DTFT Pairs confusion When I am in the DT Fourier Domain, and I want to come back to the time domain, which pair do I use? Asking because both pairs have the exact same &quot;form&quot; in the Fourier domain, and that is what is tripping me up. Do I just use pair 2 when I want to go into the Fourier domain and then I have to use pair 1 when I want to come back?? Or is there more to it than that?
I would really recommend that you compute the DTFT of the sequences $x_1[n]=a^nu[n]$ and $x_2[n]=-a^nu[-n-1]$. This is very straightforward if you use the formula for the geometric series:
$$\sum_{n=0}^{\infty}q^n=\frac{1}{1-q},\qquad |q|&lt;1\tag{1}$$
It will be important to consider the magnitude of $a$. Note that it's not for nothing that the table in your question specifies a condition on $|a|$.