Z-transform: Difference between revisions

Revision as of 16:59, 19 June 2012

Revision as of 17:06, 19 June 2012

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<math> \begin{array}{l l l l l l}

L\{x(kT)\} & = & X^{*}(s) & = & \int_0^{\infty}{\sum_{k=0}^{\infty}{x(t).\delta(t-kT)} e^{-st}dt} \\

& = & \sum_{k=0}^{\infty}{x(kT).e^{-kTs}}, & & \text{by sifting property} \\

& = & \sum_{k=0}^{\infty}{x^{*}(k).z^{-k}}, z = e^{sT} \\

\left. L\{x(kT)\}\right|_{s = \frac{\ln{(z)}}{T}} & = & \left.X^{*}(s)\right|_{s = \frac{\ln{(z)}}{T}} & = & Z\{x^{*}(k)\}

\end{array} </math>

It can be seen that the [[Laplace_Transform]] of an impulse sampled signal is the called the [[Star_transform]] and is the same as the [[Z_Transform]] of the corresponding sequence when <math>s = \frac{\ln{(z)}}{T}</math>. Similar relationship holds when a continuous time system is converted into a sampled data system by cascading an actual impulse sampler at the input and a fictitious impulse sampler at the output.

<ref name=ogata_dtcs>{{cite book|last=Ogata|first=Katsuhiko|title=Discrete-Time Control Systems|publisher=Pearson Education|___location=India|isbn=81-7808-335-3|pages=75-77,98-103}}</ref>