mathematics//signal processing//sampling//aliasing

Aliasing is the error of sampling in which content above half the sampling rate reappears in the samples at a lower frequency, indistinguishable from a real signal at that frequency, and it is the reason every measurement chain filters before it converts. The wagon wheel that turns backwards on film is the everyday case: the camera samples at 24 frames a second, and a spoke pattern that advances almost one spoke per frame looks as if it moved slightly back.

Aliasing is the error of sampling in which content above half the sampling rate reappears in the samples at a lower frequency, indistinguishable from a real signal at that frequency, and it is the reason every measurement chain filters before it converts. The wagon wheel that turns backwards on film is the everyday case: the camera samples at 24 frames a second, and a spoke pattern that advances almost one spoke per frame looks as if it moved slightly back.

The samples of a fast sine and of a slower one can coincide exactly. A component at frequency fff sampled at fsf_sfs​ shows up at the alias frequency faf_afa​.

fa=∣f−mfs∣,m=the integer nearest to f/fsf_a=\lvert f-m f_s\rvert,\qquad m=\text{the integer nearest to } f/f_sfa​=∣f−mfs​∣,m=the integer nearest to f/fs​

Every frequency therefore folds into the band from zero to fs/2f_s/2fs​/2, the Nyquist frequency of the sampling theorem. The motors of a small drone spin at tens of thousands of rpm and those of a large one at a few thousand, and both shake the frame between about 50 and 500 Hz. Sample the accelerometer at 250 Hz without filtering first and a 240 Hz vibration appears at 10 Hz, in the middle of the attitude loop's band, where the controller starts correcting a roll that exists only in the samples.

apparent frequency2 Hz Nyquist, fs/27.5 Hz step Δ7.81 mV quantization error, RMS2.57 mV A 13 Hz sine sampled at 15 Hz, reconstructed from its samples and rounded by a converter of 8 bits over ±1 V. It lies above the Nyquist frequency of 7.5 Hz, so it folds: the samples draw a 2 Hz wave that is not there.

Set the signal above half the sampling rate and watch the reconstruction draw a slow wave that is not there, then switch on the anti-alias filter: the ghost loses almost all its amplitude, and so does anything real above fs/2f_s/2fs​/2.

Once folded, the ghost cannot be separated from the signal.

After sampling, the 10 Hz alias and a real 10 Hz roll are the same numbers, so no digital filter, estimator or learned model can remove one and keep the other. The only remedies act before the converter: an anti-aliasing filter in the analog or mechanical path, or a higher rate.

Aliasing is confused with noise because both add content that was not wanted, yet an alias is deterministic and coherent: it has a definite frequency that moves when the source moves (a motor speeding up makes the alias slide), and averaging does not shrink it the way it shrinks random noise.

The danger is greatest where a real band and a folded one overlap. Vibration analysis of a bearing needs content up to several kilohertz and is sampled at tens of kilohertz with an analog filter in front; sampled at 1 kHz the same signal would fold into false low-frequency lines with no way back (vibration analysis).

Aliasing also happens in time series and logs that are decimated without a filter: keeping one sample in ten from a 1 kHz log folds everything between 50 and 500 Hz into the stored data (analog-to-digital converter for the hardware side).