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

or

Notes

^ Harris' motivation for the distinction is to distinguish between an odd-length data sequence with the indices $\left\{-{\tfrac {N-1}{2}}\leq n\leq {\tfrac {N-1}{2}}\right\},$ which he calls the finite Fourier transform data window, and a sequence on $\{0\leq n\leq N-1\},$ which is the DFT data window.

^ George Bachman, Lawrence Narici, and Edward Beckenstein, Fourier and Wavelet Analysis (Springer, 2004), p. 264
^ Morelli, E., "High accuracy evaluation of the finite Fourier transform using sampled data," NASA technical report TME110340 (1997).

Harris, Fredric J. (Jan 1978). "On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform" (PDF). Proceedings of the IEEE. 66 (1): 51–83. CiteSeerX 10.1.1.649.9880. doi:10.1109/PROC.1978.10837. S2CID 426548.
Cooley, J.; Lewis, P.; Welch, P. (1969). "The finite Fourier transform". IEEE Trans. Audio Electroacoustics. 17 (2): 77–85. doi:10.1109/TAU.1969.1162036.