Of Poles and Zeros

1 Introduction.- 2 RC Filter.- 2.1 The system diagram.- 2.2 The differential equation.- 2.3 The frequency response function and the Fourier transform.- 2.4 The transfer function and the Laplace transform.- 2.5 The impulse response function.- 2.6 The step response function.- 2.7 The frequency response function and the pole position.- 2.8 The difference equation.- 2.9 Review RC filter.- 3 General linear time invariant systems.- 3.1 Generalization of concepts.- 3.2 Graphical estimation of the frequency response function.- 3.3 The phase properties of general LTI system.- 3.4 The interpretation of the frequency response function.- 4 The seismometer.- 4.1 The solution for simple initial conditions.- 4.2 The determination of the damping constant.- 4.3 The frequency response function.- 5 The sampling process.- 5.1 The sampling of analog data.- 6 Analog-to-digital conversion.- 6.1 The principle of analog to digital conversion.- 6.2 Increasing dynamic range and resolution.- 7 From infinitely continuous to finite discrete.- 7.1 Fourier series for periodic continuous-time signals (CTFS).- 7.2 Fourier transform for aperiodic continuous-time signals (CTFT).- 7.3 Fourier transform of discrete-time signals.- 7.4 Fourier series for periodic discrete-time signals.- 7.5 The Discrete Fourier Transform (DFT).- 7.6 The z-transform and the discrete transfer function.- 7.7 The inverse z-transform.- 7.8 The z-transform and the Discrete Fourier Transform (DFT).- 7.9 The convolution of sequences.- 8 The digital anti-alias filter.- 8.1 Removing the maximum phase component of a FIR filter.- 9 Inverse and simulation filtering of digital seismograms.- 9.1 Stability problems.- 9.2 Implementation.- 9.3 Review inverse and simulation filtering.- 10 The measurement of wavelet parameters from digital seismograms.- 10.1 The determination of ground motion amplitudes.- 10.2 The properties of seismic onsets.- 10.3 Rise time and pulse duration.- 10.4 Signal moment.- References.- Appendix: Solution to problems.