Motivation

The calculation of the way in which experimental errors affect the amplitude, frequency, and phase determination of a time series signal is a nontrivial problem. In the general case, the data may not be evenly sampled in time and the noise superimposed on the (assumed) sinusoidal signal may be correlated in time as well as being non-Gaussian. In addition, there may be more than one such sinusoidal signal present, which, depending upon the aliasing, will also add uncertainty to the parameter determination To treat adequately and systematically the realistic cases which are encountered in astronomy, for example, it is probably necessary to perform numerical simulations which use the same time sampling as the data set and which incorporate a realistic model for the ``noise''. While this is certainly a project worth doing, in this present paper we wish to compute analytically the best case scenario, i.e., a lower limit on the size of the errors one can expect. Our goal is to collect in one place all the simple formulae which will allow the reader to make these estimates. For more detailed and statistically rigorous treatments of aspects of this problem, we refer to the papers by Schwarzenberg-Czerny (1991, 1999) and Koen (1999, in press). In the following section, we derive the errors in the amplitude, phase, and frequency for the case when these parameters are determined from a single data set. Another commonly encountered case is one in which the frequency is well-constrained by previous observations and can be considered known, but where the amplitude and phase are yet to be determined. In this case the derivation is somewhat simpler, and it is treated in the appendix of Breger et al. (1999, in press). The formulae derived for the errors in amplitude and phase are identical in both cases.