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.
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.