ENSC3015_2016_ExamAttachments.pdf - ENSC3015 Signals and...

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ENSC3015: Signals and Systems SEMESTER 2, 2016 EXAMINATIONS PAGE NO 3 Zero-input Response Real Roots (continuous-time systems) Assume r of the N roots are real but identical, that is ࠵༆࠵? 1 = ࠵༆࠵? 2 = = ࠵༆࠵? ࠵౟࠵? = ࠵༆࠵? with the remaining ( ࠵ు࠵? − ࠵౟࠵? ) roots being distinct, that is ࠵༆࠵? ࠵౟࠵?+1 , ࠵༆࠵? ࠵౟࠵?+2 , … , ࠵༆࠵? ࠵ు࠵? : ࠵౦࠵? 0 ( ࠵ౡ࠵? ) = ( ࠵౐࠵? 1 + ࠵౐࠵? 2 ࠵ౡ࠵? + + ࠵౐࠵? ࠵౟࠵? ࠵ౡ࠵? ࠵౟࠵?−1 ) ࠵౒࠵? ࠵༆࠵?࠵༆࠵? + ࠵౐࠵? ࠵౟࠵?+1 ࠵౒࠵? ࠵༆࠵? ࠵౟࠵?+1 ࠵༆࠵? + ࠵౐࠵? ࠵౟࠵?+2 ࠵౒࠵? ࠵༆࠵? ࠵౟࠵?+2 ࠵༆࠵? + + ࠵౐࠵? ࠵ు࠵? ࠵౒࠵? ࠵༆࠵? ࠵ు࠵? ࠵༆࠵? Real Roots (discrete-time systems) Assume r of the N roots are real but identical (= ࠵༆࠵? ) , with the remaining ( ࠵ు࠵? − ࠵౟࠵? ) roots being distinct: ࠵౦࠵? 0 [ ࠵౛࠵? ] = ( ࠵౐࠵? 1 + ࠵౐࠵? 2 ࠵౛࠵? + + ࠵౐࠵? ࠵౟࠵? ࠵౛࠵? ࠵౟࠵?−1 ) ࠵༆࠵? ࠵౛࠵? + ࠵౐࠵? ࠵ౘ࠵? ࠵༆࠵? ࠵ౘ࠵? ࠵౛࠵? ࠵ు࠵? ࠵ౘ࠵?=࠵౟࠵?+1 Convolution ࠵౥࠵? 1 ( ࠵ౡ࠵? ) ∗ ࠵౥࠵? 2 ( ࠵ౡ࠵? ) = ࠵౥࠵? 1 ( ࠵༏࠵? ) ࠵౥࠵? 2 ( ࠵ౡ࠵? − ࠵༏࠵? ) ࠵౑࠵?࠵༏࠵? −∞ ࠵౥࠵?
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