ECE634S09_HW3solutions - ECE634 Signals and Systems II...

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ECE634 Signals and Systems II, Spring 2009 – Homework 3 Solutions Daniel S. Brogan 1 Show all work and write legibly for full credit. 1. For the differential equations that you solved in problem 5 of HW2, determine the zero-input and zero-state components. ( ) ( ) y t Y s ( ) ( ) 0 dy sY s y dt ( ) ( ) ( ) 2 2 2 0 0 d y s Y s sy y dt ± ( ) ( ) x t X s ( ) ( ) 0 dx sX s x dt a) ( ) 2 2 2 d y dy dx y t dt dt dt + = if ( ) ( ) 0 0 0 y y = = ± and ( ) ( ) x t tu t = ( ) ( ) ( ) 2 2 1 0 0 0 2 2 0 s Y s s sY s Y s s s + = ( ) N N 2 initial condition terms input terms 2 2 0 Y s s s s + = + ( ) N ( )( ) zero-input component zero-state component zero-state component zero-state component 2 1 2 / 3 1/ 3 0 1 2 1 2 1 2 A B C Y s s s s s s s s s s = + = + + = − + + + + + ²³³´³³µ ²³³´³³µ ²³³´³³µ ( ) N ( ) ( ) 2 2 zero-input response zero-state response 2 1 2 1 0 1 1 3 3 3 3 t t t t y t e e u t e e u t = + + = + ²³³³´³³³µ b) ( ) ( ) 2 3 2 d y dy dx y t x t dt dt dt + + = + if ( ) 0 3 y = , ( ) 0 3 y = ± and ( ) ( ) t x t e u t = ( ) ( ) ( ) 2 1 1 3 3 3 3 2 0 1 1 s Y s s sY s Y s s s s + + = + + + ( ) N 2 input terms initial condition terms 3 2 3 12 1 Y s s s s + + = + + ²´µ ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( ) zero-input component zero-state component zero-input component zero-state component 3 12 1 1 2 1 2 1 2 1 2 s A B C D Y s s s s s s s s s + = + = + + + + + + + + + + +
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  • Spring '09
  • messner
  • Trigraph, Stability theory, 2 j, component zero-state component

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