## Chapter I. Complex Calculus
1. Definition 2. Rules for manipulation 3. Graphical representation – Polar form; Trigonometric representation – Exponential form 4. Powers and roots 5. The logarithmic function 6. Trigonometric and hyperbolic functions 7. Inverse trigonometric and hyperbolic functions 8. Polynomials, Rational functions and Partial fraction expansion 9. Functions of a complex variable, Analytic functions
## Chapter II. Fourier Analysis
A. Fourier Series 1. Periodic functions 2. Fourier series 3. Orthogonal functions 4. Evaluation of the Fourier coefficients 5. Least-squares approximation of a periodic function by a finite Fourier series 6. Convergence of the Fourier series; Parseval’s relation 7. The Dirichlet conditions 8. Differentiation and integration 9. Even and odd functions 10. Complex form of the Fourier series 11. Orthogonality of complex Fourier series functions 12. Parseval’s theorem B. Fourier Integral 13. From Fourier series to Fourier integral 14. The Fourier transform of even and odd and of real and complex functions 15. Properties of the Fourier transform: - Linearity - Scaling - Shifting - Modulation - Symmetry - Differentiation - Integration 16. Convolution 17. Parseval’s theorem 18. Correlation 19. The Fourier transform of special functions: - The unit-impulse function - The unit-step function - The sign function - Causal functions - Periodic functions 20. Calculations of moments with the Fourier transform 21. The uncertainty principle in Fourier analysis 22. The second moment as a function of amplitude and phase spectrum 23. Poisson’s summation formula C. Table of Fourier transform pairs D. Complex signal analysis: envelope, instantaneous phase and frequency
## Chapter III. The Laplace Transform
A. The uniteral or one-sided Laplace transform 1. Definition 2. Fundamental theorem 3. Examples 4. Properties of the Laplace transform: - Linearity - Scaling - Shifting - Modulation - Differentiation - Integration - Convolution 5. The relationship between the Fourier transform of a causal function and the unilateral Laplace transform 6. The inversion formula 7. The initial-value theorem B. The bilateral or two-sided Laplace transform 1. Definition 2. Properties: - Linearity - Scaling - Shifting - Modulation - Differentiation - Integration - Convolution - Product - Autocorrelation
## Chapter IV. Linear Systems
1. Definition 2. Linearity 3. Stationarity 4. Eigenfunctions of linear stationary systems 5. The impulse response and the transfer function 6. Stability 7. Causality 8. The unit-step response 9. Systems with feedback 10. Linear differential equations with constant coefficients
## Chapter V. Sampling
1. The sampling operator 2. The Fourier transform of a sampled signal 3. Reconstruction of the time-continuous signal; aliasing 4. Properties of band-limited time-continuous signals and their discrete counterparts 5. Periodic functions of time 6. Sampling in the frequency domain 7. Convolution of discrete signals in matrix notation 8. The discrete Fourier transform pair
## Chapter VI. The Z-Transform and Inverse Filters
1. From Laplace transform to z-transform 2. Cauchy’s integral formula for the inverse z-transform 3. Properties of the z-transform: - linearity - shift - modulation - differentiation - conjugation - initial value - convolution - multiplication - time reversal - correlation - Parseval’s relation 4. Amplitude and phase properties and autocorrelation of a two point wavelet and its transpose 5. An (n+1)-point wavelet; decomposition, transposed wavelet and autocorrelation 6. Digital filter networks without (FIR) and with (IIR) feedback 7. Inverse filters 8. The inverse filter of a two-point wavelet 9. The inverse filter of an arbitrary wavelet 10. Properties of minimum-phase functions in the time domain 11. Reverberation and ghost and their inverse operators 12. Dispersive filters Table of transforms
## Chapter VII. Stochastic processes and least-squares estimation
1. Signal classification 2. Elements of probability theory 3. Random variable and random process 4. Probability distribution and probability density 5. Expectation values; ensemble averages 6. Auto- and crosscorrelation and auto- and crosscovariance 7. Moments and characteristic functions 8. Independent, uncorrelated and orthogonal random variables 9. Linear least mean-squares estimation 10. Time averages and ergodicity 11. Response of a linear stationary system to a random signal 12. Stochastic differential equations 13. Properties of estimators 14. Estimates of the autocorrelation 15. Power spectrum estimation: - windowing - autoregressive spectral estimation - statistical tests 16. Recursive linear least-squares estimation 17. Statistical distributions
## Chapter VIII. Least-squares filter design
1. The least-squares criterion 2. The normal equations 3. The performance figure 4. Geometrical interpretation 5. Least-squares inverse filters 6. Least-squares inverse filters and projection 7. Prediction filtering 8. Prediction error filtering 9. Properties of least-squares zero-delay inverse filters 10. Summary of inverse filters 11. Filtering in the presence of noise 12. The Wiener filter in the frequency domain 13. The Levinson procedure 14. The matched filter and the output energy filter 15. Homomorphic deconvolution
## Chapter IX. Digital filter design
1. Zero-phase low-pass filters 2. From low-pass to high-pass, band-pass and band-stop 3. Transformations starting from low-pass filters 4. The impulse-invariant technique 5. The bilinear z-transformation 6. Prototype low-pass filters 7. Finite differences
## Chapter X. The two-Dimensional Fourier Transform
1. The Fourier transform of 2-dim functions 2. Geometrical interpretation of the 1-dim and 2-dim Fourier transform 3. Two-dimensional convolution and correlation 4. Properties of the 2-dim Fourier transform 5. From the (x,t)-domain to the (k,f)-domain 6. The 2-dim Fourier transform of a system of plane waves 7. The effect of truncation in x 8. Sampling in t and in x 9. The Fourier transform in diffraction and image formation
## Chapter XI. Multi-trace Filtering
1. The 2-dim delta-function 2. Two-dimensional linear systems 3. The 2-dim z-transform 4. Two-dimensional filters 5. Multi-trace filters 6. The velocity filter
## Chapter XII. Transforms
1. The Fourier transform (temporal and spatial) 2. The Radon transform (linear, hyperbolic, parabolic and generalized) 3. The Wavelet transform 4. The S transform 5. The Hilbert transform and the analytic signal 6. SVD and the Karhunen-Loeve transform 7. The Dreamlet transform 8. The Curvelet transform
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