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298 M. Ayub Khan Yousuf Zai et al Decomposition at level 5: s: a5+d5+d4+d3+d2+d1. S a 5 d 5 d 4 d 3 d 2 Fig. 7. Reconstructed details analyzed at level 5, residual part of ionospheric turbulence model at Pakistan air space. d 1 50 100 150 200 250 300 350 Fig. 6. Decomposition at level 5, of ionospheric turbulence at Pakistan air space. cumulative histogram of ionospheric turbulence at Pakistan air space in Fig. 6–8. Wavelet approach is a recently developed signal processing tool enabling us on several timescale of the local properties of complex signals that can present non stationary variations and also to study how it varies with time by decomposing time series into frequency space. Finally, powerful multi resolution with respect to Haar is carried out. This decomposes actual signal into different signals to be analyzed into principal and residual part. By mathematical expression
Wavelet Characterization of Turbulence 299 Residuals S A j D j j 4 2 0 -2 50 100 150 200 250 300 350 Histogram Auto correlations Cumulative histogram S, A j and D j are the signal, principal part j level and residual part j level. Mathematically, the relation between principal and residual part j level by mathematical is: A j1 A j D j The wavelet used in the study by Haar of the level 5, j the dyadic scale is a 2 for level 5 the resolution is given by 1/a, or 2 -j . In order to approach the cyclic study maximum and minimum values, we have carried out wavelet analysis of ionospheric turbulence. In Fig. 6 the variation is presented in the form of different resolutions at level 5 of Haar wavelet type in the detailed and approximated part the cyclic variation is also presented at different level. In the detailed and approximated part at the lowest resolution several peaks are appeared. The expression of Haar wavelet in continuous time series function is represented as follows: 1 { 1;0 t 2 { 0; otherwise, 1 2 t 0 f ( t) { 1; t0 t t0 The corresponding wavelet in discrete time series { 1; n 1 h { 1; n 2 h n {0;otherwise 0 The quantity h 1/ 0 2 is a constant. Fig. 7 shows wavelet 1-D analysis of ionospheric turbulence at Pakistan air space. Present data is non-linear there is local minimum in this period which depicts a positive skewness of frequency histogram: FFT - Spectrum Frequency Fig. 8. The graph of residual, autocorrelation, FFTspectrum, histogram, cumulative histogram obtained from wavelet analysis F 2 layer.for ionospheric turbulence. s a 5 d5 d 4 d3 d 2 d1 The Fig. 8 manifests the residual, autocorrelation, FFT-spectrum, histogram, cumulative histogram of ionospheric turbulence F 2 layer. For an advance and more recent analysis of ionospheric turbulence at Pakistan air space, between 1 to 365 components data point, we have constructed model of ionospheric turbulence. We have used wavelet transformation which is a common method of analyzing ionospheric plasma turbulence [8-10].
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ISSN Print : 0377 - 2969 ISSN Onlin
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C O N T E N T S Volume 49, No. 4, D
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236 M. Farooq et al impedance spect
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238 M. Farooq et al Fig. 3. Variati
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