## Advanced Topics in DSP

**Instructor**

Justin Romberg

Email: jrom@ece.gatech.edu

Office: Centergy 5227

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. Signal representations in vector spaces

– introduction to discretizing signals using a basis

– first examples: the sampling theorem, Fourier series, splines

– linear vector spaces, linear independence, and basis expansions

– norms and inner products

– orthobases and the reproducing formula

– Parseval’s theorem and the general discretization principle

– important orthobases: DCT, lapped orthogonal, wavelets

– non-orthogonal basis expansions: polynomials, B-splines

– signal approximation in an inner product space

– Gram-Schmidt and the QR decomposition

II. 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

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

– LU and Cholesky decompositions

– structured matrices: Toeplitz, diagonal+low rank, banded

– large-scale systems: steepest descent

– large-scale systems: the method of conjugate gradients

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

– recursive least-squares

– the Kalman filter

– adaptive filtering

V. Matrix approximation using least-squares

– low-rank approximation of matrices using the SVD

– total least-squares

– principal components analysis (PCA)

– signal and noise subspaces estimation in array processing

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

– approximation in non-Euclidean norms

– regularization using non-Euclidean norms

– recovering vectors from incomplete information (compressed sensing)

– recovering matrices from incomplete information (matrix complettion)