Course Notes

Course Notes

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Course Notes for ECE 6250, Fall 2011

I. Sampling and Filter banks
   Notes 1, the Shannon-Nyquist sampling theorem
   Notes 2, upsampling and downsampling
   Notes 3, changing the sampling rate digitally
   Notes 4, the Shannon and Haar filter banks
   Notes 5, quadrature mirror filter banks, orthogonality,
       vanishing moments
   Notes 6, time-frequency tilings, the discrete wavelet transform,
       and non-orthogonal filter banks

II. Signal Representations in Vector Spaces
   Notes 7, linear vector spaces and bases
   Notes 8, inner products, orthobases, and signal approximation
   Notes 9, Parseval, JPEG case study, and the Gram-Schmidt algorithm
   Notes 10, Shannon and Haar wavelet bases
   Notes 11, wavelets and filter banks
   Notes 12, wavelets, locality, and vanishing moments; 2D wavelets

III. Linear Inverse Problems
   Notes 13, examples of linear inverse problems
   Notes 14, eigenvalue decompositions, solving sym+def systems
   Notes 15, the SVD and the least-squares problem
   Notes 16, the pseudo-inverse, stability of least-squares in noise
   Notes 17, stable reconstruction with the truncated SVD, Tikhonov regularization, and linearly constrained least-squares
   Notes 18, weighted least-squares and the best linear unbiased estimator (BLUE)

IV. Computing the Solution to Least-Squares Problems
   Notes 19, structured systems: circulant, Toeplitz, and identity+low-rank
   Notes 20, steepest descent
   Notes 21, conjugate gradients
   See also Jonathan Shewchuk’s excellent manuscript on CG

V. Streaming Solutions for Least-Squares Problems
   Notes 22, recursive least squares (updated Nov 30, 10:30a)
   Notes 22a, RLS and the BLUE
   Notes 23, the Kalman filter

VI. Matrix Approximation using Least-Squares
   Notes 24, low-rank approximation using the SVD and total least squares
   Notes 25, principal components analysis