False Alarms and Aliasing

One major advantage of the periodogram method is that it generates a value called the power. The power can be converted to the probability that the period found is not true, the false alarm probability (FAP). Scargle (1982) showed that the probability distribution returned by this method is exponential, and thus the probability of a false alarm is defined as , where z is the power at a given frequency and N is the number of frequencies sampled. Unfortunately, the formulation for calculating the FAP is only valid in the case where the data are not clumped in their sampling. The data reported here are clumpy since up to four observations are taken per night, but none during the day.

To overcome this problem, Eaton et al. (1995) have suggested calculating the power of the most likely period in a set of data created by randomizing the night number associated with observations of a given star. By calculating the maximum power on a large number of randomized data, the conversion from power to FAP can be individually determined for each set of observations in a statistical manner. However, it is still possible to have several possible periods with less than 1% probability of being true. This is due to aliasing in the dataset. Aliasing is caused by periodicity in the timing observations being coupled with the periodic nature of the source. The sampling theorem shows that if the frequency of a continuous function (in this case the amplitude) is less than half the sampling frequency (the Nyquist frequency) the original signal can be recovered. Given a sampling time interval , one can identify individual observations at time t as the observations. If one were to observe an infinite number of times, one would obtain:

where: is the Nyquist frequency. This has an unfortunate side effect, Two waves, and give the same samples if they differ by 1/. This implies than any given sample can be reconstructed into an infinite number of original signals all separated by n/ in frequency space. Any power in the spectral density which exists outside the search range is moved into the search range. Two periods p1 and p2 will give the same samples, and thus be indistinguishable, if they differ by the inverse of the sampling frequency. The problem is complicated by the two sampling frequencies in the data, the diurnal cycle and the three hours between samples each night. The aliasing causes several peaks in the periodogram analysis to appear at similar power.