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, Russia2 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

Direct and inverse Fourier transform:

, and

Sliding Window Fourier transform:

, where - window

function

- 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.

dtetFF ti

)(

2

1)(

deFxF ti)(2

1)(

detWFtF i)()(),( )( tW

Wigner-Ville (VW) transform:

,

where z(t) = s(t) + jsG(t) – analytic signal; s(t) and sG(t) – real signal sample and its Gilbert conjugate.

Gilbert conjugate:

Function P(f,t) gives the distribution of the signal energy over the frequency and time.

detztztfP fi2*

22),(

))(sin()()())(cos()()( ttftsttfts G

Method descriptionAlgorithm 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)

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

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)

DrawbacksWV 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 long-term 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• WV spectrum for the test digital signal consisting of

three frequency-modulated components with • constant amplitudes with the

discretization frequency .• The signals 1 and 2 have the linear frequency

modulation:• , , where ,

• • , .• The third signal have additional phase modulation:• where ,

• the temporal shift of phase is • 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:

Df)3,2,1(),( itsi

tftf max1 )( tftf min2 )( Dff 05.0min

Dff 2.0max 0001.0

dt

tdtftf

)()()( 23

)](tan[04.0)( sttktt

)6/()( minmax fft s

2/))()(()( 212,1 tftftf 2/))()(()( 323,2 tftftf

2/))()(()( 313,1 tftftf

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

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.

AR9608 on Sept.15,2001, observed by TRACE

Parametric resonance in the solar corona

• Pumping frequency corresponds to the 5-minute photospheric oscillations.• Modulation of radio emission with periods

3.3, 5, 10 min• Mechanism can explain the origin of quasi-stationary X-ray loops with

temperatures 3-6 MK

...3,2,1,2

nn

n

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 current-carrying magnetic loop (before, during and after flare).

where l, S and are length, area and radius of a loop respectively.

A magnetic loop is considered as an equivalent electric circuit with the eigen oscillation frequency

7

48ln

1

2 2/320

0

S

lnm

cr

I

i

0r

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. Long-periodic modulations can help to definite parameters of the system and planets conditions.

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.

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.

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:• 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.

• 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.

• 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

• Methods of intermodulation effects decrease in Wigner-Ville transform. Shkelev, E.I., Kislyakov, A.G., Lupov, S, Yu., Radiophys. & Quant. Electronics, 45, 433, 2002.