**3. Data preparation and analysis methods**

Differently to the idealized infinite in time analytical modelling signal considered in section 2, the natural radio emission received from a solar active region with oscillating coronal loop(s) is essentially time-dependent. It usually begins with an impulsive phase of a solar flare and has duration of only several periods of decaying oscillations of the loop(s). Besides of that, a signal from a particular oscillating loop is quite often strongly contaminated by interfering signals from neighboring loops in the active region of interest, as well as by the radiations emitted from other solar active regions. This complicates the task of detection and diagnostics of coronal magnetic loop oscillations in microwaves and requires special data preparation procedures with consequent application of high spectral and time resolution data analysis techniques.

Since the analyzed data appear in a form of discrete counts of the signal intensity, it is natural that digital methods are applied for their processing. A basic specifics of the digital methods consists in certain limitation of dynamical range of the resulting spectra which may lead to the loss of relatively weak and short-time, but important parts of the whole spectra-temporal picture of the studied phenomenon. To avoid of that, the analyzed data pass certain pre-processing preparation which (depending on particular case and task) may include the following procedures: 1) Subtraction of a constant component of a signal, or the signal average; 2) subtraction of a slow (as compared to the analyzed oscillations) major trend of the signal; 3) slow polynomial approximation of the analyzed data with the consequent subtraction of the approximating signal; 4) signal "normalization" (will be described below);

and the nonlinear Wigner-Ville (WV) method (Cohen, 1989; Ville, 1948; Wigner, 1932). Below

<sup>151</sup> Analysis of Long-Periodic Fluctuations of Solar

The classical Fourier transform enables the analysis of a given signal in terms of separate spectral frequency components. It is applied for the study of relative distribution of energy between the spectral components in the case of sufficiently long (ideally infinite) duration of the analyzed signal. However such energetic spectrum does not provide an information on a time when each particular spectral component appears. Possible improvement of the classical Fourier transform method in that respect consists in its application within a certain interval of time Δ*t* (so called "window") and in a consequent shift of this "window" along the time axis. This approach became a standard method for the analysis of non-stationary signals. Further generalization of SWF transform method leads to the wavelet analysis, where effect of the "window" is produced by means of a certain mother-wavelet function. Wavelet transform enables judging about energy distribution over the time and frequency of an analyzed signal. Nowadays this method is also widely used for the analysis of non-stationary and impulsive signals. Its efficiency is however strongly dependent on parameters of the applied mother-wavelet function, which needs to be specially selected and adjusted to the

Recently one more spectral analysis method has been applied in astrophysics. This method is

where *z*(*t*) = *s*(*t*) + *isH*(*t*) is an analytic signal, made of the analyzed sample of real signal *s*(*t*), and its Hilbert conjugate *sH*(*t*). Function *P*(*f* , *t*) gives distribution of the signal energy over frequency *f* and time, and may be visualized in the form of a dynamical spectrum of the signal. According to its definition (5), WV transform may be also interpreted as Fourier image (relative the shifted time) of the local autocorrelation function for the analytical signal *z*(*t*). Since the time *t* appears explicitly among the arguments of the WV spectrum *P*(*f* , *t*), this method is most efficient for high-resolution spectra-temporal analysis of non-stationary signals with varying spectra, such as quasi-harmonic signals with a changing frequency, or varying impulsive signals. In these cases SWF transform and wavelet methods are less efficient, because the averaging over the analysis window (or over the wavelet) results in a decrease of spectral density of the signal components, varying within the corresponding time intervals. At the same time, the non-linearity and non-locality of WV method cause appearance of artificial inter-modulation spectral components at combination frequencies (artifacts) and may result also in suppression of weak spectral components of the signal by its more intense or noisy parts (Cohen, 1989; Shkelev et al., 2002). To compensate the drawbacks of SWF and WV data analysis methods, when they are used separately, and to keep their strong features, the methods were combined in a proper way in the SWF-WV algorithm, which uses various types of signal processing and filtration (with variable shape and size of the analysis windows) in order to eliminate possible artifacts and to provide high spectral

To avoid the appearance of spurious spectra caused by the signal edge effects in the case of analysis of real finite in time signal samples, the so called "weighting" functions with smoothed edges are used. In the present study, the SWF spectrum *Sk* of a discrete signal

<sup>−</sup>*i*2*<sup>π</sup> <sup>f</sup> <sup>τ</sup>dτ*, (5)

we outline the idea of this data analysis algorithm and its main features.

Microwave Radiation, as a Way for Diagnostics of Coronal Magnetic Loops Dynamics

type of particular analyzed signal.

based on the Wigner-Ville (WV) transform

*<sup>P</sup>*(*<sup>f</sup>* , *<sup>t</sup>*) = <sup>∞</sup>

−∞ *z t* + *τ* 2 *z*∗ *<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>* 2 *e*

and temporal resolution (Kislyakov et al., 2011; Shkelev et al., 2002).

and 5) digital filtration of contaminating components. Altogether, the "subtraction" methods 1-3 enhance visibility of weaker fluctuations of the radiation, enabling better analysis of their spectral and temporal characteristics. Note, that the digital filtration (e.g., the method 5) with an appropriate filter(s) may efficiently substitute all the "subtraction" methods. However, the inertness of a filter results in some smoothing of the radio emission fluctuations of interest. Besides of that, information about the intensity of the radiation fluctuations is lost during the frequency filtration.

The filtration algorithm consists in the application of a Gaussian window to the analyzed signal spectrum, obtained with the discrete Fourier transform (DFT), with the consequent performance of the reverse DFT. The Gaussian window for the low frequency (LF), high frequency (HF), band-pass (BP), and band-lock (BL) filters, respectively, is determined by the following expressions:

$$\begin{aligned} W\_{LF}(k) &= \exp\left\{-\frac{k^2}{2} \cdot \left(\frac{f\_1}{f\_s} \cdot d \cdot N\right)^{-2}\right\}, & k &= 0, \ldots, N/2; \\ W\_{HF}(k) &= 1 - \exp\left\{-\frac{k^2}{2} \cdot \left(\frac{f\_1}{f\_s} \cdot d \cdot N\right)^{-2}\right\}, & k &= 0, \ldots, N/2; \\ W\_{BP}(k) &= \exp\left\{-\frac{1}{2} \cdot \left(\frac{k}{N} - \frac{f\_0}{f\_s}\right)^2 \cdot \left(\frac{\Lambda f}{f\_s} \cdot d\right)^{-2}\right\}, & k &= 0, \ldots, N/2; \\ W\_{BL}(k) &= 1 - \exp\left\{-\frac{1}{2} \cdot \left(\frac{k}{N} - \frac{f\_0}{f\_s}\right)^2 \cdot \left(\frac{\Lambda f}{f\_s} \cdot d\right)^{-2}\right\}, & k &= 0, \ldots, N/2. \end{aligned} \tag{2}$$

Here *N* is the number of counts in the analyzed signal, *fs* is the frequency of discretization, *d* is decimation coefficient (Marple, 1986), *f*<sup>1</sup> is the cut-off frequency for the LF and HF filters, *f*<sup>0</sup> and Δ*f* , are the central frequency and the band, respectively, for the BP and BL filters.

Let's consider now the "normalization" method. It is based on a treatment of an analytical signal *z*(*n*) (Marple, 1986):

$$z(n) = s(n) + is\_H(n) = M(n) \exp(i\Psi(n)),\tag{3}$$

where *s*(*n*) and *sH*(*n*) are the analyzed digital signal and its Hilbert conjugate, respectively, and *n* is the count number. The parameters *M*(*n*) and Ψ(*n*) are the module and phase of the analytical signal, which are defined as the following:

$$\begin{array}{l} M(n) = \sqrt{s(n)^2 + s\_H(n)^2} \\ \Psi(n) = \arctan\left[\frac{s\_H(n)}{s(n)}\right] \end{array} \tag{4}$$

The "normalization" procedure consists in division of the analytical signal *z*(*n*) on *M*(*n*). By this, the amplitudes of all spectral components of the analyzed process *s*(*n*) become to be equalized. Note, that due to the orthogonality of functions *s*(*n*) and *sH*(*n*), the value of function *M*(*n*) never becomes zero. The "normalization" method is especially efficient for the analysis of non-stationary modulation processes. It enables to detect and to follow the variations of an "instantaneous" frequency of an oscillatory component of radiation.

An important role in the present work belongs also to the method, used for the detection of quasi-periodic features in the time records of the solar microwave radiation intensity. It consists in application of an original data analysis algorithm (Shkelev et al., 2002; Zaitsev et al., 2001b) made as a combination of the "sliding window" Fourier (SWF) transform technique 8 Will-be-set-by-IN-TECH

and 5) digital filtration of contaminating components. Altogether, the "subtraction" methods 1-3 enhance visibility of weaker fluctuations of the radiation, enabling better analysis of their spectral and temporal characteristics. Note, that the digital filtration (e.g., the method 5) with an appropriate filter(s) may efficiently substitute all the "subtraction" methods. However, the inertness of a filter results in some smoothing of the radio emission fluctuations of interest. Besides of that, information about the intensity of the radiation fluctuations is lost during the

The filtration algorithm consists in the application of a Gaussian window to the analyzed signal spectrum, obtained with the discrete Fourier transform (DFT), with the consequent performance of the reverse DFT. The Gaussian window for the low frequency (LF), high frequency (HF), band-pass (BP), and band-lock (BL) filters, respectively, is determined by the

> −<sup>2</sup>

Here *N* is the number of counts in the analyzed signal, *fs* is the frequency of discretization, *d* is decimation coefficient (Marple, 1986), *f*<sup>1</sup> is the cut-off frequency for the LF and HF filters, *f*<sup>0</sup> and Δ*f* , are the central frequency and the band, respectively, for the BP and BL filters.

Let's consider now the "normalization" method. It is based on a treatment of an analytical

where *s*(*n*) and *sH*(*n*) are the analyzed digital signal and its Hilbert conjugate, respectively, and *n* is the count number. The parameters *M*(*n*) and Ψ(*n*) are the module and phase of the

*<sup>M</sup>*(*n*) = *s*(*n*)<sup>2</sup> <sup>+</sup> *sH*(*n*)2,

The "normalization" procedure consists in division of the analytical signal *z*(*n*) on *M*(*n*). By this, the amplitudes of all spectral components of the analyzed process *s*(*n*) become to be equalized. Note, that due to the orthogonality of functions *s*(*n*) and *sH*(*n*), the value of function *M*(*n*) never becomes zero. The "normalization" method is especially efficient for the analysis of non-stationary modulation processes. It enables to detect and to follow the

An important role in the present work belongs also to the method, used for the detection of quasi-periodic features in the time records of the solar microwave radiation intensity. It consists in application of an original data analysis algorithm (Shkelev et al., 2002; Zaitsev et al., 2001b) made as a combination of the "sliding window" Fourier (SWF) transform technique

 *sH* (*n*) *s*(*n*) 

Ψ(*n*) = arctan

variations of an "instantaneous" frequency of an oscillatory component of radiation.

−<sup>2</sup> 

, *k* = 0, ..., *N*/2;

*z*(*n*) = *s*(*n*) + *isH*(*n*) = *M*(*n*) exp(*i*Ψ(*n*)), (3)

, *k* = 0, ..., *N*/2;

, *k* = 0, ..., *N*/2;

, *k* = 0, ..., *N*/2.

. (4)

(2)

frequency filtration.

following expressions:

signal *z*(*n*) (Marple, 1986):

*WLF*(*k*) = exp

*WBP*(*k*) = exp

*WHF*(*k*) = 1 − exp

*WBL*(*k*) = 1 − exp

 −*k*2 2 · *f*<sup>1</sup> *fs* · *d* · *N*

 −1 2 · *<sup>k</sup> <sup>N</sup>* <sup>−</sup> *<sup>f</sup>*<sup>0</sup> *fs* 2 · Δ*f fs* · *d* −<sup>2</sup> 

analytical signal, which are defined as the following:

 −*k*2 2 · *f*<sup>1</sup> *fs* · *d* · *N*

 −1 2 · *<sup>k</sup> <sup>N</sup>* <sup>−</sup> *<sup>f</sup>*<sup>0</sup> *fs* 2 · Δ*f fs* · *d* −<sup>2</sup>  and the nonlinear Wigner-Ville (WV) method (Cohen, 1989; Ville, 1948; Wigner, 1932). Below we outline the idea of this data analysis algorithm and its main features.

The classical Fourier transform enables the analysis of a given signal in terms of separate spectral frequency components. It is applied for the study of relative distribution of energy between the spectral components in the case of sufficiently long (ideally infinite) duration of the analyzed signal. However such energetic spectrum does not provide an information on a time when each particular spectral component appears. Possible improvement of the classical Fourier transform method in that respect consists in its application within a certain interval of time Δ*t* (so called "window") and in a consequent shift of this "window" along the time axis. This approach became a standard method for the analysis of non-stationary signals. Further generalization of SWF transform method leads to the wavelet analysis, where effect of the "window" is produced by means of a certain mother-wavelet function. Wavelet transform enables judging about energy distribution over the time and frequency of an analyzed signal. Nowadays this method is also widely used for the analysis of non-stationary and impulsive signals. Its efficiency is however strongly dependent on parameters of the applied mother-wavelet function, which needs to be specially selected and adjusted to the type of particular analyzed signal.

Recently one more spectral analysis method has been applied in astrophysics. This method is based on the Wigner-Ville (WV) transform

$$P(f,t) = \int\_{-\infty}^{\infty} z\left(t + \frac{\tau}{2}\right) z^\*\left(t - \frac{\tau}{2}\right) e^{-i2\pi f \tau} d\tau,\tag{5}$$

where *z*(*t*) = *s*(*t*) + *isH*(*t*) is an analytic signal, made of the analyzed sample of real signal *s*(*t*), and its Hilbert conjugate *sH*(*t*). Function *P*(*f* , *t*) gives distribution of the signal energy over frequency *f* and time, and may be visualized in the form of a dynamical spectrum of the signal. According to its definition (5), WV transform may be also interpreted as Fourier image (relative the shifted time) of the local autocorrelation function for the analytical signal *z*(*t*).

Since the time *t* appears explicitly among the arguments of the WV spectrum *P*(*f* , *t*), this method is most efficient for high-resolution spectra-temporal analysis of non-stationary signals with varying spectra, such as quasi-harmonic signals with a changing frequency, or varying impulsive signals. In these cases SWF transform and wavelet methods are less efficient, because the averaging over the analysis window (or over the wavelet) results in a decrease of spectral density of the signal components, varying within the corresponding time intervals. At the same time, the non-linearity and non-locality of WV method cause appearance of artificial inter-modulation spectral components at combination frequencies (artifacts) and may result also in suppression of weak spectral components of the signal by its more intense or noisy parts (Cohen, 1989; Shkelev et al., 2002). To compensate the drawbacks of SWF and WV data analysis methods, when they are used separately, and to keep their strong features, the methods were combined in a proper way in the SWF-WV algorithm, which uses various types of signal processing and filtration (with variable shape and size of the analysis windows) in order to eliminate possible artifacts and to provide high spectral and temporal resolution (Kislyakov et al., 2011; Shkelev et al., 2002).

To avoid the appearance of spurious spectra caused by the signal edge effects in the case of analysis of real finite in time signal samples, the so called "weighting" functions with smoothed edges are used. In the present study, the SWF spectrum *Sk* of a discrete signal

above SWF and WV methods provide an efficient data analysis algorithm characterized by high sensitivity, high spectral and temporal resolution, and ability to detect complex multi-signal modulations in the analyzed data records, enabling the dynamical spectra of these modulations. For successful operation of the SWF-WV algorithm, the sampling cadence of analyzed data series should provide sufficient number of the data points, e.g. ≥ 10, 000 points per realization. The length of the analyzed data series should be consistent with the time scales of considered dynamic phenomena, i.e. the duration of an analyzed data set should include at least several periods of the modulating oscillatory component. The SWF-WV method appears the most efficient for the study of signals with non-stationary complex modulations. In such cases the traditional Fourier transform and wavelet methods are less efficient. This feature of the algorithm has been, in particular, used to distinguish between the modulations, possibly caused by the large-scale transverse quasi-periodic motion of the loops, which are the subject of the present study, and the modulations with frequency

<sup>153</sup> Analysis of Long-Periodic Fluctuations of Solar

Microwave Radiation, as a Way for Diagnostics of Coronal Magnetic Loops Dynamics

For the visualization of the whole variety of the detected modulations, so called averaged spectral density plots are produced along with the dynamic spectra by the SWF-WV algorithm. These plots are obtained by averaging of multiple instantaneous cuts of the dynamic spectrum taken at given moments of time, so that short-living modulation features also become clearly seen among the longer lasting modulation lines. These both types of spectra (dynamic and averaged) enable the detection of the large-scale transverse oscillatory

The universality of SWF-WV algorithm resulted in its successful application in different branches of space physics. The algorithm was used for the diagnostics of intrinsic physical and dynamical conditions in the stellar and planetary systems, solar/stellar winds, as well as in solar and planetary radiation sources and planetary environments (Khodachenko et al., 2006; Kislyakov et al., 2006; Panchenko et al., 2009; Zaitsev et al., 2003; 2004). Nowadays, the link to SWF-WV data analysis algorithm is available for the scientific community via the on-line catalogue of models and data analysis tools (http://europlanet-jra3.oeaw.ac.at/catalogue/), developed within the JRA3-EMDAF (European modelling and data analysis facilities) activity (http://europlanet-jra3.oeaw.ac.at/) of the European FP7 research infrastructure project

**4. Diagnostics of large-scale oscillations of coronal loops by the analysis of VLF**

We analyze the VLF (< 0.01 Hz) modulations of solar microwave bursts recorded in Metsähovi Radio Observatory (Finland) with the 14-m and 1.8 m radio telescope antenna at 37 GHz and 11.7 GHz, respectively. The key selection criterion for the analyzed microwave data was their synchronism with the oscillating loops observed in extreme ultraviolet (EUV) by TRACE (Aschwanden et al., 2002). The width of the antenna beam pattern of the Metsähovi radio telescope at 37 GHz is 2.4', the sensitivity of the receiver is about 0.1 sfu (10−<sup>23</sup> W m−<sup>2</sup> Hz−1), and time resolution, 0.05 <sup>÷</sup> 0.1 s. Therefore, at 37 GHz the spatial resolution of the radio telescope is sufficient for identification of an active region that contains a radiating source. This enables to analyze microwave radiation emitted directly from the region, imaged in EUV (e.g., observed by TRACE), and to perform the comparison of the radiation features

drifts related with the electric current LCR-oscillations in the loops.

dynamics of the radiating coronal magnetic loops.

**modulations of microwave radiation**

**5. Instrumentation and modulations detection capabilities**

Europlanet-RI.

*sn* consisting of *N* counts is calculated by a discrete Fourier transform (DFT) (Allen & Mills, 2004; Marple, 1986):

$$S\_k = \sum\_{n=0}^{N-1} \text{wnd}(n) s\_n \exp\left\{-i\frac{2\pi nk}{N}\right\}, \quad k = 0, 1, \ldots, N-1\tag{6}$$

with a sliding window wnd(*n*). The following window functions are used (see also Pollock (1999)):

$$\begin{array}{ll} \text{wnd}\_1(n) = \begin{cases} 1 & 0 \le n < N \\ 0, & n \ge N \end{cases} & \text{\$-a \$\text{rectangular window}\$} \\\\ \text{wnd}\_2(n) = \begin{cases} \cos^2\left(\frac{\pi n}{2N}\right), & 0 \le n < N/2 \\ 0, & n \ge N/2 \end{cases} & \text{\$-Henning's window} \\\\ \text{wnd}\_3(n) = \begin{cases} 1 - 6 \cdot \left(\frac{2 \cdot n}{N}\right)^2 + 6 \cdot \left(\frac{2 \cdot n}{N}\right)^3, & 0 \le n < N/4 \\ 2 \cdot \left(1 - \frac{2 \cdot n}{N}\right)^3, & N/4 \le n < N/2 \\ 0, & n \ge N/2 \end{cases} & \text{\$-Parsen's window} \\\\ \text{The advantage of this method consists in its high performance speed, especially if the standard} \end{array}$$

The advantage of this method consists in its high performance speed, especially if the standard algorithms of fast Fourier transform (FFT) are used in the calculations (Allen & Mills, 2004; Marple, 1986). On the other hand, frequency resolution of SWF is reverse proportional to the number of signal counts in the applied window. Therefore the size of window should be sufficiently large. This in its turn decreases the temporal resolution of the method. In practice, the choice of particular type and width of the window is determined by dynamical features of the analyzed signal.

An algorithm of the discrete WV transform is determined by the following expression:

$$P\_{m\bar{k}} = P(m\Delta t, k\Delta f) = 2\Delta t \sum\_{n=0}^{2N-1} \left[ z\_{m+n} z\_{m-n}^\* \exp\left\{-i\frac{\pi nk}{N}\right\} \right], \text{ } \{m, k\} = 0, 1, 2, ..., 2N,\tag{8}$$

where *zn* and *z*∗ *<sup>n</sup>* are discrete values of the analyzed complex analytical signal made, as determined above, of the real discrete signal and its Hilbert conjugate; Δ*t* is a period of discretization, and Δ*f* = 1/(4*N*Δ*t*) is the frequency step. Note, that WV transform results in real values only in the case of continuous functions integrated in infinite limits (like in (5)). The discrete WV transform (8) yields a complex function *Pmk*, which is called as "pseudo-WV transform" (Cohen, 1989). In that respect in practice only a module of *Pmk* or its real part are considered.

In course of the comparative study performed in Kislyakov et al. (2011); Shkelev et al. (2002) it has been shown that WV data analysis technique enables higher spectral and temporal resolution than that of the SWF. In Shkelev et al. (2002) the efficiency of WV and SWF methods was checked with various test signals, made as combinations of impulsive and quasi-harmonic processes. By this, along with the meaningful spectrum, WV transform generated specific artificial spectral features (due to the non-linearity of the method). It has been shown in that respect that superimposing of the higher resolution WV spectra with those of lower resolution provided by SWF, may help to identify and to exclude these artificial spectral features form the consideration. Altogether, combined with each other the described 10 Will-be-set-by-IN-TECH

*sn* consisting of *N* counts is calculated by a discrete Fourier transform (DFT) (Allen & Mills,

with a sliding window wnd(*n*). The following window functions are used (see also Pollock

0 , *<sup>n</sup>* <sup>≥</sup> *<sup>N</sup>* – a rectangular window

0 , *<sup>n</sup>* <sup>≥</sup> *<sup>N</sup>*/2 – Henning's window

, 0 ≤ *n* < *N*/4

, *N*/4 ≤ *n* < *N*/2

The advantage of this method consists in its high performance speed, especially if the standard algorithms of fast Fourier transform (FFT) are used in the calculations (Allen & Mills, 2004; Marple, 1986). On the other hand, frequency resolution of SWF is reverse proportional to the number of signal counts in the applied window. Therefore the size of window should be sufficiently large. This in its turn decreases the temporal resolution of the method. In practice, the choice of particular type and width of the window is determined by dynamical features of

An algorithm of the discrete WV transform is determined by the following expression:

*<sup>m</sup>*−*<sup>n</sup>* exp

determined above, of the real discrete signal and its Hilbert conjugate; Δ*t* is a period of discretization, and Δ*f* = 1/(4*N*Δ*t*) is the frequency step. Note, that WV transform results in real values only in the case of continuous functions integrated in infinite limits (like in (5)). The discrete WV transform (8) yields a complex function *Pmk*, which is called as "pseudo-WV transform" (Cohen, 1989). In that respect in practice only a module of *Pmk* or its real part are

In course of the comparative study performed in Kislyakov et al. (2011); Shkelev et al. (2002) it has been shown that WV data analysis technique enables higher spectral and temporal resolution than that of the SWF. In Shkelev et al. (2002) the efficiency of WV and SWF methods was checked with various test signals, made as combinations of impulsive and quasi-harmonic processes. By this, along with the meaningful spectrum, WV transform generated specific artificial spectral features (due to the non-linearity of the method). It has been shown in that respect that superimposing of the higher resolution WV spectra with those of lower resolution provided by SWF, may help to identify and to exclude these artificial spectral features form the consideration. Altogether, combined with each other the described

� −*i πnk N*

*<sup>n</sup>* are discrete values of the analyzed complex analytical signal made, as

��

, *k* = 0, 1, ..., *N* − 1 (6)

– Parsen's window.

, {*m*, *k*} = 0, 1, 2, ..., 2*N*, (8)

(7)

� <sup>−</sup>*<sup>i</sup>* <sup>2</sup>*πnk N* �

*Sk* <sup>=</sup> <sup>∑</sup>*N*−<sup>1</sup> *<sup>n</sup>*=<sup>0</sup> wnd(*n*)*sn* exp

� , 0 <sup>≤</sup> *<sup>n</sup>* <sup>&</sup>lt; *<sup>N</sup>*/2

�2·*<sup>n</sup> N* �3

0 , *n* ≥ *N*/2

2*N*−1 ∑ *n*=0

� *zm*+*nz*∗

� 1, 0 <sup>≤</sup> *<sup>n</sup>* <sup>&</sup>lt; *<sup>N</sup>*

� cos<sup>2</sup> � *<sup>π</sup>*·*<sup>n</sup>* 2*N*

1 − 6 ·

2 · � <sup>1</sup> <sup>−</sup> <sup>2</sup>·*<sup>n</sup> N* �3

�2·*<sup>n</sup> N* �2 + 6 ·

⎧ ⎪⎪⎨

⎪⎪⎩

*Pmk* = *P*(*m*Δ*t*, *k*Δ*f*) = 2Δ*t*

2004; Marple, 1986):

(1999)):

wnd1(*n*) =

wnd2(*n*) =

wnd3(*n*) =

the analyzed signal.

where *zn* and *z*∗

considered.

above SWF and WV methods provide an efficient data analysis algorithm characterized by high sensitivity, high spectral and temporal resolution, and ability to detect complex multi-signal modulations in the analyzed data records, enabling the dynamical spectra of these modulations. For successful operation of the SWF-WV algorithm, the sampling cadence of analyzed data series should provide sufficient number of the data points, e.g. ≥ 10, 000 points per realization. The length of the analyzed data series should be consistent with the time scales of considered dynamic phenomena, i.e. the duration of an analyzed data set should include at least several periods of the modulating oscillatory component. The SWF-WV method appears the most efficient for the study of signals with non-stationary complex modulations. In such cases the traditional Fourier transform and wavelet methods are less efficient. This feature of the algorithm has been, in particular, used to distinguish between the modulations, possibly caused by the large-scale transverse quasi-periodic motion of the loops, which are the subject of the present study, and the modulations with frequency drifts related with the electric current LCR-oscillations in the loops.

For the visualization of the whole variety of the detected modulations, so called averaged spectral density plots are produced along with the dynamic spectra by the SWF-WV algorithm. These plots are obtained by averaging of multiple instantaneous cuts of the dynamic spectrum taken at given moments of time, so that short-living modulation features also become clearly seen among the longer lasting modulation lines. These both types of spectra (dynamic and averaged) enable the detection of the large-scale transverse oscillatory dynamics of the radiating coronal magnetic loops.

The universality of SWF-WV algorithm resulted in its successful application in different branches of space physics. The algorithm was used for the diagnostics of intrinsic physical and dynamical conditions in the stellar and planetary systems, solar/stellar winds, as well as in solar and planetary radiation sources and planetary environments (Khodachenko et al., 2006; Kislyakov et al., 2006; Panchenko et al., 2009; Zaitsev et al., 2003; 2004). Nowadays, the link to SWF-WV data analysis algorithm is available for the scientific community via the on-line catalogue of models and data analysis tools (http://europlanet-jra3.oeaw.ac.at/catalogue/), developed within the JRA3-EMDAF (European modelling and data analysis facilities) activity (http://europlanet-jra3.oeaw.ac.at/) of the European FP7 research infrastructure project Europlanet-RI.
