t e (b) ( ) 2 3 0 1 2 t t y t c e c e = + and ( ) 2 3 0 1 2 2 3 t t y t c e c e = − ± ( ) ( ) ( ) 2 0 3 0 0 1 2 1 2 0 2 y c e c e c c = + = + = ( ) ( ) ( ) ( ) 2 0 3 0 0 1 2 1 2 1 1 1 0 2 3 2 3 1 2 3 2 6 y c e c e c c c c c = − = − = − = − = ± 1 5 c = and 2 3 c = − ( ) 2 3 0 5 3 t t y t e e = 2.2-7 Repeat problem 2.2-1 for ( ) ( ) ( ) ( ) 2 1 5 6 D D D y t Dx t + + + = with ( ) 0 0 2 y = , ( ) 0 0 1 y = − ± and ( ) 0 0 5 y = ±± .
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