Periodicity
The Discrete-Time Fourier Transform (DTFT) is a periodic function with a period of 1 in continuous frequency. Indeed :
X\(\left(f\right)=\sum_{n=-\infty}^{\infty}{x\left(n\right)e^{-j2\pi nf}}\) (2)
\(X\left(f+1\right)=\sum_{n=-\infty}^{\infty}{x\left(n\right)e^{-j2\pi n(f+1)}=\sum_{n=-\infty}^{\infty}{x\left(n\right)e^{-j2\pi nf}e^{-j2\pi n}}}\) (3)
Where \(e^{-j2\pi n}=1\ \ \forall n\in Z\)
\(X\left(f+1\right)=X\left(f\right)\) (4)
Therefore, it is customary to represent it on an interval of length 1, namely \(f\in\left[-\frac{1}{2},\frac{1}{2}\right]\ or\ (0,1)\).
Therefore, it is customary to represent it on an interval of length 1, namely f∈[-1/2,1/2] or (0,1).