Download The present lecture is the final lecture on the analysis of the power

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:
1 i+ N / 2
μi =
∑ data(t j )
N + 1 j =i − N / 2
1 i+ N / 2
σ =
(data(t j ) − μi ) 2
∑
N + 1 j =i− N / 2
2
i
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.