User:Cplusplusboy/sandbox: Difference between revisions

Revision as of 16:49, 19 June 2012

Revision as of 16:50, 19 June 2012

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Consider a continuous time signal <math>x(t)</math>. Its one sided Laplace transform is defined as :

<math>L\{x(t)\} \equiv X(s) \equiv \int_0^{\infty}{x(t)e^{-st}dt}

</math>

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Now the Laplace transform of the sampled signal (discrete time) is called [[Star_transform]] and is given by :

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

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

& = & \sum_{k=0}^{\infty}{x^{*}(kkT).ze^{-kkTs}}, z& =& e^\text{sTby sifting property} \\

& = & \sum_{k=0}^{\infty}{x^{*}(kTk).ez^{-kTsk}}, &z &= \texte^{by sifting propertysT} \\

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

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

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

\end{array} </math>