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Temporal Data Analysis A.A. 2011/2012 2.1.5.3 Scaling Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.1.6 Partial Sums, Parseval Equation . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Continuous Fourier Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2.2 Theorems and Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2.2.1 Linearity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2.2.2 Shifting Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2.2.3 Scaling Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2.3 Convolution, Parseval Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2.3.1 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2.3.2 Cross Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2.3.3 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2.3.4 The Parseval Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2.4 Fourier Transformation of Derivatives . . . . . . . . . . . . . . . . . . . . . . 32 2.2.5 Fourier Transform and Uncertainty Relation . . . . . . . . . . . . . . . . . . 32 2.3 Window Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3.1 The Rectangular Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.3.1.1 Zeroes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.3.1.2 Intensity at the Central Peak . . . . . . . . . . . . . . . . . . . . . 37 2.3.1.3 Sidelobe Suppression . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.3.1.4 3 dB Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.3.2 The Triangular Window (Fejer Window) . . . . . . . . . . . . . . . . . . . . 39 2.3.3 The Gauss Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.3.4 Windowing or Convolution? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3 Temporal Analysis on Digital Data 43 3.1 Discrete Fourier Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.1.1 Even and Odd Series and Wrap-around . . . . . . . . . . . . . . . . . . . . . 44 3.1.2 The Kronecker Symbol or the Discrete δ-Function . . . . . . . . . . . . . . 44 3.1.3 Definition of the Discrete Fourier Transformation . . . . . . . . . . . . . . 45 3.2 Theorems and Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2.1 Linearity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2.2 Shifting Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2.3 Scaling Rule/Nyquist Frequency . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3 Convolution, Parseval Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.3.1 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.3.2 Cross Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.3.3 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3.4 Parseval Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.4 The Sampling Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.4.1 Aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.4.2 Application to Multi-Variable Signals and Images . . . . . . . . . . . . . . . 58 3.4.3 Geometrical Representation of the Signal . . . . . . . . . . . . . . . . . . . 58 ii M.Orlandini
A.A. 2011/2012 Temporal Data Analysis 3.5 Fast Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4 Response Properties of Physical Systems 63 4.1 Basic Dynamic Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2 Frequency Response Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.3 Single-Input Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 II Procedures 69 5 Procedures for Analyzing Random Data 71 5.1 Procedures for Analyzing Individual Records . . . . . . . . . . . . . . . . . . . . . . 71 5.1.1 Mean and Mean Square Value Analysis . . . . . . . . . . . . . . . . . . . . . 71 5.1.2 Autocorrelation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.1.3 Power Spectral Density Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.1.4 Probability Density Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.1.5 Nonstationary and Transient Data Analysis . . . . . . . . . . . . . . . . . . 73 5.1.6 Periodic and Almost-Periodic Data Analysis . . . . . . . . . . . . . . . . . . 74 5.1.7 Specialized Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.2 Procedures for Analyzing a Collection of Records . . . . . . . . . . . . . . . . . . . 74 5.2.1 Analysis of Individual Records . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.2.2 Test for Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.2.3 Test for Equivalence of Uncorrelated Data . . . . . . . . . . . . . . . . . . . 76 5.2.4 Pooling of Equivalent Uncorrelated Data . . . . . . . . . . . . . . . . . . . . 76 5.2.5 Cross Correlation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.2.6 Cross Correlation Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . 77 5.2.7 Coherence Function Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.2.8 Frequency Response Function Analysis . . . . . . . . . . . . . . . . . . . . . 78 5.2.9 Other Desired Multiple Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 78 6 Temporal Analysis in X–ray Astronomy 79 6.1 Power Spectra in X–ray Astronomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.2 Power Spectral Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.2.1 The Probability Distribution of the Noise Powers . . . . . . . . . . . . . . . 84 6.2.2 The Detection Level: The Number of Trials . . . . . . . . . . . . . . . . . . 86 6.3 The Signal Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.3.1 Sensitivity to Signal Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.3.2 The rms Variation in the Source Signal . . . . . . . . . . . . . . . . . . . . . 88 6.4 Detection of Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.5 Power Spectral Searches Made Easy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.6 Type of Variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.6.1 1/f Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.6.2 Shot Noise Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 M.Orlandini iii
Page 1 and 2: Temporal Data Analysis: Theory and
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Page 9 and 10: List of Definitions 2.1 Even and od
Page 11 and 12: List of Theorems 2.1 Convolution th
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AppendixA Examples Shown at the Bla
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