Finite Fourier transform
From HandWiki
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.[upper-alpha 1] 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]
See also
Notes
- ↑ Harris' motivation for the distinction is to distinguish between an odd-length data sequence with the indices [math]\displaystyle{ \left\{-\tfrac{N-1}{2} \le n \le \tfrac{N-1}{2}\right\}, }[/math] which he calls the finite Fourier transform data window, and a sequence on [math]\displaystyle{ \{0 \le n \le N-1\}, }[/math] which is the DFT data window.
References
- ↑ 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". Proceedings of the IEEE 66 (1): 51–83. doi:10.1109/PROC.1978.10837. http://web.mit.edu/xiphmont/Public/windows.pdf.
- 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.
Further reading
- Rabiner, Lawrence R.; Gold, Bernard (1975). Theory and application of digital signal processing. Englewood Cliffs, N.J.: Prentice-Hall. pp 65–67. ISBN:0139141014.
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