ECE 735 Signal Synthesis and Recovery Techniques, Fall 2008

9/3     Wednesday.  Course overview.  In the class notes (see above on
	how to download them), please read the Introduction (pp. 1-2) and
	Sections 1.1, 1.2, 1.3, and 1.4.
	Assigned HW#1 (distributed handout) (group homework) DUE WED 9/10.
	Matlab p-file for Problem 4: yfunFall08.p
	Matlab m-files mentioned in notes: smpthm.m, lincmb.m

9/5	One group presented a solution to Problem 3 of HW#1.  Started Section 1.5.
	Skip proof of Lemma 1.5.  Lemma 1.10 is important.  Its application is
	illustrated in Examples 1.11 (Linear Independence of the Power Functions)
	and 1.12 (Lagrange Interpolation).

9/8	One group presented a solution of Problem 2, and another group
	presented a solution of Problem 1.
	Stated and proved Lemma 1.13 (Interpolation).  Illustrated its
	application in Example 1.14 (Hermite Interpolation).  Skip the rest
	of Section 1.5.

9/10	One group presented a solution of Problem 4.  Started Section 2
	on Function Spaces.  Section 2.1: Definitions of a function, an
	inverse function.  Examples.  Definition of "onto" and "one-to-one".
	Skip Theorem 2.6 and Lemma 2.7.  Section 2.2 Vector Spaces of Mappings.
	Defined addition and scalar multiplication of functions.
	Assigned HW#2: 2-4, 2-7, 2-9 DUE WED 9/17.

9/12	Example 2.10 Methods of Proving Linear Independence.  Discussed Problem 2-10.
	Skip Proof of Lemma 2.11.  Used it to prove Theorem 2.12: A nonzero
	bandlimited waveform cannot be time limited.

9/15	The approximate dimension of spaces of bandlimited waveforms.  Derivation
	of the Sampling Theorem.  Some consequences of the sampling theorem.
	Illustrated the use of some Matlab scripts in the text.  Started
	Section 3 The Real Numbers.  Introduced least upper bound, greatest
	lower bound, supremum, infimum.  Stated Bolzano-Weierstrass Theorem
	and started its proof.

9/17	Completed proof of Bolzano-Weierstrass Theorem.  Started Section 4
	Metric Spaces.  Talked a bit about how to prove things.
	Assigned HW#3: 4-3, 4-4, 4-5, 4-6, 4-9 DUE WED 9/24.

9/19	Proved Prop. 4.6 and 4.7.  Covered Section 4.2, but skip Def. 4.13,
	Theorems 4.14 and 4.15.

9/22	Start Section 4.3 Sequences and Convergence in a Metric Space.
	Stated and proved Theorem 4.19.  Stated Theorem 4.20 (Approximation).
	Gave example of Fourier series.

9/24	Proved Theorem 4.20.  Introduced subsequences, sequential compactness,
	Cauchy sequences.  Showed that a convergent sequence must be Cauchy.
	Showed that a Cauchy sequence must be bounded.
	Assigned HW#4: 4-11, 4-12, 4-13, 4-14 DUE WED 10/1.

9/26	Definition of complete metric space.  Stated Lemma 4.25 that if a
	Cauchy sequence has a converging subsequence, that the sequence itself
	converges.  Theorem 4.26 Completeness of Rd.
	Section 4.4 Fixed Points and Contraction Mappings.

9/29	Section 4.5 Continuous Functions:  We covered through Def. 4.32
	and then jumped to Theorem 4.34, which we discussed along with
	Theorem 4.35.  Started Section 4.6 Compact Sets.  Covered Theorem 4.39
	(Finite Intersection Property), which is used later to prove
	Theorem 6.22 (Finite-Dimensional Separation).

10/1	Showed that a closed subset of a compact set is compact (Problem 4-27).
	Showed that a sequentially compact set is bounded.  Tried, unsuccessfully,
	to show that a compact set is sequentially compact (Corollary 4.42)
	without using Lemma 4.41.  Note that the definition of
	limit point=accumulation point is given in Def. 4.13.
	Assigned HW#5: 4-18(forward part only), 4-21, 4-23, 4-32 DUE WED 10/8.

10/3	Proved Theorem 4.48 that on a compact set, a continuous function is
	uniformly continuous.  Started Section 5 Normed Vector Spaces.  Gave
	several examples.  Solved Problem 5-2(b) that bounded functions under
	the uniform norm make up a Banach space (not quite finished).

10/6	Finished 5-2(b).  Covered Section 5.1 Projections.

10/8	Covered Section 5.2 Finite-Dimensional Subspaces through Problem 5-9.
	Skipped rest of Section 5.2.  Started Section 6 Inner Product Spaces.
	Covered through Corollary 6.5, but delayed proof of Prop. 6.4.
	Assigned HW#6: 5-3, 5-4, 6-3, 6-8, 6-9, 6-11 DUE WED 10/15.

10/10	Proved Cauchy-Schwarz Inequality (Prop. 6.4).  Showed how to use it
	to derive the matched filter.  Stated the Orthogonality Principle (OP)
	and indicated how it could be used to derive the Wiener filter.

10/13	Proved OP, (6.7), and (6.8).  Started Section 6.1 Projections onto
	Finite-Dimensional Spaces.  Emphasized Theorem 6.10 that knowing
	the projection is equivalent to knowing the inner products <v,w_i>
	for i=1,...,n.

10/15	Discussed Example 6.11 and the Remark following it.  Skipped
	Section 6.1.1.  Covered Section 6.2 Projections onto
	Infinite-Dimensional Spaces.
	Review Problems for Exam 1: 2-6, 2-8, 2-10,
	                            4-7, 4-22,
	                            5-6,
	                            6-13, 6-19, 6-20, 6-27, 6-28.

10/17	Section 6.3 The Projection Theorem.

10/20	Finished proof of the projection theorm.  Stated Corollary 6.15,
	Def. 6.16.  Discussed Problem 6-24 and 2 remarks following it.
	Mentioned Problems 6-26, 6-29.

10/22	Answered questions.

10/24	Exam 1 - Starts at 12:40 in 2534 EH.

10/27	Section 6.4 only through Problem 6-35.
	Suggested optional problems:
	6-29, 6-30, 6-32, 6-33, 6-34, 6-35, 6-36(b)(c)(d), 6-38, 6-39.

10/29	Covered Section 7.1.  Started Section 7.2.
	Assigned HW#7: 7-3, 7-4, 7-8, 7-19 DUE WED 11/5.

10/31	Finished Section 7.2.  Started Section 7.3.

11/3	Discussed Problem 7-25 and indicator function trick I_A(t).
	Discussed Problem 7-33, Example 7.25 and remark following it.
	Discussed Problem 7-34.  Introduced notion of a self-adjoint
	operator (Def. 7.29), positive-semidefinite (nonnegative) operator,
	positive-definite operator (Def. 7.30).  Briefly discussed Ellipses
	(Section 7.3.1).  Discussesd Section 7.3.2 Projections onto the
	range of A.

11/5	Discussed Section 7.4 on finding the minimum-norm solution of
	linear equations.  Skipped Section 7.5.  Started Chapter 8 Spectral
	Theory.
	Assigned HW#8: 7-23, 7-24, 7-26, 7-29, 7-45 DUE WED 11/12.
	Suggested optional problems: 7-25, 7-33, 7-35, 7-36, 7-46.

11/7	Mentioned results of Problem 8-12(a)(b), content of Theorem 8.1 and
	Lemma 8.2 and its corollary.  Proved Lemma 8.3.

11/10	Proved the Spectral Theorem.

11/12	Applications of the Spectral Theorem on pp. 52-53.  NOTE: We are
	skipping Sections 8.2 and 8.3.  In Section 8.4, we will cover
	ONLY the left-hand column on p. 54.  Stated the SVD in Section 8.5.
	Assigned HW#9: 7-49, 8-4, 8-10, 8-12(a)(d), 8-14 DUE WED 11/19.
	Suggested optional problems: 7-50, 7-51, 7-52, 8-1, 8-12(c), 8-13.

11/14	You can read the proof of the SVD on your own.  In class we proved
	only the last part of the SVD.  Covered Example and Application to
	Fredholm equations of the 1st kind on pp. 56-57.

11/17	Showed 1st kind equations are ill-posed, but 2nd kind equations are
	well posed (p. 57).  We are skipping Section 8.6.
	Section 8.7 Regularization.  We are skipping Section 8.8.

11/19	Chapter 13, version 1.
	Assigned HW#10: 9-1, 9-2, 9-8, 9-9 DUE WED 11/26.

11/21	Finshed Chapter 13 handout.  Started Section 9.9 The Frechet Derivative.

11/24	Finished Section 9.9.  Discussed Section 9.11 The Chain Rule and the
	Product Rule for Frechet derivatives.  We skipped the proofs.
	From Section 9.12 we covered a variation of the regularization
	problem discussed there.

11/26	Covered Section 9.12 Regularization and Quadratically Constrained
	Least Squares.  Started Section 9.13 and covered first subsection
	on In-Band and Out-of-Band Energy.
	Assigned HW#11: 9-25, 9-27, 9-29, 9-30 DUE WED 12/3.

11/28	Thanksgiving recess -- NO CLASS

12/1	Answered questions.

12/3	Continued Section 9.13.
	Assigned HW#12: download pdf file DUE WED 12/10.

12/5	Distributed teaching evaluations.  Answered questions.

12/8	Illustrated splinetool.  Introduced spline space, truncated power
	functions.

12/10	Discussed smooth subspaces of the space of all piecewise polynomials.
	Review Problems: 7-12, 7-25, 7-27, 7-32, 7-34,
	                 8- 1, 8- 2, 8-11, 8-13, 8-15,
	                 9-18, 9-19, 9-20,
                         pdf file.
	Prof. Robert Gray's Toeplitz and Circulant Matrices: A Review.

12/12	Last day of classes. Answered questions.  Finished discussion of B-splines.

12/18	Thursday.  Final Exam 7:45 am in 2534 EH.