The theoretically minimum length of a signal for fundamental frequency estimation in a noisy environment is discussed. Assuming that the noise is additive white Gaussian, it is known that a Cramér-Rao lower bound (CRLB) is given by the length and other parameters of the signal. In this paper, we define the minimum length as the length whose CRLB is less than or equal to the specific variance for any parameters of the signal. The specific variance is allowable variance of the estimate within an application of fundamental frequency estimation. By reformulating the CRLB with respect to the initial phase of the signal, the algorithms for determining the minimum length are proposed. In addition, we develop the methods of deciding the specific variance for general fundamental frequency estimation and pitch estimation. Simulation results in terms of both the fundamental frequency estimation and the pitch estimation show the validity of our approach.