## 3. The fourier transformIn this section we discuss the effects of data sampling on the Fourier transform. The Fourier transform of a continuous function is defined by and its inverse by When the function consists of The inverse discrete transform is given by In these expressions with
so that the time steps are given by
. The coefficients are
related to the frequencies so that the
resolution in frequency is given by . The
highest frequency, corresponding to the Nyquist frequency, is given by
. Note that corresponds
in our case to the total number of photons
detected during time The relation between the continuous and the discrete Fourier
transform for the datasets considered in this paper can be derived as
follows. Each dataset consists of The convolution of and
corresponds to averaging the actual photon
counts over a bin of width around time The discrete sampling of the data amounts to multiplying the function with a window function and with a sampling function . The window function is given by and the sampling function by The sampling function indicates that function is discretely sampled at times while the window function accounts for the finite duration of the time series. Let indicate a measured time series. From the discussion above it follows that . Let denote the Fourier transform and let , and . Then the Fourier transform of the measured time series is given by with From Eq. (9) we find that The relation of the continuous Fourier transform and the discrete transform is established by taking so that The components of the power spectrum are then given by By studying the power spectrum of the observed counts we obtain information about the function . Because we have that with and For the observations described in this paper , , and . A sample of 122 data points results in a power spectrum at 62 discrete frequencies (one at zero frequency). The highest frequency, the Nyquist frequency, corresponds to or a period of . Due to the discrete sampling any period in the signal shorter than 43 seconds will be aliased. In general the process of measuring the data points will automatically result in a suppression of high frequencies so that aliasing is a not a too serious problem. However in our case is only 0.128 seconds so that aliasing will not be suppressed due to the finite time a measurement takes. This can also be seen in Eq. (15). At the Nyquist frequency so that over the whole frequency range considered and hence no suppression of high frequencies occurs. So care has to be exercised for the possibility of aliasing. A second effect which occurs is caused by the data windowing.
Function is a box car function with length
A third effect which occurs is due to the use of the discrete Fourier transform. When looking for a periodic signature with frequency , the associated power is only recovered when corresponds exactly with one of the frequencies at which the power spectrum is evaluated. When is exactly in-between two frequency bins the power is distributed over the neighbouring frequency bins and even some bins further away. So the spread of power over adjacent bins is caused by two effects: 1) the finite length of the time series which results in windowing; 2) the use of the discrete Fourier transform. The effect of windowing can be suppressed by using a window function for the time series which differs from the box car (e.g. Welch, Hanning, Parzen etc.). However, we decided not to use any of the above windowing functions, as such a function broadens the secular variations due to a slow increase/decrease in the counts and therefore affects the determination of the variations we seek to analyse. The time series can be characterized as follows. The average number of photon counts per 0.128 seconds amounts to a few hundred counts, 300 - 400 in most datasets although, as we will see later, some parts of a dataset can have counts in the 1000-1500 range while other parts are in the 50-100 range. In each dataset, there are secular variations due to a slow decrease or a slow increase of the counts. This occurs on time scales in the range . Superimposed on these slow variations are faster variations. Simply looking at the time series suggests already that some variations are (quasi-)periodic. The amplitudes of the variations are larger than expected from pure Poisson statistics (e.g. or ). Because of the high average counting level and the presence of secular variations it can be anticipated that there will be significant power at low frequencies, say . This power can be reduced by subtracting some `average' from the observed counts, e.g., a first-order polynomial fit. However, there is little to be gained by this procedure. A description of the statistical properties of the power spectrum can be found in Jenkins and Watts (1968), Leahy et al. (1983), and is comprehensively summarized in van der Klis (1989). The normalization of the power spectrum (Eq. (5)) is chosen in such a way that if the noise in the data is (only) Poissonian, then the distribution is given by the distribution with two degrees of freedom (dof). The probability that exceeds a threshold power level is with For two dof the standard deviation of the noise powers is equal to
their mean value . This implies that in the
power spectrum the magnitude of the noise component is not well
defined. There exist basically two methods to decrease the noise in
the power spectrum. One method is to rebin the power spectrum by
averaging W consecutive frequency bins at the expense of a reduced
frequency resolution. The other method, which can be used in
combination with the previous, is to divide the data into M segments
of equal length. For each of the data segments the power spectrum is
determined and the resulting power spectra are then averaged. The
resulting power distribution of the noise corresponds then to a
-distribution with dof
which is scaled with a factor . In this case we
have that . The mean of the distribution is
still equal to 2 but the standard deviation has been reduced to
. We note that the noise in the power spectrum
can, effectively, only be reduced at the expense of frequency
resolution. Increasing the observing time Suppose that we have a power spectrum at N frequencies and want to establish which powers have a low probability of being caused by noise. The power at each of the frequencies can be considered as an independent trial. Define as the probability that a power exceeds detection level and is not caused by noise. For N independent powers this probability is so that the chance to exceed and to be caused by noise is for . From this it follows that the detection level is given by In this paper we use a confidence level of 99.9% () to determine . For , the detection level is given by . For , Eq. (17) results in an implicit relation for . © European Southern Observatory (ESO) 1997 Online publication: April 8, 1998 |