ch 6 summary - Summary Time-Domain Analysis of CT Systems...

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Summary – Time-Domain Analysis of CT Systems (Chapter 6) 1. For any CT system, the impulse response h t ( ) is the system output when the input is δ t ( ) . ( Examples: (i) If system input-output equation is y t ( ) = x r ( ) −∞ t dr , then the impulse response is h t ( ) = r ( ) dr = u t ( ) −∞ t . (ii) If system satisfies input-output differential equation dy t ( ) dt + a y t ( ) = x t ( ) with y t ( ) = 0 for t < 0 , then the impulse response satisfies equation dh t ( ) dt + ah t ( ) = t ( ) with h t ( ) = 0 for t < 0 . This implies that dh t ( ) dt + ah t ( ) = 0 for t > 0 , which gives h t ( ) = Ae at u t ( ) . Then at t = 0 we have d Ae at u t ( ) { } dt + a Ae at u t ( ) = t ( ) , which leads to aAe at u ( t ) + Ae at t ( ) + aAe at u ( t ) = t ( ) at t = 0 , which finally gives A = 1 . So, h t ( ) = e at u t ( ) .) 2. Every CT signal can be expanded as an integral of shifted impulses – that is: x t ( ) = x r ( ) −∞ t r ( ) dr . 3.
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ch 6 summary - Summary Time-Domain Analysis of CT Systems...

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