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MATH 416 , Spring 2016: Notes and Summary

 Lecture Date What was covered Notes Textbook Section 1 Jan. 26 Introduction canceled due to snow 1.1 2 Jan. 28 Introduction: Numbers and algebraic structures get assignment 1 1.1 3 Feb. 2 Introduction: Functions, sequences and matrices - 1.2 4 Feb. 4 Vector Spaces, Linear Independence, Spanning, Basis return assignment 1; get assignment 2 2.1 5 Feb. 9 Vector spaces: Basis theorem; Norms and Banach space - 2.1 6 Feb.11 Scalar products and Hilbert spaces; Cauchy-Schwarz inequality; Schauder basis; Orthonormal basis;Gramm-Schmidt orthonormalization; biorthogonal basis. - 2.2 7 Feb.16 Subspaces, Linear transformations, Matrix representation, Matrix Algebra return assignment 2; get assignment 3. 2.2 8 Feb.18 The adjoint operator; orthogonal projections - 2.2 9 Feb.23 Projections onto range and kernel of linear operators; direct sums and orthogonal decompositions return assignment 3; get assignment 4 2.2 10 Feb.25 Fourier series: Definition, ONB, convergence, inversion. Matlab Code 3.1.3 11 Mar. 1 Fourier transform: Definition, Plancherel Theorem, Riemann-Lebesgue Lemma, Inversion return assignment 4; get assignment 5. 3.1.3 12 Mar. 3 Continuous Fourier Transform: Windowing; smoothing vs decay 3.2 13 Mar. 8 Discrete Fourier Transforms: DFT, FFT return assignment 5 3.2 - Mar. 9 REVIEW SESSION - Wed. Mar. 9, 3:30pm-5:00pm in CSIC 4122 14 Mar.10 - First In-Class Test - 15 Mar. 22 Discrete Hartley Transform; Discrete Cosine Transforms get assignment 6 3.2 16 Mar.24 Sampling: Lagrange interpolation - 4.1.1, 4.1.2 17 Mar.29 Sampling: Bandlimited functions; Shannon's sampling formula return assignment 6; get assignment 7 4.1.3 18 Mar.31 Signal Estimation; Measurement, noise, likelihood, MLE, unbiased estimators - 4.2 19 Apr. 5 Signal Estimation: MAP and MMSE return assignment 7; get assignment 8 5.1.2,5.1.3 20 Apr. 7 The Continuous Wavelet Transform; Scalogram; Inversion of the CWT - 5.2.1 21 Apr.12 Discrete Wavelet Transform: frames to basis; MRA (1) return assignment 8; get assignment 9 5.2.1, 5.2.2, 5.2.3 22 Apr.14 MRA (2): Spaces Vj - 5.2.3 23 Apr.19 MRA (3): Spaces Wj return assignment 9; get assignment 10 5.3 24 Apr.21 MRA (4): Linear Time-Invariant Filters - 25 Apr.26 Synthesis Filter banks: scaling functions and wavelets return assignment 10; get assignment 11 26 Apr.28 Applications: Image analysis/synthesis 27 May. 3 Application: Interplay between statistical signal processing and linear transforms return assignment 11 - May. 4 REVIEW SESSION - Wed. May 4, 3:00pm-4:30pm in CSIC 4122 28 May 5 - Second In-Class Test 29 May.10 Uncertainty Inequalities: Heisenberg principle; CRLB Last Class 4.2.2 - May.10 REVIEW SESSION - Tues. May 10, 2:00pm-4:00pm in MATH B0431 - May. 12 FINAL EXAM FINAL EXAM 8:00am-10:00am in CSIC 4122