Solution, ELEC264 Final Exam W09

Solution, ELEC264 Final Exam W09 - ELEC264 Final Exam...

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ELEC264 Final Exam Signals and Systems I Concordia University 1 Winter 2009 Question 1: Consider a continuous-time system which has input of signal ) ( t x and output of ) ( ) ( ) ( t u t x t y = . a) Is this system time invariant? Justify your answer. b) Is this system linear? Justify your answer. Part a: To prove that the system is time invariant, we should show that for any input ) ( 1 t x and any time shift 0 t , we have ) ( ) ( 0 1 2 t t y t y = , where ) ( ) ( 1 1 t y t x , ) ( ) ( 2 2 t y t x and ) ( ) ( 0 1 2 t t x t x = . Otherwise, the system is time variant. Proof is as follows: ) ( ) ( ) ( ) ( ) ( ) ( 0 0 1 0 1 1 1 t t u t t x t t y t u t x t y = = ) ( ) ( ) ( ) ( ) ( 0 1 2 2 t u t t x t u t x t y = = Answer : Therefore, ) ( ) ( 0 1 2 t t y t y and the system is time variant. Part b: To prove that the system is linear, we should show that for any input ) ( 1 t x and ) ( 2 t x and any scalar a and b , we have ) ( ) ( ) ( 2 1 3 t by t ay t y + = , where ) ( ) ( 1 1 t y t x , ) ( ) ( 2 2 t y t x , ) ( ) ( 3 3 t y t x and ) ( ) ( ) ( 2 1 3 t bx t ax t x + = . Otherwise, the system is non-linear. Proof is as follows: {} ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 1 2 1 2 1 3 3 t by t ay t u t bx t u t ax t u t bx t ax t u t x t y + = + = + = = Answer : Therefore, ) ( ) ( ) ( 2 1 3 t by t ay t y + = and the system is linear. Question 2: Consider a discrete-time system which has input of signal ] [ n x and output of = ] [ 4 cos ] [ n x n y π . a) Evaluate and draw the impulse response of the above system. b) If the input to the system is 2 ] [ 2 n n x = , determine whether the output of the system ] [ n y is periodic. If ] [ n y is periodic, find its fundamental period and fundamental frequency.
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ELEC264 Final Exam Signals and Systems I Concordia University 2 Winter 2009 Part a: By definition, the impulse response is = = = ] [ 4 cos ] [ ] [ ] [ ] [ n n y n h n n x δ π and therefore, Answer : = × = = × = 0 1 0 4 cos 0 2 2 1 4 cos ] [ n n n h Part b: To prove that ] [ n y is periodic, we should find a positive integer number N such that for any n , ] [ ] [ N n y n y + = . = × = 8 cos 2 4 cos ] [ 2 2 n n n y and ( ) + + 8 cos ] [ 2 N n N n y If k and l are integer numbers, () ( ) ( ) ( ) k n N n k n N n n N n 16 2 8 8 8 cos 8 cos 2 2 2 2 2 2 ± = + ± = + = + l N k nN N k n nN N n 8 16 2 16 2 2 2 2 2 = = + ± = + + Answer : Therefore, the output ] [ n y is periodic with period of l N 8 = and the fundamental period and frequency are 8 = N and 4 8 2 2 0 ω = = = N , respectively. Question 3: Consider a continuous-time LTI system which has impulse response of {} ) 1 ( ) ( ) ( = t u t u t h . If { } ) 3 ( ) ( ) ( 2 = t u t u t t x is applied at the input of the system, evaluate the output ) ( t y of the system using convolution integral = τ d t h x t y ) ( ) ( ) ( as follows: a) Draw ) ( x and ) ( t h for different intervals of “ t ”. b) Evaluate the output ) ( t y for the intervals of “ t ” indicated in part (a).
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Solution, ELEC264 Final Exam W09 - ELEC264 Final Exam...

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