ECE 735 Signal Synthesis and Recovery Techniques, Fall 2000


 Instructor Office Hours: TU 10:30-11:30, W 9:00-10:30, or by appointment.
See what we did in Fall 1998. Fall 2000 will be quite similar.
Syllabus
Homework Solutions    Exam 1 Review Problems    Limit of Compact Operators
Class Schedule for Fall 2000
9/6     Wednesday.  Course overview.
	Assigned HW #1 (group problems) DUE WED 9/13.

9/8	Gave a brief overview of integral equations, regularization, law of
	cosines.  Covered Sections 1.1-1.3 in the notes.

9/11 Covered Sections 1.5, 2.1, and 2.2. 9/13 Student presentations. Assigned HW #2: Problems 2-4, 2-5, 2-6, 2-7, 2-8; 4-2, 4-4 DUE WED 9/20. 9/15 Covered Section 3, The Real Numbers. You may skip the proof of Theorem 3.1. Briefly discussed inner-product spaces, normed spaces, and metric spaces, including examples.
9/18 Started Section 4, Metric Spaces, pp. 7-9. 9/20 Covered Theorems 4.19 and 4.20 and their applications. Cauchy sequences. Completeness. Assigned HW #3: Problems 4-9, 4-10, 4-11, 4-12, 4-13(only =>) DUE WED 9/27. 9/22 Sequential compactness, completeness of the real numbers and finite dimensional Euclidean spaces. Section 4.4 Continuous Functions. Skip p. 11 except for Theorems 4.34 and 4.35.
9/25 Discussed Theorems 4.34 and 4.35. Worked Problem 4-14. On p. 12 the only thing you need to note is Corollary 4.40. On p. 13 it is important to be aware of Theorem 4.43. Also, Theorem 4.44 is important. Began discussion of normed vector spaces on p. 14. 9/27 Section 5, Section 5.1 Projections - Introduction. Started Section 5.2 Finite-Dimensional Subspaces. Assigned HW #4: Problems 5-1, 5-3, 5-5, 5-6, 5-7, 5-8 DUE WED 10/4. 9/29 Went over old HW. Started proof of Lemma 5.4
10/2 Finished proof of Lemma 5.4. Skipped proof of Lemma 5.7. You should read the proof of Theorem 5.8. Covered proof of Theorem 5.9. 10/4 Went over HW Problems 5-6, 5-7, and 5-8. Covered proof of Theorem 5.10. Skipped Example 5.11. Started Section 6. Assigned HW #5: Problems 6-5, 6-6, 6-8, 6-11, 6-12, 6-13, 6-14 DUE WED 10/11. 10/6 Derived Cauchy-Schwarz inequality and did matched filter example. Stated Orthogonality Principle and addressed HW#1 problem 5. Also showed that the projection of a function onto the even functions is given by taking the even part of the function.
10/9 Derived Orthogonality Principle (one direction only; you should read the whole proof in the notes). Derived the Wiener filter using the orthogonality principle. Covered Section 6.1: Projections onto Finite-Dimensional Spaces. 10/11 Covered Section 6.2, relation to Fourier series. Started Section 6.3, The Projection Theorem. 10/13 Finished proof of Projection Theorem. Used theorem to prove existence of conditional expectation. Proved Prop. 6.13. Skipped Theorem 6.14 and Lemma 6.15. Assigned HW #6: Problems 6-18, 6-20, 6-21 DUE Wed 10/18.
10/16 Discussed statement of Lemma 6.15 (Farkas). Started Section 7. Skipped proof of Prop. 7.4. Covered material up to Def. 7.8. Discussed controllability problem. Showed that point evaluation is a bounded linear functional on C[0,1] equipped with the uniform (=infinity) norm, but not with the L^1 norm. 10/18 Started with Def. 7.8. Worked Problem 7-8. Covered Section 7.2 on Linear Operators. Gave an example of a 1-1 operator that is not onto L^2[-pi,pi]. Assigned HW #7: Problems 7-4, 7-20, 7-22 DUE Wed 10/25. 10/20 Covered Section 7.3, Adjoint Operators, through Lemma 7.20. Illustrated use of Problem 7-20.
10/23 Discussed an example the uses the results in Problem 7-20. Covered Def. 7.22, Problem 7-21, Ellipses on p. 26. Started Section 7.4, Solving Linear Equations. 10/25 Finished Section 7.4.1. Also worked example (A x)(t) = \sum_{j=1}^n x_j B_j(t). 10/27 Finished Section 7, Subsections 7.4.2 and 7.4.3.
10/30 Distributed handout with revised Sections 6.1, 7.3.2, 7.4, and 7.5. Discussed the handout, including the pseudoinverse. 11/1 Start Section 8. Proved Lemma 8.2, existence of eigenpairs. You need to read over the proofs of Lemmas 8.1 and 8.3. Started discussion of the Spectral Theorem. Assigned HW #8: 8-1, 8-2 DUE Wed 11/8. 11/3 Proved the Spectral Theorem, discussed eig function in Matlab, discussed square root of an operator.
11/6 Discussed 2nd-kind Fredholm equations (p. 32.). Derived SVD from the Spectral Theorem. 11/8 Finished discussion of SVD. Showed that 1st kind Freholm equations are ill posed, while 2nd kind equations are well posed. 11/10 Discussed some review problems, Solutions to HW #8. Started Section 8.3, Regularization.
11/13 Solved problems from review sheet. 11/15 Review for exam Night Exam in 2535 EH, 7:15-8:45. Assigned HW #9: DUE Wed 11/22. 11/17 Went over Exam. Started discussion of Theorem 9.1, Lagrange multipliers.
11/20 Proved Theorem 9.1. Started Section 9.2 11/22 Worked several examples of computing Frechet derivatives. Discussed the chain rule and the product rule of differentiation. 11/24 NO CLASS, Thanksgiving Recess
11/27 Discussed Kuhn-Tucker Sufficiency Theorem and Example 9.18. 11/29 Finished discussing Example 9.18. Discussed Frank Kelly's paper. 12/1 Discussed Theorems of the Alternative.
12/4 Distributed handout (dated Dec. 4, 2000) re-organizing material from Section 9 and adding some new material. Covered Feasible Directions, Lemma 9.15 (Abadie), Kuhn-Tucker Necessity Theorems 9.23 9.24, and started 9.26. 12/6 Proved Theorems 9.23 and 9.26. Discussed Kelly's paper again. 12/8
12/11 12/13 Answered questions. 12/15 FRIDAY, Last Class Day
12/18 12/19 TUESDAY, Final Exam, 12:25 pm in 2540 EH