Aperiodic discrete signals
The discrete-time Fourier transform (DTFT) is just what it is, regardless of the length of the signal:

$$X(\omega)=\sum_{n=-\infty}^{\infty}x[n]e^{-jn\omega}\tag{1}$$

For finite length signals you just have to sum over finitely many elements in $(1)$, which means that the DTFT always exists. This is not necessarily the case when the signal's length is infinite, because then the sum in $(1)$ does not necessarily converge.