Signals and their models

Signals can be classified broadly into deterministic and stochastic signals. A deterministic signal, e.g. a periodic signal, can be predicted for arbitrary spaces of time. For a stochastic signal, no such prediction can be made beyond a certain time interval, called the correlation length l_corr. For any finite time series the classification into these two categories is ambiguous so that methods suitable for both stochastic and periodic signals could be applied to any time series with some success (e.g. quasiperiodic oscillations, Sect. 12.2.6).

Usually processes in the source of the signal (e.g. the nucleus of an active galaxy) and/or observational errors introduce a random component into the series, called noise. The analysis of such series usually aims at removing the noise and fitting a model to the remaining component of the series. Suitable models can be obtained by shifting a known series by some time lag, l, or by repeating fragments of it with some frequency, $\nu$. Accordingly, we are speaking of an analysis in the time and frequency domain. In these domains the correlation length l_corr and oscillation frequency $\nu_o$, respectively, have particularly simple meanings. It is transparent that the stochastic signals are analysed more comfortably in the time domain and periodic signals in the frequency domain.

Models usually depend on several parameters. Fitting of the model to the signal means choosing the best set of these parameters. Customarily, the observed series, X^(o), is split into the modeled series X^(m) and the residuals of the observations with respect to the model, $X^{(r)} \equiv X^{(o)} - X^{(m)}$.