# The Data Analysis BriefBook

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From The Data Analysis BriefBook http://physics.web.cern.ch/Physics/DataAnalysis/BriefBook/

## Aliasing

Used in the context of processing digitized signals (e.g. audio) and images (e.g. video), aliasing describes the effect of undersampling during digitization which can generate a false (apparent) low frequency for signals, or staircase steps along edges (jaggies) in images; Sampling Theorem. Aliasing can be avoided by an antialiasing (analogue) low-pass filter, before sampling. The term antialiasing is also in use for a posteriori signal smoothing intended to remove the effect.

## Sampling Theorem

The greyvalues of digitized one- or two-dimensional signals are typically generated by an analogue-to-digital converter (ADC), by sampling a continuous signal at fixed intervals (e.g. in time), and quantizing (digitizing) the samples. The sampling (or point sampling) theorem states that a band-limited analogue signal xa(t), i.e. a signal in a finite frequency band (e.g. between 0 and BHz), can be completely reconstructed from its samples x(n) = x(nT), if the sampling frequency is greater than 2B (the Nyquist rate); expressed in the time domain, this means that the sampling interval T is at most 1/2B seconds. Undersampling can produce serious errors (aliasing ) by introducing artefacts of low frequencies, both in one-dimensional signals and in digital images:

For more details and further reading , e.g. [Kunt80] or [Rabiner75].

Rudolf K. Bock, 7 April 1998

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