Aliasing effect

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As aliasing [ eɪliəs ] (also aliasing effects or short- aliasing ) are in the field of signal analysis referred to errors that occur when the signal to be sampled frequency occur proportions which are higher than half the sampling frequency ( Nyquist frequency ).

Aliasing can on the one hand by the disregard of the sampling theorem (insufficient sampling frequency) in the digital sampling of signals occur on the other hand, when the signal to be sampled by a noise signal is superimposed, comprising for its part frequencies which are higher than the Nyquist frequency.

In image processing and computer graphics , aliasing effects occur when images are scanned and lead to patterns that are not contained in the original image. In audio technology , aliasing effects are expressed as interference .

To prevent aliasing, low-pass filters can be used to filter out unwanted frequency components.

Signal processing

Illustration of the aliasing effect. A continuous output signal (black line) is discretized with an unsuitable sampling frequency that is less than required by the sampling theorem . A falsified signal with a period that is clearly too long (red line) is created from the measured values ​​(circles) obtained through interpolation .

In signal processing, aliasing effects occur when digitizing analog signals.

To ensure that the original signal can be correctly restored, the signal to be sampled may only contain frequency components that are smaller than the Nyquist frequency. However, if there are frequency components that are higher than the Nyquist frequency, these are interpreted as lower frequencies. The higher frequencies appear, so to speak, as a different (lower) one (see graphic), hence the name alias .

Interfering frequency components that can lead to aliasing occur with undersampling (i.e. the sampling theorem was not adhered to). But even if the sampling theorem is adhered to, aliasing can also occur if the signal to be sampled is overlaid by a noise signal that contains frequency components that are higher than the Nyquist frequency.

To avoid such aliasing effects, the input signal is filtered by a low pass ( anti-aliasing filter). The filter effect of this cutting off of the high frequencies can also be described by the terms treble lock, treble filter, high cut and treble cut. This filtering has to be done before digitization - a subsequent correction of alias effects is no longer possible.

Image capture

Alias ​​creation

In image processing and computer graphics, aliasing effects occur when images are scanned, one example being the appearance of moiré patterns .

The staircase effect that occurs when rasterizing geometric figures is often referred to as aliasing, although it is not "real" aliasing in the sense of signal analysis.

With cameras from 3 megapixels, aliasing effects are usually reliably suppressed by cleverly designed optics. The optical resolution here intentionally stays below the pixel resolution. The optics are a little blurred and thus serve as a low-pass filter.

Demonstration of the aliasing effect

The Fresnel zone plate in the illustration is intended to serve as an example of an original image that has signal components above the Nyquist frequency in its so-called spatial frequency . If it is scanned with 30 × 30 points, only the structure in the middle can be reproduced. In the edge areas, the spatial frequency of the object exceeds the Nyquist frequency, so that the object cannot be reproduced here. Instead, alias objects are created in the form of circles in the edge areas.

For a similar demonstration (but in one dimension) see frequency broom .

Moiré effect

Alias ​​signals also occur when scanning original images with changing spatial frequencies ; one speaks of a moiré effect , for example on items of clothing such as wool sweaters or jackets with thin stripes, or on images of tiled roofs. Moiré effects can often also be seen in the television picture if the corresponding textures are shown. The cause is a superposition of the spectra of the sampling function, the output signals of which are periodic with f sample .

Temporal aliasing

Aliasing effects can occur in films, which can be attributed to the composition of the film from individual images. A well-known example is the apparent reverse motion of the wagon wheels in Westerns. It occurs as soon as the wheel continues to turn from picture to picture by more than half the angle between two spokes.

If you observe the acceleration of a car in the film, the wheel initially turns in the right direction. From a certain speed on, however, the wheel seems to turn backwards, only to seemingly slow down again as the speed of the carriage increases. Then it seems to stop, only to start moving again in the right direction at an unnaturally slow speed. The apparent running forwards and backwards is repeated with further acceleration.

From a signal-theoretical point of view, the recording of the individual images represents a scanning process. The scanning frequency corresponds to the image refresh rate . The signal frequency corresponds to the frequency with which the spokes pass through an angle that corresponds to the distance between the spokes. With a refresh rate of 24 frames per second, the Nyquist criterion is violated from a rotational speed of the wheel of 12 spoke spacings per second, so that aliasing occurs.

  • If the wheel turns half a spoke further between two successive images, it is no longer possible to distinguish whether it is turning forwards or backwards (signal frequency = Nyquist frequency). The aliasing effect begins at this speed.
  • If the signal frequency is between the Nyquist frequency and the sampling frequency, the wagon wheel appears to be running backwards.
  • If the wheel moves by exactly one spoke or an integer multiple per image, it seems to stand still (signal frequency = n × sampling frequency).

Example sounds

Linearly rising tone (16 kHz sampling)
The first sound sample sounds a tone whose frequency increases linearly from approx. 100 Hz to over 8000 Hz (the original sampling frequency of 16 kHz was increased to 42 kHz during the transformation into the Ogg-Vorbis format).
Linearly rising tone (8 kHz sampling)
The second example reproduces almost the same signal, this time sampled at 8000 Hz. Undersampling incorrectly selects tones above 4000 Hz, with the result that a pitch is recorded that falls instead of rising.

See also