Where does aliasing collide?

The corruption that occurs when a continuous signal is sampled too slowly: any component whose frequency is above half the sampling rate is indistinguishable from a lower one, so it reappears in the sampled record wearing a false, lower frequency. Shannon's 1949 Communication in the Presence of Noise proved the sampling theorem that fixes this limit, showing that a signal is fully recoverable only if it is sampled above twice its highest frequency. The referent here is a frequency-domain masquerade in a sampled waveform, measured in cycles per second, and it is not a jagged edge on a screen, a tangle of confounded effects in an experiment, or two names for one piece of memory.[1]

Unit

A false low-frequency component in a sampled signal, in cycles per second

Recovery means

Prevented, not reversed: a low-pass filter before sampling removes the frequencies that would fold, but once a signal is aliased the true frequency cannot be recovered

Measured by

Detected as spurious spectral lines in the discrete Fourier transform of the sampled data, at frequencies below half the sampling rate

The visible defect that appears when a picture is drawn on a grid of pixels that is too coarse for its detail: smooth edges break into a staircase of steps, thin lines flicker, and fine textures shimmer as the scene moves. Crow's 1977 The aliasing problem in computer-generated shaded images named these artifacts and traced them to sampling a continuous image at too few points, prompting the whole family of anti-aliasing methods. The referent is a spatial artifact you can see in a rendered frame, judged by eye and by pixel geometry, not a folded frequency in an audio channel, a confounded factor in a design, or a shared memory location in a program.[2]

Unit

A staircase step or shimmer at an edge in a raster image, per pixel

Recovery means

Softened by anti-aliasing, such as supersampling or filtering, which averages many samples per pixel to smooth the steps rather than undo them

Measured by

Assessed by visual inspection and by comparing the rendered pixels against the ideal continuous edge they approximate

In a fractional factorial experiment, where only part of the full set of factor combinations is run to save effort, some effects become impossible to tell apart: each estimate is really the sum of two or more effects that the reduced design cannot separate, and those tangled effects are called aliases of one another. Box and Hunter's 1961 The 2^k–p fractional factorial designs set out the alias structure formally and showed how the chosen fraction decides which effects get confounded. The referent is a statistical confounding built into a design, described by an alias pattern, and it has nothing to do with a folded frequency, a jagged edge, or a duplicated name in memory.[3]

Unit

A pair or group of factor effects that a design cannot separate

Recovery means

Broken only by running more of the design: added runs, or a follow-up fraction, that let the confounded effects be estimated apart

Measured by

Read off the design's defining relation, which lists exactly which effects are aliased with which

The situation in a program where two or more names, such as pointers or references, denote the very same storage, so writing through one silently changes what the other reads. Landi and Ryder's 1991 Pointer-induced aliasing: a problem taxonomy classified when this can happen and why deciding it exactly is undecidable, work that underlies the alias analyses compilers use to reason about programs. The referent is a naming relation over memory in code, decided by static analysis, and it is unrelated to a folded frequency in a signal, a staircase edge in an image, or confounded effects in an experiment.[4]

Unit

A pair of program names that resolve to one storage location

Recovery means

Not applicable

Measured by

Approximated by alias analysis, which computes for each pair of names whether they may, must, or cannot refer to the same location