How do I calculate the step response of a discrete-time system?
Taking the $\mathcal Z$ transform of the impulse response,

$$H (z) = \dfrac{2}{1 - \frac 12 z^{-1}} - 1$$

Hence, the $\mathcal Z$ transform of the step response is

$$S (z) := \dfrac{H (z)}{1 - z^{-1}} = \cdots = \dfrac{3}{1 - z^{-1}} - \dfrac{2}{1 - \frac 12 z^{-1}}$$

Taking the inverse $\mathcal Z$ transform, we obtain the step response

$$s (n) = \begin{cases} 3 - 2 \left( \frac 12 \right)^n &amp; \text{if } n \geq 0\\ 0 &amp; \text{if } n &lt; 0\end{cases}$$