The present lecture is the final lecture on the analysis of the power spectrum. The coming lectures will deal with correlation analysis of non sinusoidal signals. We will of course continue using the discrete form of the power spectrum described via calculation of α and β. Throughout the course we have discussed the use of statistical weights and we have especially considered weights that are estimated as 1/variance. In this lecture I will discuss how to obtain weights even if we do not have access to individual scatter estimates. Earlier in the course I showed this example of how statistical weights can be used to reduce the noise level in the amplitude/power spectrum significantly.
In the present time series the scatter per data point is known and this will directly allow allocation of statistical weights to each data point. The efficiency of the statistical weighting is clearly seen in this and the next figure... noise is significantly reduced after applying the weights. An example of a situation where scatter may be calculated for individual data points is a time series of radial velocity measurements where the velocity is obtained by cross correlating the observed spectrum with a reference. The cross correlation will not only give the wavelength shift (radial velocity) between the spectrum and the reference it also allows an estimate of the error/scatter on the individual data point.
As an example of radial velocity measurements I show here the time series for the radial velocity of the stars α Centauri B. Each point in time we have a radial velocity measurements and a scatter values for that given point. In blue I show the non-weighted spectrum while the red contain the weighted spectrum (1/VAR). Shown here again for a zomm of the spectrum.. In blue I show the non-weighted spectrum while the red contain the weighted spectrum (1/VAR).
The statistical weights are only valid if they represent the noise at the frequency of interest. The weights that are calculated for individual data points correspond normally to the scatter at high frequencies and we should therefore verify that the signal is indeed represented by the noise level at high frequencies. In case of α Centauri B this seems to be the case. Let me then turn to an example of a star where we have no direct information on the scatter of the individual data points and we need to calculate those. An example of this is the time series on the sdb star PG 1325+101. In this figure I show he raw data. I then run CLEAN and remove the 100 major peaks. The signal contained by the 100 major peaks is the following. and the residual noise with no signal is plotted in this figure.
If I look at the signal zooming in at a narrow point in time one can clearly see that this star is a multi periodic pulsating star. Zooming even more also demonstrated that there is a main oscillation and a series of lowamplitude oscillations that are seen as a modulation on top of the primary oscillation. which is also seen in this figure. In order to improve the SNR of low-amplitude pulsators we need to use statistical weights. On this figure you see the same area as shown above but this time including the scatter on the signal.
The idea is now to use the residual signal (where the major signal is CELANed away) to estimate the error signal throughout the series and use this scatter for the statistical weighting. We will now calculate the weights via a box car filter that is moved in time through the time series. The scatter is estimated as N is an even number (the number of data points used in the box car filter). We the calculate: μ = i i+ N / 2 1 N + 1 j= i N / 2 data( t j ) σ i+ N / 2 2 1 i = ( data( t j N + 1 j= i N / 2 ) μ ) i 2 The best value for N depends on sampling and how fast the data quality is changing throughout the series. A typical value for N is 50-100. Based on the scatter (σ) we then calculate the weights. That there is a correlation between weights and scatter via this type of calculation can be clearly seen in the present figure.
I then show how those new weights will improve the signal-to-noise ration in the time series. Top is after applying the weights. A close inspection show how efficient this technique is.
Of course using weights that are very variable will have other effects on the time series. One way the data is affected is via a change in the window function. As shown before a uninterrupted time series will show a window that is represented by the sincfunction. Any gabs or non-uniform weights will give rise to other peaks... and this can be seen if we calculate the window function. First it is seen in case of no weights.... and here with the weights. The sampling is clearly more non-uniform that in case of no weighting.
.. and here shown in amplitude.. and zoomed in.
However the degraded window function will not necessary be a problem. We will locate the oscillation modes by use of CLEAN and this will in case of the improved signal-to-noise make thing much better. In fact after using the weights we can extract 24 different oscillation modes in the time series... and there seem to be some more modes that contain the power seen in the spectrum after CLEAning.
The final thing I will discuss is an example of how the bad pass filter can be used to isolate oscillation properties of single oscillation modes. In the present example I will look at solar oscillation (they are damped and re-excited). Using band-pass we can isolate power from a single mode. In the following I will consider 5 days of GOLF (SoHO) data.
This time series contain clearly the oscillations.. but what is seen is the combined signal of all modes. If we calculate the power spectrum of this signal.. and isolate the power for one frequency and then from the α and β values calculate time series corresponding to this signal we find.... a signal where it is clear that the mode is not coherent. The lifetime is shorten than one day for this mode.
If we take another mode (with lower frequency) we find in the same way.. a signal with a much longer lifetime... clearly seen when the amplitude for those two modes a shown. In this way we can map the variation in power for different modes as a function of time.