Sinusoidal and quasiperiodic waves

5.2 Sinusoidal and quasiperiodic waves

Some candidate sources radiate periodic or quasiperiodic waves. From an observational viewpoint a signal is effectively a sinusoid if the change in wave frequency over the duration of the observation T is significantly less than a resolution bandwidth 1/T. Since in a search experiment the signal phase is not known, spectral analysis [67 , 101

, 84

] is appropriate. In the absence of a signal, the real and imaginary parts of the time series’ Fourier transform are Gaussian and uncorrelated, so the Fourier power is exponentially distributed [83 ]. In the presence of a signal, the Fourier power is “Rice-squared” distributed [91

]. Formal tests for statistical significance [52 , 101 , 4

, 18

, 7

, 5

, 84 ] involve comparing the power in a candidate line with an estimate of the level of the local noise-spectrum continuum. Since the frequency of the signal is also not known a priori, a range of frequencies must be searched. The spectral power is approximately independent between Fourier bins, so the joint probability density function of the power in $n$ Fourier bins is the product of the individual bin pdfs. This can be used to set statistical confidence limits for the sensitivity of a search experiment over multiple candidate signal frequencies [18

A signal is effectively a linear chirp if the change in frequency over the observation interval is $>$ 1/T but the curvature of the signal’s trajectory in a frequency-time plot is negligible over the observing interval. In this case, signal power is smeared in (frequency, time) and simple spectral analysis is inappropriate. In a chirp-wave analysis the Doppler data are first passed through a software preprocessor which tunes the signal to compensate for the linear chirping. With the correct tuning function, the chirp is converted to a sinusoid in the output. Spectral analysis is then used to search for statistically significant lines. The tuning function $exp (iπ βt2)$ is complex. The parameter $β$ , an estimate of $df∕dt$ , is unknown and must be varied. (In an idealized situation, this procedure resolves the signal into three lines, with frequency separation depending on $β$ and T₂ [5 ]. The observability of all three lines is unlikely in real observations, however.) The situation is different from the sinusoidal case in that an arrow of time has been introduced by the software dechirping; the positive and negative frequency components of the dechirped spectrum contain different information. In principle, an ensemble of chirping signals, each too weak to be detected individually, could be identified by noting differences in the statistics of the positive and negative frequency components of the dechirped spectrum.

Waveforms which are more complicated than linear chirps arise, e.g., from binary systems near coalescence. To do proper matched filtering [58 ] the waveform and the source location on the sky are needed. If one assumes the time evolution of the phase, the time series can be resampled at unequal times [96 , 5 ] so that (in terms of the resampled phase variable) the signal is periodic. This suboptimum technique can be used in pilot analyses to pre-qualify candidates for exact matched filtering. Nonsinusoidal periodic waves, generated, e.g., by non-circular binary systems, can have rich Fourier content [120 ]. Searches for these waveforms have included folding the data with assumed periods [18 ].