Minimal dictionaries for spanning periodic signals

Recently, several high dimensional dictionary representations were proposed for discrete time periodic signals. These dictionaries could span any periodic signal whose period lies in a given range 1 ≤ P ≤ Pmax. Such dictionaries were used in various ways to estimate unknown periods. In this work, we derive some fundamental properties that any such dictionary must satisfy. For example, we derive bounds on the minimum size of such dictionaries, necessary conditions on their composition, and so on. Our results also demonstrate a natural connection between the well-known Euler Totient function (φ-function) from number theory, and periodicity analysis.1