EEG bands, separated

A signal can contain slow and fast changes at the same time. Toggle frequency ranges to see which parts of this synthetic trace remain, and inspect individual samples.

Illustrative demo data — educational model, not applicant research.

Static figure and worked values are available below. Interactive controls appear when JavaScript is available.

Static worked example and model values

Sample rate 128 Hz; 256 samples; frequency-bin spacing 0.5 Hz. Every band includes its lower bound and excludes its upper bound.

Tone (Hz)	Amplitude (µV)	Phase (rad)
2	15	0
6	8	0.2
10	10	0.4
20	4	0.6
36	2	0.8

What you are looking at

The muted line is the original signal; the darker line is the sum of selected bands. The vertical cursor identifies the sample reported above the plot. Play moves this cursor through a fixed recording; it does not generate new data. Turning every band off produces zero. Turning them all on reconstructs this particular original signal because all its components lie inside these bands.

This is a sum of five sinusoids, not a scalp EEG recording. Frequencies, amplitudes, phases and sample rate are documented in the static worked example. The clean separation is intentional: the tones fall exactly on Fourier bins. Real recordings rarely cooperate so neatly.

A frequency-domain teaching filter

For $N$ samples $x_n$, the discrete Fourier transform is

$$X_k=\sum_{n=0}^{N-1}x_ne^{-2\pi i kn/N}.$$

For selected frequency ranges, the demonstration keeps the corresponding positive and negative frequency coefficients and sets the others to zero. The inverse transform gives

$$\tilde x_n=\frac1N\sum_{k=0}^{N-1}M_kX_ke^{2\pi i kn/N},$$

where $M_k$ is either zero or one. Symmetric selection preserves a real-valued result. The 128 Hz sample rate and 256-sample interval give 0.5 Hz bin spacing; the Nyquist frequency is 64 Hz.

Band intervals include the lower endpoint and exclude the upper endpoint. Here they are delta [0.5, 4), theta [4, 8), alpha [8, 13), beta [13, 30) and a displayed portion of gamma [30, 45) Hz. These are declared conventions, not universal biological boundaries; gamma extends beyond this view.

Interpretation needs context

Slow delta activity is prominent in deep non-REM sleep; theta often appears with drowsiness. Posterior alpha is often visible during relaxed wakefulness with eyes closed. Beta and gamma cover faster activity, but neither is a direct label for a thought or a universal measure of concentration. Location, age, state, reference and artifacts affect interpretation. Muscle activity can strongly contaminate higher frequencies.

Why this is not a preprocessing prescription

The DFT treats the interval as periodic. Abrupt frequency cutoffs can cause ringing; an unaligned component leaks across bins. The whole interval is used at once, so this is an offline, noncausal operation. A deployable EEG pipeline needs an explicit filter design, transition bands, edge handling, artifact strategy and validation on its own task. The sleep-staging project records those choices separately; this synthetic figure is not an illustration of its results.

References

• MNE: background information on filtering.
• American Epilepsy Society EEG atlas: the normal EEG.