Finite Fourier transform explained

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.  So J.Cooley (pp. 77–78) describes the implementation as discrete finite Fourier transform.

or

• another name for the Fourier series coefficients.

or

• another name for one snapshot of a short-time Fourier transform.

See also

• Fourier transform

References

3. Harris . 10.1109/PROC.1978.10837 . Harris . Fredric J. . On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform . Proceedings of the IEEE . 66 . 1 . 51–83 . Jan 1978 . 10.1.1.649.9880. 426548 .
4. Cooley . Cooley . J. . Lewis . P. . Welch . P. . The finite Fourier transform . IEEE Trans. Audio Electroacoustics . 17 . 2 . 77–85 . 1969 . 10.1109/TAU.1969.1162036.

Further reading

• Rabiner, Lawrence R.; Gold, Bernard (1975). Theory and application of digital signal processing. Englewood Cliffs, N.J.: Prentice-Hall. pp 65–67. .