Why does large space correspond to low wavenumber in Fourier Transform?

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SUMMARY

The discussion clarifies that in Fourier Transform, large spatial features correspond to low wavenumbers due to the relationship between feature size and sinusoidal contributions. Specifically, larger features in the object space lead to significant contributions from sinusoidal terms with longer wavelengths, which equate to lower frequencies or wavenumbers. This principle is fundamental in understanding how Fourier Transform analyzes functions based on their spatial characteristics.

PREREQUISITES
  • Understanding of Fourier Transform principles
  • Knowledge of sinusoidal functions and their properties
  • Familiarity with the concept of wavenumber and wavelength
  • Basic grasp of spatial frequency analysis
NEXT STEPS
  • Study the mathematical foundations of Fourier Transform
  • Explore the relationship between spatial features and frequency components
  • Learn about applications of Fourier Transform in signal processing
  • Investigate the implications of wavenumber in image analysis techniques
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Students and professionals in physics, engineering, and signal processing who seek to deepen their understanding of Fourier Transform and its applications in analyzing spatial features and frequencies.

shirin
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Hi
In making Fourier Transform of a function, why is it said that large space (r) corresponds to low wavenumber(k)?
 
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Actually it's more about the feature size in the object space rather than the absolute position in that space. You can roughly picture it with the help of the fact that Fourier transform is an infinite sum of weighted sinusoidal disturbance. When there is a large feature in the object space, a large contribution to it must come from sinusoidal terms which have large wavelength (slow variation), or in term of frequency or wavenumber the lower ones.
 

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