Finite Fourier transform

In mathematics the finite Fourier transform may refer to either

another name for discrete-time Fourier transform (DTFT) of a finite-length series. E.g., F.J.Harris (pp. 52–53) describes the finite Fourier transform as a "continuous periodic function" and the discrete Fourier transform (DFT) as "a set of samples of the finite Fourier transform". In actual implementation, that is not two separate steps; the DFT replaces the DTFT.^[A] So J.Cooley (pp. 77–78) describes the implementation as discrete finite Fourier transform.

or

another name for the Fourier series coefficients.^[1]

or

another name for one snapshot of a short-time Fourier transform.^[2]

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[A]

[1]

[2]