Sampling and Aliasing: The Rule You Cannot Break

An analogue-to-digital converter looks at a voltage every $T_s$ seconds and writes down a number. Between measurements it has no idea what happened, and it never will. The sample rate is $f_s = 1/T_s$: 44,100 samples per second on a CD, 48,000 in video, 16,000 in most speech models.

The question this lesson answers is when that gap costs you nothing. It has an exact answer, which is unusual and worth appreciating: under one stated condition, the samples contain everything, and the original continuous signal can be reconstructed perfectly. Outside that condition the loss is not a gentle degradation. It is a specific, predictable corruption that no later processing can undo.

The condition, from Nyquist's 1928 telegraph-transmission work and Shannon's 1949 formulation, is a single inequality. If a signal contains no frequency at or above $B$, it is completely determined by samples taken at any rate

$f_s > 2B$

Half the sample rate, $f_s/2$, is the Nyquist frequency: the highest frequency the system can represent.

Definition: A signal with no energy above some frequency is called band-limited. The theorem says nothing at all about signals that are not, and no real signal ever is exactly. Band-limiting a signal is therefore something you must do to it, not something you can hope it already satisfies.

The factor of two has a simple reading: a sinusoid needs at least two samples per cycle, one for the peak and one for the trough, or you cannot tell how fast it is turning.

A frequency above Nyquist does not vanish and does not distort into noise. It reappears at a different, lower frequency, and once it does, nothing distinguishes it from a real signal that was genuinely there.

Sample a sinusoid at $f$ with rate $f_s$ and the recorded frequency is $f$ folded into the range $[0, f_s/2]$: subtract multiples of $f_s$, then reflect any negative result. At the CD rate of 44,100 Hz:

Up to 22,050 Hz the line is honest. Past it, it folds back down and keeps zigzagging. A 40 kHz ultrasonic component lands at 4,100 Hz, squarely in the middle of speech.

Aliasing is not an audio curiosity. It is what happens whenever a periodic thing is observed at intervals.

• Wagon wheels in film. A 24 frames-per-second camera samples a spinning wheel. Above 12 rotations per second the spoke pattern folds, and the wheel appears to slow, stop, or run backwards.
• Moire in photographs. A fine fabric weave has a spatial frequency above the sensor's pixel Nyquist limit, and folds down into broad coloured bands that were never in the room.
• Strobe lights. A strobe deliberately undersamples a rotating machine so that the alias sits at 0 Hz and the part appears frozen.
• Aliased plots. Charting one point per hour from data with a strong 90-minute cycle produces a slow drift that does not exist.

Key idea: Every one of these is the same equation. The alias is real in the recording, so it cannot be removed afterwards. The wheel genuinely does turn backwards in the film.

This is the practical heart of the lesson. Aliasing is not a rendering artefact you can post-process away, it is information destroyed at the moment of capture. Two different inputs produced the same sample sequence, and no algorithm can recover which one you had.

Since the damage happens during conversion, the remedy has to sit before it: an analogue low-pass filter in the signal path ahead of the converter, removing everything above Nyquist while it is still continuous. It is called an anti-alias filter, and it is the reason every serious ADC front end has one.

Gotcha: A digital filter cannot do this job. By the time the signal is a list of samples, the folding has already happened. This ordering trips people up constantly, in audio, in imaging, and in sensor pipelines, because the digital filter is the easy one to add and it looks like it should work.

The same rule applies inside a purely digital pipeline. Decimating by keeping every fourth sample lowers $f_s$ by four, which lowers Nyquist by four, so anything now above the new limit will fold. scipy.signal.decimate low-pass filters before discarding samples for exactly this reason; a bare slice x[::4] does not, and quietly aliases.

import numpy as np
from scipy import signal

fs = 48000
t = np.arange(fs) / fs
x = np.sin(2*np.pi*1000*t) + np.sin(2*np.pi*9000*t)

# WRONG: 9 kHz is above the new Nyquist of 4 kHz, so it folds to 3 kHz
naive = x[::6]

# RIGHT: filter to below 4 kHz first, then keep every 6th sample
safe = signal.decimate(x, 6, ftype="fir")

After the naive slice the new sample rate is 8 kHz, Nyquist is 4 kHz, and the 9 kHz component folds to $|9000 - 8000| = 1000$ Hz, landing exactly on top of the tone you wanted to keep. The correct version removes it beforehand, so it is simply gone rather than disguised.

Human hearing tops out around 20 kHz, so the theorem asks for anything above 40,000 samples per second. The CD standard chose 44,100. The extra is not a safety margin against the mathematics, which is exact; it is room for the filter.

An ideal filter that passes 20 kHz untouched and blocks 20,001 Hz entirely does not exist in analogue components. Real filters roll off over a finite width, so you buy that width by sampling faster than the minimum. That, rather than any audible content above 20 kHz, is the honest engineering argument for high-rate recording.

Reconstruction is the other half of the theorem, and it is not "join the dots". The exact formula sums a scaled sinc function centred on every sample:

$x(t) = \sum_{n=-\infty}^{\infty} x[n] \, \mathrm{sinc}\!\left(\frac{t - nT_s}{T_s}\right)$

Each sinc is 1 at its own sample and exactly 0 at every other sample instant, so the sum passes through all the measured points while filling the gaps with the unique band-limited curve that fits them.

In practice: A real digital-to-analogue converter cannot use an infinitely wide sinc, so it holds each sample for one period, a zero-order hold that produces a staircase, and then smooths it with a reconstruction filter. The staircase you sometimes see drawn as "what digital audio looks like" is an intermediate stage inside the converter, not the output.

Everything in this lesson reduces to four questions you can ask of any pipeline that turns a continuous quantity into numbers.

1. What is the highest frequency present in the source? Not the highest you care about, the highest that exists. Ultrasonic sensors, switching power supplies and fluorescent lighting all inject content people forget about.
2. Is there an analogue filter before the converter? If not, everything above Nyquist is folding in right now.
3. Does any stage change the rate? Every decimation is a new Nyquist limit and needs its own filter first.
4. Does the sample rate leave room for a realisable filter? The theorem's bound is a floor, not a target.

Aliasing is unusual among engineering failures in that it is entirely predictable, completely preventable, and utterly unfixable once committed.