# 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, signal and spectral estimation, sparse approximation and compressed sensing.

## Instructional Objectives

**Inverse problems and matrix theory:**linear inverse problems; orthogonal projections; left and right inverses; minimum-norm least squares solutions; Moore-Penrose pseudoinverse; singular value decomposition; conditioning and regularization, Eckart and Young theorem; total least squares; principal components analysis**General linear vector spaces**: finite and infinite dimensional vector spaces; Hilbert spaces; projection theorem; inverse problems in infinite dimensional vector spaces; approximation and Fourier series; pseudoinverse operators; iterative methods for optimization and inverse problems; bases and frames for signal representation;**Hilbert space of random variables**(*6 hours*): random processes; least-squares estimation; Wiener filtering; Wold decomposition; 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
**Applications in signal processing**: deconvolution, optimal filter design, temporal and spatial spectrum estimation, tomography, harmonic retrieval, subspace methods, sensor array processing, extrapolation of band-limited sequences, generalized sampling, wavelets, splines, subset selection, sparse approximation and compressed sensing.

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

ECE 513 (Vector Space Signal Processing) covers 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, signal and spectral estimation, sparse approximation and compressed sensing. Course Information: Prerequisite: ECE 310, ECE 313, and Math 415.