Statistical and Transform Methods in Geophysical Signal Processing

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ii Contact: ¨¥© ¢¡¤£¦¥§ ¡¢¢ ¢¡¢¢¡ ¢¢¨¢ ©¢¢¥¡¥ ¥§¥§¡¥¢¢ c 2002 by M.D.Sacchi
Contents 1 Fourier Analysis 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Orthogonal Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.1 Properties of the FT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.2 The FT of some signals . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.3 Truncation in time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 Living in a discrete World . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 Z-transform and Convolution 21 2.1 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.1 Discrete convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.1.2 An algorithm to compute the convolution sum . . . . . . . . . . . . 27 2.2 The Z transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2.1 Convolution and the Z-transform . . . . . . . . . . . . . . . . . . . . 30 2.2.2 Deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.3 Elementary Signals: Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3.1 Minimum phase dipoles . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3.2 Maximum phase dipoles . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.3.3 Autocorrelation function of dipoles . . . . . . . . . . . . . . . . . . . 39 iii
Page 1: Statistical and Transform Methods i
Page 5 and 6: CONTENTS v 4.2.3 Deconvolution in t
Page 7 and 8: CONTENTS vii 7.1.3 The sampling the

Statistical and Transform Methods in Geophysical Signal Processing

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