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As explained in Section 1.1, the FT is used to transform continuous signals from one domain to another. It is usually used to describe the continuous spectrum of an aperiodic time signal. To be able to utilize the FT while working with digital signals, the digital or DFT has to be used.

Let s(t) be a continuous periodic time signal with a period of T_o = 1/f_o. Then, its sampled (or discrete) version is s[n] ≜ s(nT_s) with a period of NT_s = T_o where N is the number of samples in one period. Then, the Fourier integral in Eq. 1.1 will turn to a summation as shown below.

(1.21)

Dropping the f_o and T_s inside the parenthesis for the simplicity of nomenclature and therefore switching to discrete notation, DFT of the discrete signal s[n] can be written as

(1.22)

In a dual manner, let S(f) represent a continuous periodic frequency signal with a period of Nf_o = N/T_o and let S[k] ≜ S(kf_o) be the sampled signal with the period of Nf_o = f_s. Then, the IDFT of the frequency signal S[k] is given by

(1.23)

Using the discrete notation by dropping the f_o and T_s inside the parenthesis, the IDFT of a discrete frequency signal S[k] is given as

(1.24)