Why can sampled sound leave an ambiguous frequency?

A recorder can measure every sampled value perfectly and still confuse two different repeating signals. The problem can occur before any rounding of the measurements. Sampling records amplitude at selected times; it does not directly preserve everything that happened between them. Different continuous curves can pass through every recorded point.

Take a deliberately slow mathematical example with eight samples each second. The sample times are 0, 1/8, 2/8 of a second, and so on. A unit-amplitude cosine repeating seven times per second gives exactly the same sample values as a cosine repeating once per second, provided both start at their maximum. The first eight values are 1, about 0.707, 0, about −0.707, −1, about −0.707, 0, and about 0.707. Between those instants, the seven-cycle curve moves very differently.

The identity continues beyond the displayed second. At sample number n, the fast curve has value cos(2π × 7n/8). Since 7n/8 equals n minus n/8, its angle differs from the negative of the slower curve’s angle by a whole number of turns. Cosine repeats after a whole turn and has the same value at opposite angles. Thus both curves give the same value at every integer n. This is exact aliasing, even though some printed decimals are rounded.

The slower curve is a possible explanation of the samples, not proof that a second physical sound was present. Without information restricting the original signal, the dots alone cannot identify which continuous curve produced them. Increasing the rate changes the ambiguity: at twelve samples per second, the same zero-phase 7 Hz cosine aliases with a 5 Hz cosine instead of a 1 Hz cosine.

For ideal recovery of a general low-pass signal, its frequencies must be restricted to below half the sampling rate, the Nyquist frequency. Eight samples per second give a 4 Hz boundary. Equality needs care: a 4 Hz sine sampled at those times gives zero at every sample, despite moving between them. Staying strictly below the boundary avoids this particular phase problem under the ideal sampling theorem’s assumptions.

Those assumptions include exact uniform sampling, a known band limit, and the full sample sequence with ideal reconstruction. A short window of eight dots does not independently prove unique recovery of an arbitrary signal. In practical recording, filtering before sampling reduces unwanted high-frequency content, while the sampling rate leaves room for the desired signal. More precise amplitude numbers cannot fix distinctions that the timing has already lost.

The example uses low frequencies to make the curves readable rather than to represent an ordinary musical note. Its consequence extends to digital sound and other measurements: faithful recording depends on both what is measured and when it is measured. A record of exact instants can become an accurate account of change only when the missing intervals are constrained in the right way.