Digital signals spectra

 

To determine the spectrum of a digital signal, the formula used is the discrete Fourier transform. This digital Fourier transform is deduced from the analogue expression mainly by replacing the integral by a discrete sum S.

Fourier transform of a discrete signal (FTDS)

This expression is a periodic continuous function of the frequency f. This expression recalls the expression of the Fourier series development, with the frequency and the time having "exchanged their roles". It is a consequence of the time-frequency duality

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Inverse Fourier transform of a discrete signal (IFTDS)

 

This expression also looks like the definition of the cn coefficients of the Fourier series development. It is again a consequence of the time-frequency duality.