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EE301HW2 - a t x t y t y dt d t y dt d =-2 2 2 b dt t dx t...

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1 EE 301 Signals & Systems Homework #2 Due: November 5, 2009 Problem 1 Using the given input-output relations where x and y denote the input and output of the system respectively, determine with full justification , whether each of the following systems is linear, time-invariant, causal and stable. a) ) 2 ( ) 2 ( ) ( t x t x t y - + - = b) ) ( ) 3 (cos ) ( t x t t y = c) - + < = 0 ), 2 ( ) ( 0 , 0 ) ( t t x t x t t y d) [ ] [ ] [ ] > = < + = 0 , 0 , 0 0 , 1 n n x n n n x n y e) [ ] [ ] { } 2 + = n x Ev n y , f) [ ] [ ] [ ] [ ] [ ] < + = 0 , 0 , 1 n x n x n x n x n y where {} . Ev means the “even part”. Problem 2 Let ( ) ) 6 ( ) 4 ( - - - = t u t u t x and ( ) ) ( 2 t u e t h t - = . a) Compute ), ( ) ( ) ( t h t x t y * = and ). ( ) ( ) ( t h dt t dx t g * = How is ) ( t g related to ) ( t y ? b) Now compute . ) ( ) ( ) ( - * = t t h dt t dx t z How is ) ( t z related to ) ( t y ? Problem 3 Let [ ] = elsewhere , 0 9 0 , 1 n n x and [ ] = elsewhere , 0 0 , 1 N n n h where 9 N is an integer. Determine the value of N given that [ ] [ ] [ ] n h n x n y * = and [ ] 5 4 = y , [ ] 0 14 = y .
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2 Problem 4 Consider the causal LTI systems described by the following differential equations:
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Unformatted text preview: a) ( ) ( ) ( ) ( ) t x t y t y dt d t y dt d = +-2 2 2 b) ( ) ( ) ( ) ( ) dt t dx t y t y dt d t y dt d = + + 3 6 3 2 2 c) ( ) ( ) ( ) ( ) 1 5 2 2 2-=-+ t x dt d t y t y dt d t y dt d For each system, find the impulse response and determine whether it is stable. Problem 5 Consider the causal LTI systems described by the following difference equations: a) ] [ ] 1 [ 2 1 ] [ n x n y n y =-+ b) ] 2 [ 3 ] 1 [ 2 1 ] [-=-+ n x n y n y c) ] [ 3 ] [ 4 1 ] 2 [ n x n y n y = + + d) [ ] ] [ 2 4 ] 1 [ 4 ] [ n x n y n y n y =-+-+ Find the impulse response for each of the above systems and determine whether it is stable....
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