Aliasing

 ( Digital Signal Processing ) 

Upper right: The continuous Fourier transform of the sinusoid (not the samples). The single non-zero component, depicting the actual frequency, means that there is no ambiguity.

Lower right: The discrete Fourier transform of just the available samples. The presence of two components means that the samples can fit at least two different sinusoids, one of which is with the true frequency (upper-right). Another sinusoid is with an alias frequency $f-f\_\{\backslash rm\; s\}$. (Here the absolute value of it is shown.)

Lower left: Using the same samples (now in orange), the default reconstruction algorithm produces the lower-frequency sinusoid.]]

Sine wave are an important type of periodic function, because realistic signals are often modeled as the summation of many sinusoids of different frequencies and different amplitudes (for example, with a Fourier series or transform). Understanding what aliasing does to the individual sinusoids is useful in understanding what happens to their sum.

When sampling a function at frequency (i.e., the sampling interval is ), the following functions of time yield identical sets of samples if the sampling starts from $t=0$ such that $t=\backslash frac\{1\}\{f\_s\}n$ where $n\; =\; 0,1,2,3$, and so on:

$$\backslash \{\; \backslash sin(2\backslash pi(f+Nf\_s)t+\backslash varphi),\; N=0,\backslash pm1,\backslash pm2,\backslash pm3,\; \backslash ldots\; \backslash \}.$$

A frequency spectrum of the samples produces equally strong responses at all those frequencies. Without collateral information, the frequency of the original function is ambiguous. So, the functions and their frequencies are said to be aliases of each other. Noting the sine functions as odd functions :

+\sin(2\pi (f+Nf_{\rm s})t + \phi),
& f+Nf_{\rm s} \ge 0 \\
-\sin(2\pi |f+Nf_{\rm s}|t - \phi),
& f+Nf_{\rm s} < 0 \\
     

thus, we can write all the alias frequencies as positive values: $f\_\{\_N\}(f)\; \backslash triangleq\; \backslash left|f+Nf\_\{\backslash rm\; s\}\backslash right|$. For example, a snapshot of the lower right frame of Fig.2 shows a component at the actual frequency $f$ and another component at alias $f\_\{\_\{-1\}\}(f)$. As $f$ increases during the animation, $f\_\{\_\{-1\}\}(f)$ decreases. The point at which they are equal $(f=f\_s/2)$ is an axis of symmetry called the folding frequency, also known as Nyquist frequency.

Aliasing matters when one attempts to reconstruct the original waveform from its samples. The most common reconstruction technique produces the smallest of the $f\_\{\_N\}(f)$ frequencies. So, it is usually important that $f\_0(f)$ be the unique minimum. A necessary and sufficient condition for that is $f\_s/2\; >\; |f|,$ called the Nyquist condition. The lower left frame of Fig.2 depicts the typical reconstruction result of the available samples. Until $f$ exceeds the Nyquist frequency, the reconstruction matches the actual waveform (upper left frame). After that, it is the low frequency alias of the upper frame.

The filtered signal can subsequently be reconstructed, by interpolation algorithms, without significant additional distortion. Most sampled signals are not simply stored and reconstructed. But the fidelity of a theoretical reconstruction (via the Whittaker–Shannon interpolation formula) is a customary measure of the effectiveness of sampling.

The first written use of the terms "alias" and "aliasing" in signal processing appears to be in a 1949 unpublished Bell Laboratories technical memorandum by John Tukey and Richard Hamming. That paper includes an example of frequency aliasing dating back to 1922. The first published use of the term "aliasing" in this context is due to Blackman and Tukey in 1958. In their preface to the Dover reprint of this paper, they point out that the idea of aliasing had been illustrated graphically by Stumpf ten years prior.

The 1949 Bell technical report refers to aliasing as though it is a well-known concept, but does not offer a source for the term. Gwilym Jenkins and Maurice Priestley credit Tukey with introducing it in this context, though an analogous concept of aliasing had been introduced a few years earlier in fractional factorial designs. While Tukey did significant work in factorial experiments and was certainly aware of aliasing in fractional designs, it cannot be determined whether his use of "aliasing" in signal processing was consciously inspired by such designs.

Spatial aliasing, particular of angular frequency, can occur when reproducing a light field or sound field with discrete elements, as in 3D displays or wave field synthesis of sound.

This aliasing is visible in images such as posters with lenticular printing: if they have low angular resolution, then as one moves past them, say from left-to-right, the 2D image does not initially change (so it appears to move left), then as one moves to the next angular image, the image suddenly changes (so it jumps right) – and the frequency and amplitude of this side-to-side movement corresponds to the angular resolution of the image (and, for frequency, the speed of the viewer's lateral movement), which is the angular aliasing of the 4D light field.

The lack of parallax on viewer movement in 2D images and in 3-D film produced by Stereoscopy glasses (in 3D films the effect is called "yawing", as the image appears to rotate on its axis) can similarly be seen as loss of angular resolution, all angular frequencies being aliased to 0 (constant).

The aliasing distortion in the lower frequencies is increasingly obvious with higher fundamental frequencies, and while the bandlimited sawtooth is still clear at 1760 Hz, the aliased sawtooth is degraded and harsh with a buzzing audible at frequencies lower than the fundamental.