ECE 513 - Vector Space Signal Processing

Semesters Offered

Official Description

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. Course Information: Prerequisite: ECE 310, ECE 313, and MATH 415.

Subject Area

Signal Processing

Course Director

Description

Fundamentals of linear least squares estimation of discrete-time signals and their spectra: minimum-norm least squares and total least squares solutions; singular value decomposition; Wiener and Kalman filtering; autoregressive spectral analysis; and the maximum entropy method.

Topics

  • 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

Detailed Description and Outline

Topics:

  • 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

Texts

Class notes.

Recommended:
B. Porat, Digital Processing of Random Signals, Prentice-Hall, 1994.

Last updated

2/13/2013