Index DigitalSampling & Aliasing

Sampling & Aliasing

See what a sample rate can capture, and where a tone lands when it is too high.

Nyquist limit22.05kHz

Recorded as16.1kHz

Samples per cycle1.57

VerdictAliased

Time domain

Input tone
Samples
What the DAC rebuilds

Spectrum fold

Input

Tone frequency

f

Sample rate (kHz)

How it works

The sampling theorem

A signal sampled $f_s$ times per second can be rebuilt exactly as long as it contains nothing at or above half that rate. That boundary is the Nyquist frequency:

f_N = \frac{f_s}{2}

What happens above it

Sample values of a tone above $f_N$ are identical to those of a lower tone, so the converter cannot tell them apart. The energy folds back into the audio band at

f_{alias} = \left| f - k\,f_s \right|,

k = \operatorname{round}\!\left(\frac{f}{f_s}\right)

The red trace in the first plot passes through every sample dot just as the input does. That is the alias, and no later processing can remove it.

Why it matters for hi-res

An ADC needs an anti-alias filter that passes 20 kHz yet is fully closed by $f_N$ . At 44.1 kHz that leaves about 2 kHz of room, which demands a very steep filter. At 96 kHz there are 28 kHz to work with, so the filter can be gentler and its ripple and ringing move far away from the audible band.

References

H. Nyquist, “Certain Topics in Telegraph Transmission Theory,” Trans. AIEE, vol. 47, no. 2, pp. 617–644, 1928. IEEE
C. E. Shannon, “Communication in the Presence of Noise,” Proc. IRE, vol. 37, no. 1, pp. 10–21, 1949. IEEE