Examination of time series through randomly broken windows

: In order to determine the Fourier transform of a quasi-periodic time series (linear problem), or the power spectrum of a stationary random time series (quadratic problem), it is desirable that data be recorded without interruption over a long time interval. In practice, this may not be possible. The effect of regular interruption such as the day/night cycle is well known. We here investigate the effect of irregular interruption of data collection (the 'breaking' of the window function) with the simplifying assumption that there is a uniform probability p that each interval of length tau, of the total interval of length T = N tau, yields no data. For the linear case we find that the noise-to-signal ratio will have a (one-sigma) value less than epsilon if N exceeds (1/p)(1-p)(1/sq epsilon). For the quadratic case, the same requirement is met by the less restrictive requirement that N exceed (1/p)(1-p)(1/epsilon). It appears that, if four observatories spaced around the earth were to operate for 25 days, each for six hours a day (N = 100), and if the probability of cloud cover at any site on any day is 20% (p = 0.8), the r.m.s. noise-to-signal ratio is 0.25% for frequencies displaced from a sharp strong signal by 15 micro Hz. The noise-to-signal ratio drops off rapidly if the frequency offset exceeds 15 micro Hz. (Author)