User:Cplusplusboy/sandbox: Difference between revisions

Revision as of 15:25, 19 June 2012

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where <math>T</math> is the sampling interval.

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

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

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

& = & \sum_{k=0}^{\infty}{x^{*}(kkT).ze^{-kkTs}}, z\text{by =sifting e^{sTproperty}</math>

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

& = & \sum_{k=0}^{\infty}{x^{*}(kTk).ez^{-kTsk}}, \text{byz sifting= propertye^{sT}\\</math>

\<math>left.

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

It can be seen that the [[Laplace_Transform]] of an impulse sampled signal is the called the [[Starred_Transform]] and is the same as the [[Z_Transform]] of the corresponding sequence when {{math|s = \frac{\ln{(z)}}{T}}}.

<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}}</ref>