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Temporal Data Analysis A.A. 2011/2012 6.3 Confidence detection level as a function of the number of trials . . . . . . . . . . 87 6.4 Relations between the different power levels . . . . . . . . . . . . . . . . . . . . . . 89 6.5 Effect of choosing the binning size in detecting weak features: the case of the kHz QPO in 4U 1728–34 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.6 Noise classification in astronomical power spectra . . . . . . . . . . . . . . . . . . . 91 6.7 Examples of 1/f noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.8 1/f noise in natural systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.9 The shot noise process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.10 Power spectra in νP ν form for the black hole candidates XTE 1118+480 and 1E 1724–3045 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.11 QPO observed in the low-mass X-ray binary and atoll source 4U 1735-44 . . . . . 98 6.12 Example of the Lomb-Scargle algorithm in action . . . . . . . . . . . . . . . . . . . 100 6.13 Pulse period of the X–ray binary pulsar GX 301–2 obtained by means of the epoch folding technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.14 Measuring the phase shift with respect to a template profile . . . . . . . . . . . . . 103 6.15 General procedure to derive information on the source from the measured TOAs 103 6.16 The time of arrivals of pulses from celestial objects are referenced to the nearly inertial reference frame of the Solar System barycenter . . . . . . . . . . . . . . . . 104 6.17 Definition of the orbital parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.18 Delays of the TOA in Her X–1 due to its orbital motion . . . . . . . . . . . . . . . . 106 A.1 The triangular function, the weighting function cos(πt/T ), and their product . . 118 A.2 The normalized Gaussian function and its Fourier transform . . . . . . . . . . . . 120 A.3 The bilateral exponential function and its Fourier transform . . . . . . . . . . . . . 121 B.1 XTE/PCA 1 hour light curve of the X–ray binary pulsar Cen X–3 . . . . . . . . . . 135 B.2 Power spectrum of the XTE/PCA observation of Cen X–3 . . . . . . . . . . . . . . . 138 B.3 Determination of the pulse period by epoch folding . . . . . . . . . . . . . . . . . . 141 B.4 Folding of data at the best pulse period . . . . . . . . . . . . . . . . . . . . . . . . . 143 B.5 Pulse period search for XTE J1946+274: effect of pulse period variability . . . . . 146 B.6 Cen X–3 pulse period change along the orbit . . . . . . . . . . . . . . . . . . . . . . 156 B.7 Doppler Shift of the Cen X–3 pulse profile along its orbit. . . . . . . . . . . . . . . 158 B.8 Power spectrum of the X–ray binary pulsar XTE J1858+034 . . . . . . . . . . . . . . 161 B.9 Power spectrum of the low-mass X–ray binary 4U1728−34 showing kHz QPO . . 164 B.10 Power spectrum of the low-mass X–ray binary 4U1728−34 showing high frequency oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 vi M.Orlandini
List of Definitions 2.1 Even and odd functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Forward Fourier transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4 Inverse Fourier transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.5 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.6 Cross correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.7 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.8 Energy and distances for a signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1 Discrete Fourier transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 Discrete convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3 Discrete cross correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 6.1 Discrete Fourier transform in X–ray astronomy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79 6.2 Power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 6.3 Detection level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.4 1/f noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.5 Lomb-Scargle normalized periodogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 vii
Page 1 and 2: Temporal Data Analysis: Theory and
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Page 11 and 12: List of Theorems 2.1 Convolution th
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Chapter6 Temporal Analysis in X-ray
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AppendixA Examples Shown at the Bla
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