Critical data length for period estimation

We address the following question in this paper: Given that the period of a discrete time periodic signal belongs to a set Ρ = {Ρ1, Ρ2,…, Ρκ}, what is the minimum duration of the signal necessary to identify its period? It will be shown that the following number of samples is both necessary and sufficient: max Pi + Pj — gcd (Pi, Pj), where gcd is the greatest common divisor, and the maximization is over all pairs Pi, Pj e P. Sufficiency is shown via a constructive proof, leading to a new period estimation algorithm.1