ECE634S09_HW3solutions

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. ( ) ( ) yt Ys () () 0 dy sY s y dt ⇔− () () () 2 2 2 00 dy sY s sy y dt ⇔−− ± ( ) ( ) x tX s 0 dx sX s x dt a) () 2 22 d y dy dx dt dt dt +− = if ( ) ( ) 0 yy −− = = ± and ( ) ( ) x tt u t = 2 2 1 0 2 2 0 s Y s s s ⎛⎞ ⎡⎤ −−+ = ⎜⎟ ⎣⎦ ⎝⎠ N N 2 initial condition terms input terms 2 20 Ys s s s +− = + N ( ) zero-input component zero-state component zero-state component zero-state component 21 2 / 3 1 / 3 0 12 AB C ss s s s s s =+ = + + = + + −+ ²³³´³³ µ² ³ ³ ´ ³ ³ µ ²³³´³³µ N zero-input response zero-state response 01 1 33 y t ee u t u t + = + ⎢⎥ ²³³³´³³³µ b) 2 32 d y dy dx xt dt dt dt ++ = + if ( ) 03 y = , ( ) y = ± and ( ) ( ) t x te u t = 2 11 333 0 s s = + + ( ) N 2 input terms initial condition terms 31 2 1 Ys s s s ++= + + ²´µ ( ) ( ) zero-input component zero-state component zero-input component zero-state component 2 1 1 2 1 2 sA B C D s s s s + =+= + + + + + + + µ²³³´³³ ³ ³ ´³³ µ²³ ³ ´³³µ
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ECE634 Signals and Systems II, Spring 2009 – Homework 3 Solutions Daniel S. Brogan 2 () ( ) ( ) zero-input component zero-state component zero-input component zero-state component 3 1 2 1 9611 12 1 2 1 2 s Ys ss s s s s + =+= + ++ + + + + ±²²³²² ´±²²³²² ´± ² ²
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This note was uploaded on 03/05/2009 for the course ECE 651 taught by Professor Messner during the Spring '09 term at New Hampshire.

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ECE634S09_HW3solutions - ECE634 Signals and Systems II,...

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