94下信號與系çµ&plusmn

94下信號與系統

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Unformatted text preview: Signals and Systems, Final Exam Solutions (Draft) Spring 2006, Edited by bypeng 1. (10) Consider the linear constant-coefficient second-order differential equation: 2 2 2 2 ( ) 2 ( ) ( ) ( ) n n n d d y t y t y t x t dt dt ζω ω ω + + = . (a) Find the frequency response ( ) H j ω of the system. (b) For 2 / 2 ζ < < , what is the frequency m ω where ( ) m H j ω has a maximum value? (c) What is the maximum value of ( ) m H j ω at the frequency in (b)? Solution: (a) We have 2 2 2 ( ) ( ) 2 ( ) ( ) ( ) ( ) n n n j Y j j Y j Y j X j ω ω ζ ω ω ω ω ω ω ω + + = so 2 2 2 ( ) ( ) ( ) ( ) 2 ( ) n n n Y j H j X j j j ω ω ω ω ω ζ ω ω ω = = + + (b) To make ( ) H j ω maximum, we need the dominator (or the downstairs) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 ( ) 2 ( ) 2 4 n n n n n n j j j ω ζω ω ω ω ω ζω ω ω ω ζ ω ω + + = − + = − + to be minimum. So ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 8 4 2 4 ( 1 2 ) n n n n n n n d d ω ω ζ ω ω ω ω ω ζ ω ω ω ω ω ω ζ ω ω ω ζ ω ⎡ ⎤ − + = − − + ⎢ ⎥ ⎣ ⎦ = − + + = − − = which holds when ω = or 2 1 2 n ω ω ζ = ± − , which is real since 2 / 2 ζ < < , and of which the positive value is adopted. Considering that ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 ( 1 2 ) 4 ( 1 2 ) 8 12 4(1 2 ) n n n n n d d d d ω ω ζ ω ω ω ω ζ ω ω ω ω ζ ω ω ω ζ ω ⎡ ⎤ ⎡ ⎤ − + = − − ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ = − − + = − − which is positive (minimum occurred) when 2 1 2 n ω ω ζ = − and is negative (maximum occurred) when ω = , we have 2 1 2 m n ω ω ζ = − . (c) We have ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 2 4 4 4 2 2 2 2 ( ) 4 ( 1 2 ) 4 ( 1 2 ) 4 4 8 1 2 1 2 1 n n m n m n m n n n n n n n n n n H j ω ω ω ω ω ζ ω ω ω ω ζ ζ ω ω ζ ω ζ ω ζ ω ζ ω ω ζω ζ ζ ζ = = − + − − + − = + − = = − − 2. (10) Consider a continuous-time LTI system with frequency response ( ) ( ) ( ) j H j H j H j e ω ω ω = ) and real impulse ( ) h t . Suppose that we apply an input ( ) sin( ) x t t ω φ = + to the system. The resulting output can be shown to be the form ( ) ( ) y t A x t t = − , where A is a nonnegative real number representing an amplitude-scaling factor and t is a time delay....
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94下信號與系統

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