Physical meaning of a fourier transform?

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Discussion Overview

The discussion revolves around the physical meaning and applications of the Fourier transform, particularly in "natural" contexts. Participants explore various real-world examples and the conceptual understanding of how Fourier transforms relate to physical phenomena in fields such as optics, signal processing, and imaging.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants inquire about "real" examples of Fourier transforms in natural applications, expressing a desire for understanding beyond theoretical contexts.
  • One participant highlights the utility of representing signals as sums of sinusoids in signal processing, particularly for Linear, Time-Invariant (LTI) systems.
  • Another participant suggests that Fourier transforms serve primarily as a change of basis, allowing different representations of the same object.
  • Examples provided include the far-field scattering pattern of light being the Fourier transform of the aperture and the Laue pattern in crystallography representing the charge distribution within a unit cell.
  • Participants mention applications in optics, such as the relationship between optical fields at different planes, and in electrical signal processing, including equalizers and guitar effects.
  • One participant notes the importance of Fourier transforms in frequency analysis of vibrations, particularly in contexts like jet engine monitoring.
  • Another participant expresses a need for examples where the Fourier transform occurs automatically in nature, referencing the optical field and MRI applications.

Areas of Agreement / Disagreement

Participants generally agree on the usefulness of Fourier transforms in various applications, but there is no consensus on the interpretation of their physical meaning or the nature of their applications in "natural" contexts.

Contextual Notes

Some limitations include the varying interpretations of what constitutes a "natural" application and the dependence on specific fields of study, such as optics and signal processing, which may influence the examples provided.

Who May Find This Useful

This discussion may be of interest to individuals studying physics, engineering, signal processing, or anyone curious about the practical applications of Fourier transforms in various scientific fields.

dst
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Are there any "real" examples of a Fourier transform being applied? When we see that something accelerates and then moves we can say its acceleration is being "integrated" to get a velocity, but what meaning does a Fourier transform have? I understand it's used in spectroscopy but I mean "natural" applications only.
 
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maybe it's just because i work in signal processing, but the concept of breaking a virtually arbitrary signal into a sum of sinusoids (more specifically e^{i \omega t}) is an extremely useful application and does a good job of representing the physical situation, particularly for Linear, Time-Invariant (LTI) systems.
 
dst said:
Are there any "real" examples of a Fourier transform being applied?
In every application I know, a Fourier transform is nothing more than a change of basis; it simply allows you to represent the same object in a different way.
 
The far-field scattering pattern for light is (under certain conditions) the Fourier transform of the aperture. In crystallography and such, the scattering pattern (Laue pattern) is the Fourier transform of the charge distribution within a unit cell.

Fourier transforms are used a lot in optics. It has a very physical application- the optical field at a focal plane is the Fourier transform of the optical field at the opposite pupil plane.

Electrical signal processing is another real-world use: the equalizer on a stereo, or windoze media player. Guitar effects can work via manipulating the frequency content.
 
dst said:
Are there any "real" examples of a Fourier transform being applied? When we see that something accelerates and then moves we can say its acceleration is being "integrated" to get a velocity, but what meaning does a Fourier transform have? I understand it's used in spectroscopy but I mean "natural" applications only.
In MRI (Magnetic Resonance Imaging) the voltage that is physically induced in the receive coil is the Fourier transform of the tissue magnetization.
 
Along what rbj mentioned, without an FFT, frequency analysis of vibrations on a running jet engine would be impossible. Looking at the frequency domain instead of the time domain is about a factor of 1 bazillion times easier.
 
Thanks, good examples. It's hard to visualise what would be meant by such a thing. I was asking especially for cases where it happens automatically, i.e. by nature like:

the optical field at a focal plane is the Fourier transform of the optical field at the opposite pupil plane

the voltage that is physically induced in the receive coil is the Fourier transform of the tissue magnetization

Nonetheless, my mind is at rest :)
 

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