Infinite Dihedral Group - Aliasing

Aliasing

A concrete example of infinite dihedral symmetry is in aliasing of real-valued signals; this is realized as follows. If sampling a signal (signal processing term for a function) at frequency f_s, then the functions sin(ft) and sin((f + f_s)t) cannot be distinguished (and likewise for cos), which gives the translation (r) element – translation by f_s (the detected frequency is periodic). Further, for a real signal, cos(−ft) = cos(ft) and sin(−ft) = −sin(ft), so every negative frequency has a corresponding positive frequency (this is not true for complex signals), and gives the reflection (f) element, namely f ↦ −f. Together these give further reflection symmetries, at 0.5f_s, f_s, 1.5f_s, etc.; this phenomenon is called folding, as the graph of the detected signal "folds back" on itself, as depicted in the diagram at right.

Formally, the quotient under aliasing is the orbifold, with a Z/2 action at the endpoints (the orbifold points), corresponding to reflection.