Post History

By choosing $n=1$, Kuechel et al. actually use the special case of the _Rayleigh statistic_ $R^2$, which only looks at the fundamental harmonic. In general, however, the $Z_n^2$ statistic is ideal for searching for **non-sinusoidal periodic signals**, which requires looking at contributions from higher harmonics and hence typically choosing $n>1$ ([Belanger 2017](https://arxiv.org/abs/1712.00734), [Buccheri 1983](https://ui.adsabs.harvard.edu/abs/1983A%26A...128..245B/abstract)). The ability to look beyond the fundamental can be quite handy when looking at, among other things, the distinctly non-sinusoidal x-ray or gamma ray pulses from pulsars.

If you know _a priori_ what your signal should look like, you maybe able to determine how many harmonics to use without much trouble. In this context within x-ray astronomy, $n=2$ may actually maximize the signal-to-noise ratio of an observation, as it provides sensitivity to unknown pulses with a range of different broadnesses. An optimal number of harmonics, however, may be determined by applying the $H$-test of [de Jager et al. 1989](https://ui.adsabs.harvard.edu/abs/1989A%26A...221..180D/abstract). Here, we define the $H$-statistic by
$$H\equiv\max_{1\leq n\leq20}(Z_n^2-4n+4)$$
Realistically, an infinite number of harmonics could be searched, but the authors argue that $n\leq20$ is typically an adequate truncation, at least for these purposes. $H$ itself can also in fact be used as a statistical test in place of $Z_n^2$.