Overview
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
