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