ch 7 summary - Summary Time-Domain Analysis of DT...

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Summary – Time-Domain Analysis of DT Systems (Chapter 7) 1. For any DT system, the impulse response h n [ ] is the system output when the input is δ n [ ] . ( Examples: (i) If system input-output equation is y [ n ] = x n [ ] + 1 2 x n 1 [ ] , then impulse response is h [ n ] = n [ ] + 1 2 n 1 [ ] . (ii) If system satisfies input-output difference equation y [ n ] a y n 1 [ ] = x n [ ] with y n [ ] = 0 for n < 0 , then impulse response satisfies equation h [ n ] ah n 1 [ ] = n [ ] with h n [ ] = 0 for n < 0 . This implies that h 0 [ ] 0 = 1 h 0 [ ] = 1 ; and that h [ n ] 1 [ ] = 0 for n > 0 , which leads to h 1 [ ] = a , h 2 [ ] = a 2 , etc. – that is, to h n [ ] = a n u n [ ] .) 2. Every DT signal can be expanded as a sum of shifted impulses – that is: x n [ ] = x k [ ] k = −∞ n k [ ] . 3. From (1) and (2) it follows that if the system is LTI, then the system output due to input x n [ ] can be written as y n [ ] = x k [ ] k = −∞ h n k [ ] . That is, the output is the convolution of the input and impulse response (shorthand notation: y n [ ] = x n [ ] * h n [ ] ).
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This note was uploaded on 10/27/2011 for the course ECE 313 taught by Professor Muschinski during the Fall '08 term at UMass (Amherst).

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ch 7 summary - Summary Time-Domain Analysis of DT...

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