Ztransform_note - EE338DiscretetimeSignalsandSystems...

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EE 338 Discrete time Signals and Systems Part 3: The Z Transform 1. The Z transform Definition: ܺሺݖሻ ൌ ∑ ݔሺ݊ሻݖ ሼି௡ሽ ௡ୀିஶ , Region of convergence (ROC) ൌሼݖ: ∑| ݔሺ݊ሻ||ݖ| ሼି௡ሽ ௡ୀିஶ ∞ሽ . ROC removes the ambiguity in the time domain representation of the signal. Rational Z transform: ܺሺݖሻ ൌ ௉ሺ௭ሻ ொሺ௭ሻ , where P(z) and Q(z) are polynomials of z (or ݖ ିଵ ) o Zeros of X(z): roots of P(z) ; poles of X(z): roots of Q(z) ROC analysis: ROC is a ring/disk centered around the origin, which does not include any poles. o ݔሺ݊ሻൌݔ ሺ݊ሻ൅ݔ ሺ݊ሻ , where ݔ ሺ݊ሻ is the anti causal part and ݔ ሺ݊ሻ is the causal part. ܺሺݖሻൌܺ ሺݖሻ൅ܺ ሺݖሻ,ܴܱܥ ൌܴܱܥ תܴܱܥ o To find the ROC of ܺ ሺݖሻ , let ሼܿ ,ܿ ,…,ܿ be the poles of ܺ ሺݖሻ , ± If ݔ ሺ݊ሻ
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This note was uploaded on 12/20/2010 for the course E E 338 taught by Professor Vicky during the Spring '10 term at University of Alberta.

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