Z-transforms

Solved Example:

Z-01

The z -transform of the time function: \[\sum ^{\infty }_{k=0}\delta \left( n-k\right)\]

Solution:
\begin{align*} x\left( n\right) &=\sum ^{\infty }_{k=0}\delta \left( n-k\right) \\ &=\delta \left( n\right) +\delta \left( n-1\right) +\delta \left( n-2\right) +\ldots .\\ x\left( n\right) =u\left( n\right) \\ x\left( z\right) =z\left( u\left( n\right) \right) =\dfrac{z}{z-1} \end{align*}

Correct Answer: C

Solved Example:

Z-02

The z-transform of the function $f(nT) = a^{nT}$ is

Solution:
The z-transform is given by: \begin{align*} z\left[ f\left( nT\right) \right] &=\sum ^{\infty }_{n=-\infty }f\left( nT \right) z^{-n}\\ &=\sum ^{\infty }_{n=-\infty }a^{nT}z^{-n}\\ &=\sum ^{\infty }_{n=-\infty }\left( a^{T}z^{-1}\right) ^{n}\\ &=\dfrac{1}{1-a^{T}z^{-1}}\\ &=\dfrac{z}{z-a^{T}} \end{align*}

Correct Answer: A