Digital Filters Design for Signal and Image Processing

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 ...