ECE 513 Course Goals and Instructional Objectives
Course Goals
Mathematical tools in a vector space framework, including: finite and infinite dimensional vector spaces, Hilbert spaces, orthogonal projections, subspace techniques, least-squares methods, matrix decomposition, conditioning and regularizations, bases and frames, the Hilbert space of random variables, random processes, iterative methods; applications in signal processing, including inverse problems, filter design, sampling, interpolation, sensor array processing, and signal and spectral estimation.
Instructional Objectives
- Matrix inversion: orthogonal projections; left and right inverses; minimum-norm least squares solutions; Moore-Penrose pseudoinverse; reularization; singular value decomposition; Eckart and Young theorem; total least squares; principal components analysis
- Projections in Hilbert space: Hilbert space; projection theorem; normal equations, approximation and Fourier series; pseudoinverse operators, application to extrapolation of bandlimited sequences
- Hilbert space of random variables: spectral representation of discrete-time stochastic processes; spectral factorization; linear minimum-variance estimation; discrete-time Wiener filter; innovations representation; Wold decomposition; Gauss Markov theorem; sequential least squares; discrete-time Kalman filter
- Power spectrum estimation: system identification; Prony's linear prediction method; Fourier and other nonparametric methods of spectrum estimation; resolution limits and model based methods; autoregressive models and the maximum entropy method; Levinson's algorithm; lattice filters; harmonic retrieval by Pisarenko's method; direction finding with passive multi-sensor arrays
Office Hours: Thursday 11-12 or by appointment.
No other question has ever moved so profoundly the spirit; no other idea has so fruitfully stimulated the intellect; yet no other concept stands in greater need of clarification than that of the infinite. David Hilbert