Characterization of Frequency Stability: Uncertainty due to the Finite Number of Measurements

We consider the characterization of the frequency stability of signal generators by means of the Allan variance of their fractional frequency fluctuations. As the number m of measurements is always finite, one can only get an estimate of this variance, which is a random function of m. We calculate the variance of the Allan variance for commonly encountered spectral densities of frequency fluctuations and deduce the relative uncertainty in the evaluation of frequency stability, as a function of m. The theory is experimentally checked for white noise of phase and flicker noise of frequency. The main hypothesis of the calculations, i.e., the Gaussian character of frequency fluctuations is verified. The m dependence of the uncertainty (one sigma) in the evaluation of frequency stability agrees with the predicted value.