I usually start my signal processing classes with this example. Stand me next to a tape measure and I come out at 1.85 metres. But my height isn’t really “1.85”. 1.85 is how my height is expressed in the metric system. Measure the same height in inches and you get about 72.8. I haven’t changed by a millimetre; only the unit system has.
We use exactly the same logic when we process signals.
Time Is A Unit System Too
When you look at an EEG recording on a screen, you’re really seeing the signal in one particular unit system: time. There’s a value for every instant. So many microvolts at this moment, so many at the next. That description is correct, but it doesn’t make every question easy. Ask “how much alpha rhythm is in this signal?” and it’s hard to answer by staring at thousands of numbers along a time axis.
Fourier, Or Changing The Unit System
In the 19th century, Joseph Fourier’s idea was this: however complex a signal is, it can be written as a sum of simple waves oscillating at different speeds, that is, sines. The Fourier transform is the way to express a signal in this new unit system. Instead of asking “what’s the value right now?”, we ask “how much of the wave that oscillates 10 times a second is in this signal? How much of the one that oscillates 20 times?” We call this new description the frequency domain, or spectrum.
Imagine pressing three keys on a piano at once. You hear a single chord, but a good musician can name every note inside it. The Fourier transform does the same thing: it separates the notes inside a mixed sound, the way a prism splits white light into colours.
The key point: the signal didn’t change. Just like my height. We simply switched to a unit system that makes some questions much easier to answer.
Spectral Features
Once the signal is in the frequency domain, we can summarise it with a few meaningful numbers:
- Band power: how much energy is in each of the delta, theta, alpha, beta and gamma bands? (We described these rhythms in our EEG post.)
- Relative power: a band’s share of the total. It reduces the effect of overall signal strength, which varies from person to person.
- Band ratios: for example, theta power divided by beta power. Often used in attention studies.
- Peak alpha frequency: exactly where in the alpha band the peak sits. It behaves almost like a personal signature.
- Spectral entropy: is the energy concentrated at a few frequencies, or spread everywhere? The spectrum of a regular signal is spiky; the spectrum of noise is flat.
Speed And Limits
Computers can calculate the Fourier transform quickly thanks to the fast Fourier transform (FFT) algorithm published in 1965. Today we can compute the spectrum of an EEG recording in a tiny fraction of a second.
Fourier has one limit: it tells you how much of a frequency is in a signal, but not when it appears. Because brain signals change all the time, we split the recording into short windows and compute the spectrum of each one separately. The resulting time–frequency map can be processed like an image, as described in our CNN post. In one of our lab’s smell studies, we placed spectral features on a map of the head and classified them with a CNN.
In the next post, we’ll stay in the time domain, without changing the unit system, and see what we can learn there.
References
- Akan, A., Chaparro, L. F. (2024). Frequency analysis: the Fourier transform. Elsevier (book chapter). DOI: 10.1016/b978-0-44-315709-7.00015-x
- Cooley, J. W., Tukey, J. W. (1965). An algorithm for the machine calculation of complex Fourier series. Mathematics of Computation, 19(90), 297–301. DOI: 10.1090/S0025-5718-1965-0178586-1
- Yeganli, F., Akan, A., Sadıkzade, R. (2025). Olfactory Emotion Recognition Using EEG Spectral Topographic Heatmaps and CNNs. IEEE TIPTEKNO. DOI: 10.1109/tiptekno68206.2025.11270101