2.4 The discrete-time fourier transform

The discrete-time fourier transform

The (non-normalized) DTFT is simply a special case of the $z$ -transform for the case $\left|z\right|=1$ , i.e., $z=e^{j\omega }$ for some value $\omega \in [-\pi ,\pi ]$

The picture you should have in mind is the complex plane. The $z$ -transform is defined on the whole plane, and the DTFT is simply the value of the $z$ -transform on the unit circle, as illustrated below.

This picture should make it clear why the DTFT is defined only for $\omega \in [-\pi ,\pi ]$ (or why it is periodic). Using the normalization above, we also have the inverse DTFT formula: