1. Introduction
The Fourier transform is one of the most powerful and widely used mathematical tools. It decomposes any signal into a sum of sinusoids of different frequencies. In the complex plane, each sinusoid corresponds to a rotation — but not necessarily to a spiral.
2. The Fourier transform
The Fourier transform of a function f(t) is defined by:
The kernel e^(−iωt) is a complex exponential. By Euler's formula, e^(−iωt) = cos(ωt) − i·sin(ωt). In the complex plane, this kernel traces a circle of radius 1 traversed at angular velocity ω.
3. The complex plane
In the complex plane, multiplication by e^(iθ) is a rotation of angle θ. The Fourier transform can therefore be seen as a process that "winds" the signal around the origin of the complex plane at different angular velocities.
When the winding speed corresponds to a frequency present in the signal, contributions accumulate on one side of the complex plane and the transform takes a large value. When the speed corresponds to no frequency, contributions cancel and the transform is close to zero.
4. When does the spiral appear?
The kernel e^(−iωt) alone traces a circle, not a spiral. A spiral appears when the amplitude varies at the same time as the phase. For example, e^(−(α+iω)t) = e^(−αt) · e^(−iωt) is a logarithmic spiral: the radius decreases exponentially while the phase rotates.
The Laplace transform — a generalization of the Fourier transform — uses precisely this type of spiral kernel. It is particularly suited to the analysis of dynamic systems with damping.
5. Applications
- Audio signal processing: equalization, MP3 compression
- Image processing: JPEG compression, filtering
- Nuclear magnetic resonance: MRI image reconstruction
- Quantum mechanics: representation in momentum space
- Spectral analysis: identifying frequencies in a signal
6. Conclusion
The Fourier transform reveals the hidden frequency structure in any signal. In the complex plane, it corresponds to a circular winding process — a rotation, not automatically a spiral. The spiral appears when generalizing to the Laplace transform, which simultaneously integrates rotation and damping.