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Unformatted text preview: Preliminary Exam 1 (ECE220 - Fall 2007) Name: Solution NetID: QED This exam is purely technical and does not leave much space for the imagination. Write the equations that lead to the result rather than trying to guess. Only after you have gone methodically through the derivation, try to see if it makes sense intuitively. Reasoning and work must be shown to gain full/partial credit. This exam is closed book and closed note. WRITE YOUR NAME AND NET ID ON ALL PAGES HANDED IN! Name: Solution NetID: QED 1. (60 points) Convolution: The following system has input x [ n ], impulse response h [ n ], and output y [ n ]. x [ n ] h [ n ] y [ n ] Let the impulse response of the system be h [ n ] = [ n ]- [ n- 1]. (a) (15 points) Prove that this LTI system is causal. Solution: y [ n ] = h [ n ] * x [ n ] = inf k =- inf h [ k ] x [ n- k ] = 1 k =0 ( [ k ]- [ k- 1]) x [ n- k ] = x [ n ]- x [ n- 1] Clearly the system is causal, because the output only depends on the input at that...
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