Adaptive and Array Signal Processing - Interactive curriculae of TU Ilmenau

The fundamental concepts of adaptive filters and array signal processing are developed in class. The students understand the relationships between temporal and spatial filters, as well as the principle of high-resolution parameter estimation, and they are able to adapt their knowledge to other scientific disciplines. The students are able to develop or improve algorithms and to evaluate their performance in an analytical manner or by simulations. Futhermore, the students are enabled to read and understand current research publications in the areas of adaptive filters and array signal processing and they can use these concepts and results for their own research.

1 Introduction
- Adaptive Filters
- Single channel adaptive equalization (temporal filter)
- Multi channel adaptive beamforming (spatial filter)

2 Mathematical Background

2.1 Calculus
- Gradients
- Differentiation with respect to a complex vector
- Quadratic optimization with linear constraints (method of Lagrangian multipliers)

2.2 Stochastic processes
- Stationary processes
- Time averages
- Ergodic processes
- Correlation matrices

2.3 Linear algebra
- Eigenvalue decomposition
- Eigenfilter
- Linear system of equations
- Four fundamental subspaces
- Singular value decomposition
- Generalized inverse of a matrix
- Projections
- Low rank modeling

3 Adaptive Filters
3.1 Linear Optimum Filtering (Wiener Filters)
- Principle of Orthogonality
- Wiener-Hopf equations
- Error-performance surface
- MMSE (minimum mean-squared error)
- Canonical form of the error-performance surface
- MMSE filtering in case of linear Models

3.2 Linearly Constrained Minimum Variance Filter
- LCMV beamformer
- Minimum Variance Distortionless Response (MVDR) spectrum: Capon's method
- LCMV beamforming with multiple linear constraints

3.3 Generalized Sidelobe Canceler

3.4 Iterative Solution of the Normal Equations
- Steepest descent algorithm
- Stability of the algorithm
- Optimization of the step-size

3.5 Least Mean Square (LMS) Algorithm

3.6 Recursive Least Squares (RLS) Algorithm

4 High-Resolution Parameter Estimation
- Data model (DOA estimation)
- Eigendecomposition of the spatial correlation matrix at the receive array
- Subspace estimates
- Estimation of the model order

4.1 Spectral MUSIC
- DOA estimation
- Example: uniform linear array (ULA)
- Root-MUSIC for ULAs
- Periodogram
- MVDR spatial spectrum estimation (review)

4.2 Standard ESPRIT
- Selection matrices
- Shift invariance property

4.3 Signal Reconstruction
- LS solution
- MVDR / BLUE solution
- Wiener solution (MMSE solution)
- Antenna patterns

4.4 Spatial smoothing

4.5 Forward-backward averaging

4.6 Real-valued subspace estimation

4.7 1-D Unitary ESPRIT
- Reliability test
- Applications in Audio Coding

4.8 Multidimensional Extensions
- 2-D MUSIC
- 2-D Unitary ESPRIT
- R-D Unitary ESPRIT

4.9 Multidimensional Real-Time Channel Sounding

4.10 Direction of Arrival Estimation with Hexagonal ESPAR Arrays

5 Tensor-Based Signal Processing

5.1 Introduction and Motivation

5.2 Fundamental Concepts of Tensor Algebra

5.3 Elementary Tensor Decompositions
- Higher Order SVD (HOSVD)
- CANDECOMP / PARAFAC (CP) Decomposition

5.4 Tensors in Selected Signal Processing Applications

6 Maximum Likelihood Estimators

6.1 Maximum Likelihood Principle

6.2 The Fisher Information Matrix and the Cramer Rao Lower Bound (CRLB)
- Efficiency
- CRLB for 1-D direction finding applications
- Asymptotic CRLB

• Moodle course
• Skript, Overheadprojektor, Beamer

• T. Kaiser, A. Bourdoux, H. Boche, Smart Antennas State of The Art.
  Hindawi Publishing Corporation, 2005.
• A. H. Sayed, Fundamentals of Adaptive Filtering.
  John Wiley & Sons, Inc., New York, NY, 2003.
• T. K. Moon and W. C. Stirling, Mathematical Methods and Algorithms for Signal Processing.
  Prentice-Hall, 2000.
• S. Haykin and M. Moher, Modern Wireless Communications.
  Pearson Education, Inc., 2005.
• S. Haykin, Adaptive Filter Theory.
  Prentice-Hall, 4th edition, 2002.
• A. Paulraj, R. Nabar, and D. Gore, Introduction to Space-Time Wireless Communications.
  Cambridge University Press, 2003.
• H. L. V. Trees, Optimum Array Processing.
  John Wiley & Sons, Inc., New York, NY, 2002.
• M. Haardt, Efficient One-, Two-, and Multidimensional High-Resolution Array Signal Processing.
  Shaker Verlag GmbH, 1996, ISBN: 978-3-8265-2220-8.
• G. Strang, Linear Algebra and Its Applications.
  Thomson Brooks/Cole Cengage learning.
• G. Strang, Introduction to Linear Algebra.
  Wellesley - Cambridge Press, Fifth Edition.
• L. L. Scharf, Statistical Signal Processing.
  Addison-Wesley Publishing Co., 1991.
• S. M. Kay, Fundamentals of Statistical Signal Processing, Estimation Theory.
  Prentice-Hall, Englewood Cliffs, N.J., 1993.
• M. Haardt, M. Pesavento, F. Roemer, and M. N. El Korso, Subspace methods and exploitation of special array structures.
  in Academic Press Library in Signal Processing: Volume 3 - Array and Statistical Signal Processing (A. M. Zoubir, M. Viberg, R. Chellappa, and S. Theodoridis, eds.), vol. 3, pp. 651 - 717, Elsevier Ltd., 2014, Chapter 15, ISBN 978-0-12-411597-2 ISBN: 978-3-8265-2220-8.