Nyquist & aliasing

Chapter 01 ended at a cliff edge. The sample-rate slider felt forgiving. A 440 Hz tone survived 4 kHz, even 1 kHz, sounding rougher but recognizably itself. That forgiveness has a floor, and the floor is exact: to capture a wave you need more than two samples per cycle. The intuition is almost physical. A cycle goes up and comes down; if you want to know it happened, you must catch it at least once on the way up and once on the way down. Sample any slower and entire swings of the wave happen between your measurements, unrecorded.

Turned around, that floor becomes a ceiling. A converter running at a sample rate fs can faithfully capture any frequency below fs/2. That boundary is called the Nyquist frequency, after Harry Nyquist, the Bell Labs engineer who worked out the limit in the 1920s (Claude Shannon proved the full theorem in 1949). At 48 kHz the ceiling is 24 kHz; at the 8 kHz rate telephones use, it is 4 kHz. Below the ceiling, the samples pin down one and only one wave. The theorem’s promise is stronger than it first sounds: not “a decent approximation,” but perfect reconstruction, in principle, of everything under the limit.

Here is the strange part — the part that makes this a cliff and not a slope. Feed a converter a tone above its ceiling and the samples don’t degrade. They lie. A 5,000 Hz tone sampled at 8 kHz produces a list of numbers that is, sample for sample, identical to the list a 3,000 Hz tone would produce. The frequency folds back around the ceiling like a ruler creased at the 4 kHz mark: the recorded frequency is |f − fs·round(f/fs)|. This is aliasing. The high tone travels under an assumed name, and nothing in the data betrays it.

Sweep the slider. Below 4,000 Hz there is one wave and twenty dots riding it. Cross the ceiling and a second sine appears, dashed: that is the alias. Notice what the figure is not doing: it isn’t drawing an error band or an approximation. Both curves are honestly computed, and both pass through every dot exactly, because at 8,000 samples per second they are the same list of numbers. Then press play. The tone is synthesized at 48 kHz, where 7,000 Hz exists comfortably, and then decimated to 8 kHz by keeping every sixth sample — no filter, which is precisely the mistake. Sweep upward past 4 kHz and the pitch you hear stops rising and comes back down; by 7,800 Hz the converter swears it heard 200 Hz.

The information is gone

The reflex is to ask which algorithm fixes this. None does, and it’s worth sitting with why: aliasing isn’t distortion layered on top of a signal, like hiss or clipping, where the original is still in there somewhere, damaged. Two different waves produced the same numbers. Once only the numbers remain, no computation — not a cleverer FFT, not machine learning, nothing — can decide which wave made them, for the same reason you can’t un-add two numbers knowing only their sum. The information was never captured. Every stage downstream of the converter, in this pipeline and any other, inherits whatever fiction the samples tell.

You have seen aliasing with your own eyes. A film camera is a sampler too, running at 24 frames per second, and a wagon wheel or helicopter rotor spinning faster than 12 revolutions per second folds back exactly the same way, appearing to slow, stop, or turn lazily backward on screen. Same theorem, different converter.

So real systems refuse to let the crime happen. In front of every analog-to-digital converter sits an anti-alias filter: an analog low-pass that removes everything above fs/2 before it is ever measured. It must live in the analog world: by the time software could act, the folding is already in the numbers. (Modern converters actually sample far above the target rate and do most of the filtering digitally where it’s cheap and precise, but the principle is untouched: filter first, then commit to samples.) The figure’s play button skips this filter deliberately; your audio hardware never does.

The margin this pipeline runs on

What does the ceiling mean for this codebase? At the pipeline’s default 48 kHz, everything below 24 kHz is safe, comfortably past the ~20 kHz edge of human hearing, with the anti-alias filtering handled by the audio interface before a single Float32Array reaches the engine. And the signals this project decodes sit nowhere near the edge. DTMF’s highest tone is 1,633 Hz, a fourteenth of the ceiling. That is not an accident of generosity; it is inherited history. DTMF was engineered for telephone lines, whose 8 kHz sampling and ~3.4 kHz voice band meant every signaling tone had to live below a 4 kHz ceiling — a story chapter 08 tells properly. The figure’s converter runs at 8 kHz for exactly that reason: it is the rate at which this ceiling shaped real engineering for a century.

The theorem also quietly labels an axis you are about to meet. When chapter 03 turns a window of samples into a spectrum, the frequency axis will run from 0 up to fs/2 and stop — not because the math gets tired, but because you now know there is nothing past that line for the samples to say. The Nyquist frequency isn’t a limit on the FFT. It is a limit on what the numbers ever knew.