Although measured astronomical data are time domain data, a commonly
applied method works in the frequency domain by analyzing the power
spectrum of the time series. As the observational window function is
convoluted to the true spectrum of the source, artefacts might be produced
in the power spectrum, which make a proper interpretation more difficult
(Papadakis & Lawrence 1995; Priestley 1992). In
most cases, the power spectra are fit by a power law function with an offset
described as , with values of ranging from 0 to 2
and a mean of about 1.5 (Lawrence & Papadakis 1993). The
value *c* is often denoted as the "observational noise floor" which
describes the random process comprising the observational errors whereas the
"red noise" component is the signal of interest. In the case of long AGN
observations, however, a flattening at low frequencies occurs which cannot
be modelled by the -model (McHardy 1988).

The -model is an ad hoc description of the measured periodogram, without any direct physical motivation. However, it is possible to generate time series with a -spectrum using self-organized criticality models simulating the mass flow within an accretion disc of the AGN (Mineshige et al. 1994). Such models produce a stationary time series that exhibits a -power spectrum by limiting the timescales occurring in the simulated accretion process. A -model without limited timescales would be stationary only if the power law slope is smaller than unity (Samorodnitsky & Taqqu 1994). The observed time series is composed by the superposition of single luminosity bursts. The slope of the -spectrum of data simulated in that way is about 1.8, significantly higher than those measured from real data (Lawrence & Papadakis 1993). If the inclination of the accretion disk is brought in as an additional model parameter the slope can be diminished, but not in a way that leads to convincing results (Abramowicz et al. 1995). Another point that contradicts this assumption is that there is no correlation between the spectral slope and the type of the Seyfert galaxy (Green et al. 1993). This correlation should be present since the Seyfert type is believed to be caused by the inclination of the line of sight (Netzer 1990).

The periodogram which is used to estimate the true source spectrum is difficult to interpret in the presence of non-equispaced sampling time series arising from real astronomical data (Deeter & Boynton 1982 and references therein). The estimation of the -spectrum is hampered even in the absence of data gaps. This is due to the finite extent of the observed time series. Therefore, the transfer function (Fourier transform of the sampling function) is a sinc-function which will only recover the true spectrum if this is sufficiently flat (Deeter & Boynton 1982; Deeter 1984). In the case of "red noise" spectra the sidebands of the transfer function will cause a spectal leakage to higher frequencies which will cause the spectra to appear less steep (the spectral slope will be underestimated).

Even periodograms of white noise time series deviate from a perfectly flat distribution of frequencies as the periodogram is a -distibuted random variable with a standard deviation equal to the mean (Leahy et al. 1983). Thus the periodograms fluctuate and their variances are independent of the number of data points in the time series. Due to the logarithmic frequency binning, AGN periodograms will always show this strong fluctuation due to the low number of periodogram points averaged in the lowest frequency bins (see Fig. 1 (click here)).

**Figure 1:** **a)** EXOSAT ME X-ray lightcurve of the quasar 3C 273
(Jan. 1986), **b)** corresponding periodogram. Each dot represents the
spectral power at its frequency, stepped with . The
periodogram is binned logarithmically (squares indicates a single point
within the frequency bin)

Furthermore, additional modulations can be created in white noise periodograms if the time series consists of parts which slightly differ in their means and variances, respectively (Krolik 1992). In the case of the EXOSAT ME X-ray lightcurves this effect is due to the swapping of detectors as each detector has its own statistical characteristics which cannot be totally suppressed (Grandi et al. 1992; Tagliaferri et al. 1996). Figure 1 (click here)a shows a typical X-ray lightcurve which mainly consists of uninterrupted 11 ksec observation blocks before detectors are swapped. If the periodogram frequency corresponds to the observation block length, the calculated sum of Fourier coefficients equals its expected white noise value of due to the constant mean and variance within the entire oscillation cycle. At other, mainly lower, frequencies the Fourier sum yields non-white values due to temporal correlations caused by different means and variances of observation blocks located in the test frequency cycle. These deviations from a flat spectrum will be very strong at frequencies which correspond to twice the observation block length. The arrows in Fig. 1 (click here)b clearly show this minimum feature at and another shortage of power at which corresponds to the long uninterrupted 72 ksec observation block starting at the second half of the EXOSAT observation (Fig. 1 (click here)a).

Consequently a model is required which operates in the time domain and avoids any misleading systematical effects occuring in power spectra.

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