## 3.3. The discrete Fourier transform (DFT)

### 3.3.1. *Expressing the Fourier transform of a discrete sequence*

Let us look at the signal *x*_{s}(*t*) coming from the sampling of *x*(*t*) at the sampling frequency *f*_{s}:

According to equation (3.24), the Fourier transform of the signal *x*_{s}(*t*) verifies the following relation:

If we introduce *f*_{r}, the frequency reduced or normalized in relation to the sampling frequency , we will have:

The Fourier transform of a discrete sequence is one of the most commonly used spectrum analysis tools. It consists in decomposing the discrete-time signal on an orthonormal base of complex exponential functions.

*X*_{s} (*f*_{r}) is generally a complex function of the reduced frequency *f*_{r}, as we see in the following expression:

Among the properties of the Fourier transform, we can first of all consider that:

Then, using equation (3.54), we have:

Secondly, we can verify that the Fourier ...