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Combined Sliding-Window Fourier (SWF)
and Wigner-Ville (WV) transform: an efficient
algorithm for astronomical analysis
K.G.Kislyakova1, M.L.Khodachenko2,
A.G.Kisljakov1, H.Lammer, S.Yu.Lupov1,
V.V.Zaytsev1,3
1 Lobachevsky
State University of Nizhni Novrorod, Russia
2 Space Research Institute, Austrian Academy of Sciences, Austria
3Institute of Applied Physics, Russian Academy of Sciences,
Russia
Europlanet JRA3 Workshop, UCL, April 22-24, 2010
The algorithm for data analysis
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Wigner E.P., Phys. Rev., 40, 749, 1932; Ville, J.,
Cables Transm., 2A 61, 1948;
Cohen, L., Proc. IEEE, 77, 72, 1989;
Shkelev, E.I., Kislyakov, A.G., Lupov, S, Yu., Radiophys. & Quant. Electronics, 45, 433, 2002.
Wigner-Ville
(VW)Fourier
transform:
Direct
and inverse
transform:
F ( ) 
1
2

1i 2f


it




*
F
(
x
)

F
(

)
e
F
(
t
)
e
dt
,
and
P ( f , t )   z  t   z  t  e2  d , d


2
2






it

Sliding Window Fourier transform:
where z(t) = s(t) + jsG(t) – analytic signal;
s(t) and sG(t)
– real signal sample and its Gilbert conjugate.
i
where
- window
F
(t , conjugate:
)   F ( )sW
d(t ,)) 
t ) (t ))
(t )( f t(t))ecos(
sG (t ) W
 f(
(t )
sin(
Gilbert

Function
functionP(f,t) gives the distribution of the signal energy over the frequency
and time.
Method description
Algorithm outcome:
-
Dynamical spectra of Long Periodic (LP) signals which modulate physical data
records (radiations, parameters of solar wind: magnetic field; density, speed,
etc.)
Analysis of these LP spectra allows to judge about dynamics of the physical
system and perform its diagnostics (including the external & internal
influences)
Advantages of the SWF-WV algorithm:
- High sensitivity: detection of the modulating LF signals appeared below the noise level
- High spectral and temporal resolution
- Ability to detect complex multi-signal modulations (multi-planetary systems and/or complex
(multi-mode) stellar seismology processes)
Technical requirements:
- The length of the analyzed time series should include at least several periods of the
LP modulating component;
- Sampling cadence of analyzed time series  104 equidistant points per realization;
- Input data format: ASCII
Drawbacks
WV method allows to analyze temporal and frequency distribution of
the energy of the low frequency pulsations modulating the analyzed
signal. WV method appears the most efficient for the analysis of cases
with the quasi-harmonic and impulsive modulating signals with a
constant or changing frequency.
-
-
BUT some drawbacks are to be mentioned:
Non-local character of the WV method; in the case of analysis of the longterm signals, comparatively weak components of the signal can be
suppressed in certain regions of the dynamic spectrum by an intense (or
noisy) signal part;
Nonlinearity of WV transform results in appearance of artificial
intermodulation spectral components (artifacts) at the combination
frequencies;
Different types of the signal and spectral processing and filtration are
realized in our combined SWF-WV algorithm in order to eliminate possible
artifacts and provide high spectral and temporal resolution.
Drawbacks: illustration
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WV spectrum for the test digital signal consisting of
three frequency-modulated components with
constant amplitudes si (t ), (i  1,2,3) with the
discretization frequency f D .
The signals 1 and 2 have the linear frequency
modulation:
f1 (t )  f max  t , f 2 (t )  f min  t , where f min  0.05 f D ,
f max  0.2 f D ,   0.0001 .
The third signal have additional phase modulation:
d (t ) where  (t )  0.04t tan[ k (t  t )] ,
f 3 (t )  f 2 (t ) 
s
dt
the temporal shift of phase is t s  ( f max  f min ) /( 6 )
and k=0.01. Dimensionless time is measured in the
count numbers N.
1,2,3 - the spectral lines of the test signal. They
represent the signal power distribution over the
time and frequency. Thus, the intensity of these
lines changes smoothly with N.
4,5,6 - Intermodulation components (artifacts) at
combination frequencies:
f 2,3 (t )  ( f 2 (t )  f 3 (t )) / 2
f1, 2 (t )  ( f1 (t )  f 2 (t )) / 2
f1,3 (t )  ( f1 (t )  f 3 (t )) / 2
Applications of the combined SWF-WV
algorithm: Sun observations
Above: realization and spectrum of
sun burst on July 13, 1992
observed at 11.7 GHz in
Metsähovi, Finland.
Below: simulation of spectrum with
the model of two inductively
coupled loops.
Studying of
low-frequency modulations of solar
microwave emission can be used
for coronal magnetic loops and
loops current diagnostics.
Post-flare modulations of
microwave radiation
AR9608 on Sept.15,2001, observed by
TRACE
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Detected periods:
12 min (also TRACE)
7 min (small loops)
3.3 min
4.4 min
55 min (seismology)
According to our considerations,
these ”modulation pairs” could be
an indication of the kink-oscillating
loops with the periods
corresponding to the main
frequencies in the pairs.
Parametric resonance in the solar corona
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Pumping frequency corresponds to the 5-minute photospheric oscillations.
Modulation of radio emission with periods
n


, n  1,2,3...
n
3.3, 5, 10 min
2
Mechanism can explain the origin of quasi-stationary X-ray loops with
temperatures 3-6 MK
The limb flare of November 2, 1992 observed
by Nobeyama Radioheliograph (17 GHz)
Linear frequency modulation can be
explained in terms of increasing and
decreasing current in the currentcarrying magnetic loop (before,
during and after flare).

I0

2 3/ 2
2cr0 
1
 8l 4 
nmi  ln
 
S 7 

where l, S and r0 are length, area
and radius of a loop respectively.
A magnetic loop is considered as an
equivalent electric circuit with the
eigen oscillation frequency 
Extrasolar planet search: analysis of the
CoRoT light curves
Application of the algorithm to the analysis of the light curves can be very useful both
in extrasolar planet search and in investigations of already discovered systems. Longperiodic modulations can help to definite parameters of the system and planets
conditions.
Period of the main frequency
(2.3h) is close to the whole
duration of the transit impulse.
Higher frequencies (except of its
second and third harmonics)
emerge due to star activity.
Studying of the CoRoT light
curves gives a possibility of
seismological stars activity
investigation using planets transit
observations. It should be a
powerful means to judge about
planet-star interaction and star
magnetic activity.
Summary of studied physical processes
using the combined SWF-WV algorithm
• Sun observations: detection of parametric resonance in the
solar corona, studying of plasma heating in the magnetic
loops, investigation of low-frequency modulations of
inductively coupled close loops, studying of loops eigen
oscillations via radio emission modulation (Nobeyama), etc.
• Investigation of flaring stars activity (AD Leo)
• SKR (Saturnian kilometric radiation) – diagnostics of solar
wind and planetary systems
• Solar wind magnetic field diagnostics (Ulysses 1995-2006)
• Extrasolar planet research: CoRoT light curves analysis.
Investigation of planet-star interaction and hot-Jupiters
systems parameters
Analysis to be continued…
Basic papers:
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Parametric Resonance in the Solar Corona. Cosmic Research, 2008, Vol. 46, No. 4,
pp. 301–308. © Pleiades Publishing, Ltd., 2008. Original Russian Text © V.V. Zaitsev,
A.G. Kislyakov, K.G. Kislyakova, 2008, published in Kosmicheskie Issledovaniya,
2008, Vol. 46, No. 4, pp. 310–317.
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Plasma Heating during the Parametric Excitation of Acoustic Waves in Coronal
Magnetic Loops. Astronomy Reports, 2010, Vol. 54, No. 4, pp. 367–373. Pleiades
Publishing, Ltd., 2010. Original Russian Text V.V. Zaitsev, K.G. Kislyakova, 2010,
published in Astronomicheskiy Zhurnal, 2010, Vol. 87, No. 4, pp. 410–416.
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Detection of large-scale kink oscillations of coronal loops manifested in modulations
of solar microwave emission. M.L. Khodachenko1, K.G. Kislyakova , T.V.
Zaqarashvili1, A.G. Kislyakov2, M. Panchenko, V.V. Zaitsev and H.O. Rucker.
Submitted to A&A, March 2010
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Methods of intermodulation effects decrease in Wigner-Ville transform. Shkelev, E.I.,
Kislyakov, A.G., Lupov, S, Yu., Radiophys. & Quant. Electronics, 45, 433, 2002.