## Introduction

Aliasing is a potential problem whenever an analog signal is point sampled to convert it into a
digital signal. It can occur in audio sampling, for example, in converting music to digital form to be
stored on a CD-ROM or other digital device. Alising happens whenever an analog signal is not
sampled at a high enough frequency. In audio, aliasing manifests itself in the form of spurious
low frequencies. An example is shown below of two sin waves.

In the top sin wave, the sampling is fast enough that a reconstructed signal (the small circles) would have the same frequency as the original sin wave. In the bottom wave, with a higher frequency but the same point sampling rate, a reconstructed signal (the small circles) would appear to be a sin wave of a lower frequency, i.e., an *aliased* signal.

From **Point Sampling Theory** it turns out that to accurately reconstruct a signal, the signal must be sampled at a rate greater than or equal to two times the highest frequency contained in the signal. This is called the Nyquist Theorem and the highest frequency that can be accurately represented with a given sampling rate is called the Nyquist limit. For example, to produce a music CD-ROM the analog signal is sampled at a maximum rate of 44 Khz, therefore the highest possible audio frequency is 22khz. Any audio frequencies greater than 22Khz must be removed from the input signal or they will be aliased, i.e., appear as low frequency sounds.

Aliasing occurs in *computer graphics*, since we are point sampling an analog signal. The mathematical model of an image is a continuous analog signal which is sampled at discrete points (the pixel positions). When the sampling rate is less than Nyquist Limit then there are aliasing artifacts that are called "jaggies" in computer graphics. In general, aliasing is when high frequencies appear as low frequencies which produce regular patterns easy to see.
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