aliasing

Aliasing is the high frequencies in original signal masquerade as low frequencies after reconstruction (due to undersampling).

The name “aliasing” comes from the fact that the high-frequency signal becomes indistinguishable from low frequency signals (they become aliases).

Example: Image

Images can be decomposed into “frequencies.” And thus is also subject to aliasing.

Spatial Aliasing Example

Here is an example of spatial aliasing on the function :

Temporal Aliasing example

An example is the “wagon-wheel effect”:

The Nyquist-Shannon theorem states that any frequency above the Nyquist frequency (half the sampling rate) can produce an alias.

Definition

Because we can write periodic signals as linear combination of sine waves, it is often convenient to think about aliasing just in term of sinusoids . ¹

Definition

Given a sampling rate , two frequencies and are aliases of each other if for some integer ,

We can see that when sampled at rate , a wave at frequency and a wave at frequency will produce identical samples: ¹

Proof

Let and

Footnotes

1. 2.2. Aliasing — Digital Signals Theory ↩ ↩²