Discussion
What is the Z transform of given discrete time signal? Its ROC = \(n\,u(n)\).
Answer: A
The Z transform of the unit ramp \(n\,u(n)\) is found from the geometric series by differentiating. Since \(\sum_{n=0}^{\infty} z^{-n} = \dfrac{z}{z - 1}\) for \(|z| \gt 1\), multiplying by \(n\) in the time domain corresponds to \(-z\,\dfrac{d}{dz}\) in the z domain.
\[ \begin{aligned} Z\{n\,u(n)\} \\ &= -z\frac{d}{dz}\left(\frac{z}{z-1}\right) \\ &= \frac{z}{(z-1)^2},\quad |z| \gt 1 \end{aligned} \]
The region of convergence is outside the unit circle because the signal is right sided. The options with \(|z| \lt 1\) are for left sided signals, and \(z/(z-1)\) is the transform of the unit step. The answer is option A.
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