Use samples of Fourier transform as DFT?
Check my answer on a similar question : H(z) as N-DFT

Only thing to add it that for both the systems, it is possible to compute $H[k]$ by figuring that the numerator and denomenator are as follows :


Numerator = $[1]$, Denominator = $[1, -\frac{1}{2}]$. 
Numerator = $[1]$, Denominator = $[1, -2]$.


Take $N_b$-DFT of numerator and $N_b$-DFT of Denominator and point-wise division of these will give $H[k]$. Take IDFT to get $N_b$ samples of $h[n]$. Convolve with $x[n]$ to get output.

This is possible for stable and unstable both systems, because we are truncating $h[n]$, which will inherently make the system stable. The main requirement is that the system is causal.