Difference between revisions of "2010 Fall ECE 438 Examples" - Rhea

Latest revision as of 13:47, 30 April 2015

Examples of Z transform computations

1. Unit step

When $ x[n]=1 (n{\ge}0) $
  x[n] = 0(n < 0)

$ X(z)=\sum_{n=0}^{\infty}x[n]z^{n}=\sum_{n=0}^{\infty}1\cdot z^{-n}=\frac{1}{1-z^{-1}} $ , ROC : |z|>1

2. Power series

x[n]=aⁿ,

$ X(z)=\sum_{n=0}^{\infty}x[n]z^{n}=\sum_{n=0}^{\infty}a^{n} z^{-n}=\frac{1}{1-az^{-1}} $ , ROC : |z|>a

3. Exponential funtion

x[n]=e^-an,

$ X(z)=\sum_{n=0}^{\infty}x[n]z^{n}=\sum_{n=0}^{\infty}e^{-an} z^{-n}=\sum_{n=0}^{\infty}[e^{-a} z^{-1}]^{n}=\frac{1}{1-e^{-a}z^{-n}} $ , ROC : |z|>e^-a

4. Sinusoidal function

x[n]=sinwn,

$ X(z)=\sum_{n=0}^{\infty}x[n]z^{n}=\sum_{n=0}^{\infty}\frac{e^{jn{\omega}} -e^{-jn{\omega}}} {2j} z^{-n} $ $ =\frac{1}{2j} (\frac{1}{1-e^{j\omega}z^{-1}}-\frac{1}{1-e^{-j\omega}z^{-1}}) $ $ =\frac{1}{2j} (\frac{-e^{-j\omega}z^{-1}+e^{j\omega}z^{-1}}{1-e^{-j\omega}z^{-1}-e^{j\omega}z^{-1}+z^{-2}}) $ $ =\frac{z^{-1}sin(\omega)}{1-2z^{-1}cos(\omega)+z^{-2}} $

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