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Unformatted text preview: EE113: Digital Signal Processing Prof. Mihaela van der Schaar Final Exam (Version A) Spring 2009 Solutions: 1. (a) We first observe that x 1 ( n ) = x 2 ( n )+ x 3 ( n +4). Because the system T is time-invariant, the output of the system is y 3 ( n + 4) when the input is x 3 ( n + 4). If the system were linear, then we should have y 1 ( n ) = y 2 ( n ) + y 3 ( n + 4). This is clearly not true, and therefore the system is not linear . (b) If x ( n ) = ( n ) = x 3 ( n +4), then y ( n ) = y 3 ( n +4) because the system T is time-invariant. (c) We can evaluate the response of the system T to x 1 ( n- k ), x 2 ( n- k ), and x 3 ( n- k ) for all integers k 1 2. (a) Since x ( n ) = 16 parenleftbigg sin 8 n n parenrightbigg 2 cos 4 n + sin 4 n n The DTFTs of the signals at points A, B, and C (i.e. X ( e j ), V ( e j ), and Y ( e j ), respectively) can be evaluated as illustrated below....
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