PlaygroundSignals · 03
Uncertainty
Step 1
One window, one compromise.
The test sound holds four things: two low notes a whole tone apart that play all the time, a short high blip, a click, and a rising sweep. A spectrogram shows it through a sliding window of time: time runs left to right, pitch bottom to top. Change the window length and watch the click and the two low notes.
What one window can resolve
Wide and flat, or tall and thin: the area never changes.
Step 2
What is a wavelet?
A Fourier transform compares the sound with sine waves that go on forever. A wavelet is a short burst of wave instead: a few oscillations that swell and fade.
To look for a low note you stretch the wavelet; to look for a high one you squeeze it. It always keeps the same number of oscillations, so its length follows the pitch: long for low notes, where you need patience to pin down the pitch, short for high sounds, where you need precision in time.
A spectrogram uses one window length for every pitch. Wavelets choose a different one for each, which is also roughly how your ear works.
High pitch
short: precise in time
Middle
Low pitch
long: precise in pitch
Step 3
Wavelets adapt.
The same sound, analysed with wavelets. The click stays a sharp line and the two low notes stay apart: no single window length in step 1 manages both.
How they divide time and pitch
Sharp click and separate notes, in one picture.
Every tile still has the same area: the uncertainty principle is not broken, only spent differently at each pitch.
What's going on
Any way of looking at a sound over time has to decide how long to listen. Short looks are precise in time and vague in pitch; long looks are the reverse. The two precisions multiplied together can never drop below a fixed limit: the time-frequency uncertainty principle, the same mathematics as Heisenberg's.
Wavelets do not escape the limit. They spend it differently at every pitch, in the way that suits music: fast detail up high, slow notes and harmony down low.
Why it matters: how you look at a sound decides what a model can learn from it. Multi-resolution representations like wavelets are one of the directions our members explore; see Research.
The maths behind itFor the curious · formulas and details
The uncertainty principle
A window of length Δt can only tell apart frequencies that are roughly 1/Δt apart. With both measured as standard deviations, the Gabor limit (the same maths as Heisenberg's) is:
Δt · Δf ≥ 1 / 4π
The slider runs from 32 samples (4 ms, frequency resolution about 250 Hz) to 2048 samples (256 ms, about 4 Hz) at 8,000 samples per second. The two notes are 196 and 220 Hz, 24 Hz apart; with a Hann window they only separate from about 1024 samples (128 ms) on.
Wavelets
The Morlet wavelet is a complex sine under a Gaussian. It is stretched by a scale s and shifted by a time τ, and the transform correlates the signal with every stretched copy:
ψ(t) = e^(iω₀t) · e^(−t²/2) W(s, τ) = (1/√s) ∫ x(t) · ψ*((t − τ) / s) dt
Every copy holds the same number of cycles (24 here), so its length in time is inversely proportional to its frequency: constant relative bandwidth, which is also roughly how the inner ear resolves pitch. The tiling sketches in step 3 are schematic (linear axes), but the areas are honest: every tile covers the same Δt · Δf.