# Physical meaning of a fourier transform?

• dst
In summary, the concept of Fourier transform is extremely useful in representing physical situations, particularly for Linear, Time-Invariant (LTI) systems. Examples of its applications include the far-field scattering pattern for light, crystallography and optical fields in optics, and voltage induction in MRI. Without the use of Fourier transform, frequency analysis in various fields would be impossible.
dst
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.

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 :)

## 1. What is the physical meaning of a Fourier transform?

The Fourier transform is a mathematical tool that decomposes a signal into its individual frequency components. This can provide insight into the underlying physical processes that generated the signal.

## 2. How is a Fourier transform used in science?

A Fourier transform is used in various scientific fields, such as physics, engineering, and signal processing, to analyze and manipulate signals and data. It is especially useful for studying periodic or oscillating phenomena.

## 3. Can the physical meaning of a Fourier transform be explained intuitively?

Yes, the physical meaning of a Fourier transform can be understood by considering a signal as a combination of different waves with different frequencies and amplitudes. The Fourier transform allows us to see the individual components of a signal and how they contribute to the overall signal.

## 4. What is the relationship between a Fourier transform and a Fourier series?

A Fourier transform is a continuous version of a Fourier series, which is used to represent a periodic function as a sum of sinusoidal functions. The Fourier transform extends this concept to non-periodic functions by allowing for a continuous range of frequencies.

## 5. How does a Fourier transform help in solving differential equations?

Fourier transforms are used in solving differential equations by converting them into algebraic equations. This allows for easier manipulation and analysis of the equations, making it a powerful tool in engineering and physics applications.

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