File:FTcontinuousToDiscrete3.png
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Captions
Captions
Summary
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| Description |
English: Schematic showing continuous to discrete functions, periodic or not, and their Fourier transforms. The original of this figure was put into the public domain by Steve Byrnes, but is here completely regenerated with labels, new functions, and shading. "Even functions" (symmetric about the origin) have been chosen so that the Fourier transform coefficients are real rather than complex.
The yellow shaded regions show the limited data regions (when finite) needed to generate the plots, from either the direct time/space coordinate (top) or the reciprocal frequency coefficient (bottom) side. The horizontal axis on the top row is in units of space or time, while the horizontal "frequency" axis on the bottom row is in cycles per unit space or time. The vertical displacement/amplitude scales are normalized to keep the scales similar as well. Discrete sampling (in direct and/or reciprocal space) matches the realities of digital sampling, and opens the door to rapid "Fast Fourier Transform" calculations, but entails the often artificial assumption of periodicity (in reciprocal and/or direct space respectively). This assumption is therefore something that we are learning to live with, at least as a useful approximation. |
| Date | |
| Source | Own work |
| Author | P. Fraundorf |
Discussion
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This "Fourier transform type" figure shows how the DFT (discrete Fourier transform) gives rise to other cases as limiting forms:
- For the DFT, the time/space sampling increment Δx = L/N for -L/2 ≤ x < L/2 where L is the coordinate range, while the frequency sampling increment Δf = 1/L for -B/2 ≤ f < B/2 (or Nyquist limit) where the frequency range B = N/L. Here of course N is the number of sample points in both direct and reciprocal space.
- If we let Δf go to zero but hold Δx constant, then the coordinate range ±L/2 and the number of sample points N go to infinity while the frequency range ±N/(2L) stays the same, taking us from the DFT case to the DTFT (Fourier time/space series) limit.
- If we let Δx go to zero but hold Δf constant, then the coordinate range ±L/2 is constant but the sample point number N and the frequency range ±B/2 go infinite, taking us from the DFT to the DFFT (Fourier series) limit.
- If we let both Δf and Δx go to zero, then both the coordinate range ±L/2 and the frequency range ±B/2 along with sample point number N go infinite, taking us from the DFT to the continous Fourier transform (Fourier integral) case.
Licensing
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 00:29, 18 February 2022 | | 2,555 × 814 (122 KB) | Unitsphere (talk | contribs) | Uploaded own work with UploadWizard |
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