The minimum number of samples that must be taken from a sinusoidal signal affected by white Gaussian noise, in order to find its frequency with a predetermined maximum error, is derived. This analysis is of interest in evaluating the performance of velocity-measurement systems based on the Doppler effect. Specifically, in laser Doppler anemometry (LDA) it is usual to receive bursts with a poor signal-to-noise ratio, yet high accuracy is required for the measurement. In recent years special attention has been paid to the problem of monitoring the temporal evolution of turbulent flows. In this kind of situation averaging or filtering the data sequences cannot be allowed: in a rapidly changing environment each one of the measurements should rather be performed within a maximum permissible error and the bursts strongly affected by noise removed. The method for velocity extraction that will be considered here is the spectral analysis through the squared discrete Fourier transform, or periodogram, of the received bursts. This paper has two parts. In the first an approximate expression for the error committed in LDA is derived and discussed. In the second a mathematical formalism for the exact calculation of the error as a function of the signal-to-noise ratio is obtained, and some universal curves for the expected error are provided. The results presented here appear to represent a fundamental limitation on the accuracy of LDA measurements, yet, to our knowledge, they have not been reported in the literature so far.