Question 22.27: Find the z transform of the unit step function u(t) and the ......
Find the z transform of the unit step function u(t) and the shifted unit step u(t — 2T), sampled at intervals of T seconds.
Step-by-Step
If the function u(t) is sampled at intervals T then we are concerned with finding the z transform of the sequence u[k]. This has been derived earlier: \mathcal{Z}\{u[k]\}=\frac{z}{z-1}. If u(t — 2T) is sampled, we have
u[k-2]= \begin{cases}1 & k=2,3,4, \ldots \\ 0 & \text { otherwise }\end{cases}Therefore, by the second shift theorem,
\begin{aligned} \mathcal{Z}\{u[k-2]\} & =z^{-2} \mathcal{Z}\{u[k]\} \\ & =z^{-2} \frac{z}{z-1} \\ & =\frac{1}{z(z-1)} \end{aligned}Related Answered Questions
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