siutulysr5

2022-01-22

Find the Z transform of the sequence
$\left\{k{\left(\frac{1}{2}\right)}^{k}\right\}$

Joy Compton

Expert

We have:
$\left\{k{\left(\frac{1}{2}\right)}^{k}\right\}$
$Z\left[k{\left(\frac{1}{2}\right)}^{k}\right]=?$
Remember that $Z\left[{a}^{n}\right]=\frac{Z}{Z-a}$
$Z\left[{a}^{n}f\left(n\right)\right]=f\left(\frac{Z}{a}\right)$
Thus, we have
$Z\left[k{\left(\frac{1}{2}\right)}^{k}\right]=\left[Z\left[k\right]\right]$, $Z\to \frac{Z}{\frac{1}{2}}$
$\left[\frac{Z}{{\left(Z-1\right)}^{2}}\right]$, $Z\to \frac{Z}{\frac{1}{2}}$
$=\frac{2Z}{{\left(2Z-1\right)}^{2}}$
$=\frac{Z}{2{\left(Z-\frac{1}{2}\right)}^{2}}$

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