The Discrete Fourier Transform (DFT) analyzes the frequency content of discrete-time signals.

It maps the N-sampled discrete time-domain sequence to its discrete frequency-domain sequence, where k represents the frequency index from 0 to N minus one.

For a discrete-time sequence, the z-transform is defined by summing the sequence terms multiplied by powers of the inverse of z.

Consider the z-transform of a discrete sequence. For a causal sequence, the z-transform simplifies to a finite summation.

By sampling the z-transform at equally spaced points on the unit circle, represented by complex exponentials, these values are substituted into the z-transform.

The resulting expression matches the definition of the DFT, showing that the DFT of the sequence is a sampled version of its z-transform on the unit circle.

The DFT is a sampled version of the z-transform evaluated at specific points on the unit circle in the complex plane, linking time-domain sequences to their frequency-domain representations.

This shows that the DFT is a specific case of the z-transform evaluated on the unit circle.