113_1_Week04_part2

113_1_Week04_part2 - EE 113: Digital Signal Processing Week...

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EE 113: Digital Signal Processing Week 4 1. Zero input and zero state responses 2. The z-transform
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The Z Transform ± Powerful tool for analyzing & designing DT systems ± Definition ± Example G ( z ) = Z gn [] {} = g [ n ] z n n =−∞ Z Transform
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Region of Convergence (ROC) ± Critical question: Does summation converge (to a finite value)? ± In general, depends on the value of z Region of Convergence : Portion of complex z -plane for which a particular G ( z ) will converge G ( z ) = x [ n ] z n n =−∞ z -plane Re{ z } Im{ z } ROC | z | > λ
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ROC Example ± e.g. x [ n ] = λ n μ [ n ] ± Σ converges for | z -1 | < 1 i.e. ROC is | z | > | | ± | | < 1 (e.g. 0.8) - finite energy sequence ± | | > 1 (e.g. 1.2) - divergent sequence, infinite energy, but still has ZT when | z | > 1.2 (ROC) X ( z ) = n z n n = 0 = 1 1 − λ z 1 n -1 1234 -2
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About ROCs ± ROCs always defined in terms of | z | circular regions on z -plane (inside circles/outside circles/rings) z -plane Re{ z } Im{ z } 1
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ROC is necessary!
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This note was uploaded on 06/01/2010 for the course EE EE113 taught by Professor Mihaela during the Spring '10 term at UCLA.

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113_1_Week04_part2 - EE 113: Digital Signal Processing Week...

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