It is the automorphism group of the graph consisting of a path infinite to both sides.
An example of infinite dihedral symmetry is in aliasing of real-valued signals.
Thus, the detected value of frequency f is periodic, which gives the translation element r = fs.
Noting the trigonometric identity: we can write all the alias frequencies as positive values:
For example, with f = 0.6fs and N = −1, f + Nfs = −0.4fs reflects to 0.4fs, resulting in the two left-most black dots in the figure.
In one dimension, the
infinite dihedral group
is seen in the symmetry of an
apeirogon
alternating two edge lengths, containing reflection points at the center of each edge.
When periodically sampling a sinusoidal function at rate
f
s
, the abscissa above represents its frequency, and the ordinate represents another sinusoid that could produce the same set of samples. An infinite number of abscissas have the same ordinate (an equivalence class with the
fundamental domain
[0,
f
s
/2]
), and they exhibit dihedral symmetry. The many-to-one phenomenon is known as
aliasing
.
