Topical Outline:
Basic techniques for modeling discrete-time sequences (3 weeks)
Problem formulation
The direct (least squares) method
The Pad\'{e} approximation
Prony's method
Shanks' method, iterative prefiltering
All-pole modeling and linear prediction
The autocorrelation and covariance methods
FIR least squares inverse filter design
Applications and examples
Fast algorithms for solving Toeplitz equations (3 weeks)
The Levinson-Durbin recursion
Step-up, step-down, inverse Levinson-Durbin recursion
Minimum phase property of PEF
Cholesky decomposition of autocorrelation matrix and its consequences
Lattice filters
The Levinson recursion
The Trench algorithm and the Schur recursion - optional
Split Levinson recursion and line spectral pairs - optional
Fast covariance algorithm - optional
Applications and examples
Lattice methods (2 weeks)
Lattice filters (FIR, all-pole, and pole/zero)
Forward and backward covariance methods
The Burg recursion and the modified covariance algorithm
Examples
Application - wave propagation in layered material
Wiener filtering (2 weeks)
Review of Discrete-time random processes
FIR Wiener filters
Noncausal IIR Wiener filters
Causal Wiener filters
Applications - Linear prediction, deconvolution, smoothing
Power spectrum estimation (4 weeks)
Classical methods
The minimum variance method
The maximum entropy method and relation to minimum variance method
Parametric spectrum estimation
Comparison of methods
Subspace methods
Applications
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