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# ECE 6250, Advanced Topics in DSP

## Fall 2011

**Instructor**

Justin Romberg

Office: Centergy 5245

Phone: 404-894-3930

**Description**

ECE 6250 is a general purpose, advanced DSP course designed to follow an introductory DSP course. The central theme of the course is the application of tools from linear algebra to problems in signal processing.

**Outline**

I. Sampling and filter banks

the Shannon-Nyquist sampling theorem

multirate digital signal processing

filter banks and the discrete wavelet transform

II. Signal representations in vector spaces

linear vector spaces, linear independence, and basis expansions

norms and inner products

orthobases, the reproducing formula, and Parseval’s theorem

signal approximation in an inner product space

Gram-Schmidt and the QR decomposition

III. Linear inverse problems

introduction to linear inverse problems, examples

the singular value decomposition (SVD)

least-squares solutions to inverse problems and the pseudo-inverse

stable inversion and regularization

weighted least-squares and linear estimation

least-squares with linear constraints

IV. Computing the solutions to large-scale least-squares problems

structured matrices

steepest descent

the conjugate gradient method

V. Low-rank updates for streaming solutions to least-squares problems

recursive least-squares

the Kalman filter

VI. Matrix approximation using least-squares

low-rank approximation of matrices using the SVD

total least-squares

principal components analysis

VII. Beyond least-squares (topics as time permits)

norm approximation problems

non-smooth regularization for linear inverse problems

independent components analysis

linear programming for L1 and Linfinity problems