Obtaining the Impulse response from the frequency of a discrete system
As the frequency response of the system is asked, the RoC woulbe be $|z|&gt;\frac{1}{2}$. Thus, the impulse response is right sided and is given by $$h[n] = \left(\frac{1}{2}\right)^n u[n]$$ We obtain the frequency response by replacing $z = e^{j\omega}$ in $H(z)$. Thus, $$H(e^{j\omega}) = \dfrac{1}{1-\frac{1}{2}e^{-j\omega}}$$