ECE 735 Signal Synthesis and Recovery Techniques, Fall 1998
See
what we did in Fall 1996. Spring
1998 will be quite similar.
m1.m
m2.m
m3.m
rootfcn.m
vprint.m
Copy these files. Then in MATLAB run the scripts m1, m2, and m3.
Hgen.m
Hsol1.m
Hsol3.m
yvec.dat
Copy these files. Then in MATLAB run the scripts Hgen, Hsol1, m2, and Hsol3.
mods1.m
mods3.m
Copy these files. Then in MATLAB run the scripts Hgen, mods1, and mods3.
Spline Interpolation M-
file
Function to be Interpolated M-file
Least Squares Spline Approximation M-file
Syllabus
Class Schedule for Fall 1998
9/2 Wednesday. Broke class into groups for HW#1 due Wed. 9/9.
Introduction to fields and vector spaces. Concepts of metric,
norm, and inner product for two-dimensional Euclidean space.
Examples of function spaces.
9/7 Labor Day - No class.
9/9 Groups discussed their solutions to HW #1 in class.
We have now covered Sections 1-6 in "ECE 735 Course Notes".
We just started Section 7.
Assigned HW problem on board.
Also assigned Problems 2-1 and 7-1 DUE Wed. 9/16.
9/14 Covered pages 5-7 through Theorem 7.20.
9/16 Covered p. 8. Worked Problem 7-10 on p. 9. Skipped all text
after Def. 7.31 until Theorem 7.33. Discussed Theorem 7.33.
Assigned HW Problems 7-8 and 7-9. Also: Prove that if
f is continuous, then it is convergence preserving.
DUE Wed. 9/23.
9/21 Th. 7.34 - a continuous function on a sequentially compact set
achieves its max and min values. Skip p. 10 and most of 11.
Briefly discussed uniform continuity. Section 8: Normed vector
spaces, projections, especially onto finite dimensional subspaces.
Assigned HW handout and Problem 8-3 DUE Wed. 9/30.
9/23 Discussed Example 8.10 on uniform polynomial approximation.
Inner product spaces. Cauchy-Schwarz ineq., parallelogram law,
polarization identity. Orthogonality Principle (OP). Used OP
to solve the continuous-time Wiener filter problem.
Assigned HW Problems 9-1, 9-2 DUE Wed. 9/30.
9/28 Section 9.1 Projections onto finite-dimensional spaces.
Apply linear estimation of random vectors, exponential transients,
L^2 polynomial approximation, finite Fourier series.
Section 9.2 Projections onto infinite-dimensional spaces.
9/30 Continued projections onto infinite-dimensional spaces. Application
to Fourier series. The projection theorem.
Assigned HW Handout,
including Problems 9-10, 9-12, 9-13, 9-14, 9-15 DUE Wed. 10/7.
10/5 Class attended seminar by Gerald Matz on "A Time-Frequency Approach
to Minimax Robust Estimation of Nonstationary Random Signals," held in
4610 Engr. Hall.
10/7 Went over HW solutions. Section 10: Linear functionals and the
Riesz Representation for Hilbert Spaces.
Assigned HW Problems 9-16, 10-5 DUE Wed. 10/14.
10/12 Section 11: Linear Operators, Section 12: Adjoint Operators.
Assigned HW on blackboard DUE Wed. 10/21.
10/14 Finished Section 12. Covered Section 13.1.
Assigned HW Problems 12-2, 12-6 DUE Wed. 10/21.
10/19 Section 13.2, Linear Equations with Many Solutions.
Class Notes on Lagrange Multipliers.
10/21 More Lagrange multipliers - derivatives.
Examples of computing Frechet derivatives.
Assigned HW Handout DUE Wed. 10/28.
10/26 Convexity and Lagrange multipliers. Showed that if f is convex,
then f(y) >= f(x) + (Df)(x,y-x). Worked capacitor example and
signal design problem. Proved Theorem 12.
10/28 Worked more derivative examples - l^1, nonlinear Gateaux derivative
Assigned HW Handout (convexity problem and showing that a matrix
operator is a bounded linear operator) DUE Wed. 11/4.
10/30 (Make up lecture) Lecture on first and second derivatives,
gradient, derivative of the gradient.
11/2 Reviewed lecture of 10/30 and revised Lagrange Multiplier notes.
Started notes on Spectral Theory of Bounded Linear Operators.
11/4 Proved existence of eigenvalues and the Spectral Theorem.
11/9 Go over Exam 1 Review.
11/11 Exam 1 - in class.
11/16 Go over exam 1. Proved Lemmas 4 & 5 in Spectral Theorem notes.
Went over square root of an operator.
11/18 Solving 2nd kind Fredholm equations. The singular-value decomposition.
11/23 Class canceled.
11/25 Applications of SVD to solving A x = y, to regularization,
and to min ||y-A x|| subject to ||x||^2 <= c, a Lagrange
multiplier problem.
11/30 Talked about MATLAB files m1.m, m2.m, m3.m
and Hgen.m, Hsol1.m, and Hsol3.m
Piecewise Polynomial Functions notes, sections 1 and 2.
Assigned: Handout p. 1, 1-7 and Run m1, m2, m3 (nothing to turn in).
Also run Hgen, Hsol1, m2, Hsol3 with c = 15 and c = 20.
In each case plot xlambda and give energy in xlambda.
Plot yvec and ylambda DUE Wed. 12/9.
12/2 Section 3: B-Splines
Assigned: Now run mods1 and mods3 in place of Hsol1 and Hsol3.
First try rank r = 35 and plot xlambda.
Then try rank r = 33 and plot xlambda. DUE Wed. 12/9.
12/7 Teaching evaluations.
Collocation and Galerkin methods for solving integral equations.
Generalized eigenvalue problem.
12/9 Go over Exam 2 Review.
12/14 Exam 2 - in class.