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Digital Filters Design for Signal and Image Processing by Mohamed Najim

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3.3. The discrete Fourier transform (DFT)

3.3.1. Expressing the Fourier transform of a discrete sequence

Let us look at the signal xs(t) coming from the sampling of x(t) at the sampling frequency fs:

images

According to equation (3.24), the Fourier transform of the signal xs(t) verifies the following relation:

images

If we introduce fr, the frequency reduced or normalized in relation to the sampling frequency images, we will have:

images

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.

Xs (fr) is generally a complex function of the reduced frequency fr, as we see in the following expression:

images

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

images

Then, using equation (3.54), we have:

Secondly, we can verify that the Fourier ...

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