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ECE298-ComplexLinearAlg-F21

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 University calendar; Academic calendar; Fall 2021
Summary of class status as of Oct 24, 2021

ECE 298 ComplexLinearAlg-F21 Schedule (Fall 2021)

Part I: Lecture + videos: Complex algebra (Calendar week 42-44; 8 Lecs)
WeekMWF
42L1: Intro + Overview (Lec1-360-F20 Starts @ 6:30 min with 360 reset @16 min;)L2: Roots of Polynomials Lec2-360-F21@6:30 min; summary of 360 problems @5 mins; (Lec2-360-S21 @1:25; Lec2-3600-F20 @3 min)L3: Companion Matrix + Examples Lec3-360-F21 @6:20 min, Summary of 360 issues @2min (Lec3-360-S21-NoAudio @5:30; Lec3-360-F20 @2 min)
43L4: Eigen-analysis, analytic solution Lec4-360-F21 @0:10, (Lec4-360-S21 @1:43; Lec4-360-F20)L5: Eigen-analysis Lec5-360-F2 @00:151;, (Lec5-360-S21; Lec5-360-F20 @2 min)L6: Eigen-analysis; Taylor series & Analytic functions Lec6-360-F21, (Lec6-360-F20 @2, Lec6-360-F20)
44L7: 3.9,.1 \({\cal FT}\) of signals vs. systems Lec 7 (No Audio),
360-L7-360.F21 S21 @5:30, (F20, Lec6-360@1:45, ECE-493: F20, L11-360 @10 min); STFT (Example: Remote Control)
L8: Impedance (L8-360 S21 @9:45, F20: L8-360)L9: Exam I (NS1, AE1, Schwarz inequality (p. 118, 124); probs AE3 (#9, #10) relevant HW solutions:
Room: ECEB 2015: Time 1-4 PM
L D Date Lecture and Assignment

Part I: Introduction to 2x2 matricies (8 Lectures)
1 M 10/18 Lecture: Introduction & Overview: (Read \(\S\)1 (Chap. 1, p. 1-17) & Intro + history; \(\S\)3.1 (p.69-73);
Homework 1: NS-1: (not graded) pdf, NS1-sol.pdf,NS-3: Prob #3, #4 pdf, NS3-sol.pdf
NS-1 & NS-3 Due on Lec 4
2 W 10/20 Lecture: Roots of polynomials; Matlab Examples; Allm.zip;
Read: \(\S\)3.1 (p. 73-82) Roots of polynomials+monics; Newton's method.
3 F 10/22 Lecture: Companion Matrix;
Pell's equation: \(x^2_m-N y^2_n=1\) with \(x_n,y_n,n,N\in{\mathbb N}\), Fibonacci Series \(f_{n+1} = f_n + f_{n-1}\), with \(n,f_n \in{\mathbb N}\) (p. 57-64),
move or merge with Lec 5: Companion matrix; (@39 mins): Pell & Fibonocci companion Matrix (p. 84-86) + solutions;
Read: \(\S\)3.1.3,.4, (p.84-88) More on Monic roots; (Lec-3 video @23 mins)
4 M 10/25 Lecture: Eigen-analysis; Examples + analytic solution (Appendix B.3, p. 310)
Next time merge Lec 4 & 5 too much overlap
Read: \(\S\)3.2,.1,.2, B1, B3 Eigen-analysis, (p. 88-93)
Homework 2: AE-1: Problems: pdf, Due on Lec 8; AE1-sol.pdf
AE-1 Due on Lec 8
NS-1 Due
5 W 10/27 Lecture: Solution of Pell's and Fibinocci's Eqs., (p. 57-64); (See Lec 3 @44 min)
Read: \(\S\)3.2.3;Eigen-analysis (Appendix B.3)
6 F 10/29 Lecture: Eigen-analysis; Taylor series (\(\S3.2.3\)) & Analytic functions (\(\S3.2.4\));
History: Beginnings of modern mathematics: Euler and Bernoulli, The Bernoulli family: natural logarithms; Euler's standard circular-function package (log, exp, sin/cos);
Brune Impedance \(Z(s) = z_o{M_m(s)}/{M_n(s)}\) (ratio of two monics) and its utility in Engineering applications; Examples of eigen-analysis.
Read: \(\S\)3.2,.3,.4, Eigen-analysis: Taylor series (\(\S3.2.3\), p. 93-8) & Analytic Functions (p. 98-100)
Homework 3: AE-3: Problems: #1, 2, 3: 2x2 complex 2-port matrices; #4: FTA; #5-8: Complex algebra & scalar products; #9, 10: Schwarz Inequality
AE3.pdf, Due on Lec 10, AE3-sol.pdf
7 M 11/1 Lecture: Fourier transforms for signals vs. Laplace transforms for systems:
Read: \(\S\)3.9 (p. 152-6); Fourier Transform (wikipedia);
Notes on the Fourier series and transform from ECE 310 pdf including tables of transforms and derivations of transform properties;
Classes of Fourier transforms pdf due to various scalar products.
Read: \(\S\)3.10
8 W 11/3 Lecture: The important role of the Laplace transform re impedance: \(z(t) \leftrightarrow Z(s)\);
Read: \(\S\)3.2.5,.3.10, Impedance (p. 100-1) & \(\cal LT \);
Read: Hamming Digital filters: The idea of an Eigenfunction,; Impedance and Kirchhoff's Laws
Fundamental limits of the Fourier vs. the Laplace Transform: \(\tilde{u}(t)\) vs. \(u(t)\)
The matrix formulation of the polynomial and the companion matrix
Complex-analytic series representations: (1 vs. 2 sided); RoC of \(1/(1-s), 1/(1-x^2), -\ln(1-s)\)
1) Series; 2) Residues; 3) pole-zeros; 4) Continued fractions; 5) Analytic properties
AE-1 Due
AE-3 Due Lec 10
9 F 11/5 Exam 1: 1-4 PM: Via zoom (ONL) or 2015-ECEB (CLA); Exam Emailed via zoom or Download from Gradescope; Paper or pdf copy provided upon request.
Topics: Homework 1-3: NS-1 & NS-3, AE-1, AE-3 (selected problems) Problems: FT vs LT; Residues; Power series, RoC;
Part II: Lecture + videos: Complex algebra (12 Lecs)
WeekMWF
45L10: Integration in complex plane: FTC & FTCC F21:L10-360 (no public), F21:L10-360 (?broken?), F21:L10-360 (@6:00min) ( login), (S21: L9-360 (@2m), F20: L9-360)L11: 3.2,.4,.5 Complex Taylor series, Residues, Convolution; More on FTCC?: L11-360.F21 @1:00 (L10-360, S21, L10-360, F20 @4:00)L12: 3.10,.1-.3 Complex analytic functions L12-360-F21: Audio OFF >@2:00 & ON@16:21 etc,
(L11-360 @5m, S21; L11-360, F20)
46L13: 3.11,.1,.2 Multi-valued functions; Domain coloring L13-360, F21, (L13-360, S21; L13-360, F20)L14: 3.5.5, 3.6,.1-.5 1) Riemann's extended plane 2) Schwarz inequality; 3) Triangle inequality; L14-360, F21, (L14-360,F20)L15: Cauchy's intergral thms CT-1,2,3; L15-360.F21 @3:30; L15-360.S21 (audio @18:00m); (L15-360, F20)
47L16: Transmission line problem Lec16-360, F21 No Audio, (Lec16-360, S21; Lec16-360, F20)L17: Wave function \(\kappa(s)\) when sound speed depends on frequency; L17-360,F21 Audio@2:45, Video@31:30; (Lec17-III-360, S21; Lec17-360, F20 @4min (Inv LT: \(t<0\)))L18: LT (t>0) Lec18 360-F21 @3:00; Lec18 360-S21; Lec18 360-F20
48L19: S21: Review for Lec19-360-F21 @3:00; Final Exam (360-S21);
L20: LT Properties: (Lec20-III-360-F20)Thur: Reading Day: Optional review for final Student Q&A 1-2 PM Review F20
L D Date Lecture and Assignment
Part II: Complex analytic analysis (6 Lectures)
10 M 11/8 Lecture: Fundamental theorem of real vs. complex calculus; Differentiation in the complex plane: Complex Taylor series;
Cauchy-Riemann (CR) conditions and differentiation wrt \(s\) (p. 180-182)

Discussion of Laplace's equation and conservative fields: (1, 2)
Homework 4: DE-1: Problems: #1-4 Complex series; #5-6 Cauchy-Riemann (CR) Equations; #7-8 Branch Cuts & Riemann Sheets;
DE-1 (pdf), Due on Lec 15 DE1-sol.pdf;
Read: \(\S\)4.2
AE-3 Due; AE3-sol.pdf

11 W 11/10 Lecture: Integration in the complex plane: FTC vs. FTCC; CR conditionsComplex analytic functions, e.g.: \(z(t) \leftrightarrow Z(s)\); FTC, FTCC (\(\S\)4.1, 4.2), Analytic vs complex analytic functions and Taylor formula and Taylor Series (p. 93-98); Calculus of the complex \(s=\sigma+j\omega\) plane: \(dF(s)/ds\); \(\int F(s) ds\) (Boas, p. 8);
Text \(\S\)3.2.3: The convergent of the analytic power series (ROC): Residues; power series, RoC; LT;
Read: \(\S\)3.2.6 (p. 101-3)
12 F 11/12 Lecture: Continue Lec 11
Homework 5: DE-2 Problems: Integration, differentiation wrt \(s\); Cauchy theorems; LT;
DE-2 pdf, Due on Lec 15; DE2-sol.pdf
DE-1 due on Lec15
13 M 11/15 Lecture: Multi-valued functions: Visualizing complex valued functions Domain coloring, Branch cuts & points, Riemann Sheets, Colorized plots of rational functions; Software: Matlab: Working with Octave/Matlab: 3.1.4 (p. 86): zviz.m, zviz.zip, python
Read: \(\S 3.11\) (p. 167)
14 W 11/17 Lecture: Riemann’s extended plane: The Riemann sphere (1851) pdf; Multi-valued functions; Branch points and cuts;Mobius Transformation: (youtube, HiRes), pdf description
Mobius composition transformations, as matrices
Read: \(\S 3.11.1\) (p. 170)
15 F 11/19 Lecture: Cauchy’s Integral theorems (CT-1,-2,-3) & Formula:
Read: \(\S\)4.6.3 (p. 203-209)
Thanksgiving Break
16 M 11/29 Lecture: Scalar wave equation & WHEN (p. 246-248)
Homework 6: DE-3: Problems: Inverse LT; Impedance; Transmission lines;
DE3 pdf, Due on Lec 19; DE3-sol.pdf
Read: \(\S 3.5.4, 5.2\) (p. 123-125
DE-1 & DE-2 due
17 W 12/1 Lecture: Train-mission problem (ABCD matrix method); More on the Cauchy Residue theorem;
Read: \(\S\)4.8.3 (p. 224-226)
18 F 12/3 Lecture: Analysis of the wave propagation function \(\kappa(s)\in\mathbb{C}\) when speed of sound depends on frequency \(s=\sigma + \jmath\omega\).
Examples of the inverse Laplace transform for \(F(s)=1/s, e^{-s T_o}/s, 1/s^2\) using the Cauchy residue theorem
Read: \(\S\)3.1 (p. 69-73)
19 M 12/6 Lecture: Inverse Laplace transform via the Residue theorem \(t > 0\) and \(t < 0\); Case for causality Laplace Transform,
Examples: Convolutions by the step function:LT \(u(t) \leftrightarrow 1/s\) vs. FT \(2\tilde{u}(t) \equiv 1+ \mbox{sgn}(t) \leftrightarrow 2\pi \delta(\omega) + 2/j\omega\)
Read: \(\S\)5.7 (p. 216-220)
DE-3 Due
20 W 12/8 Lecture: Properties of the Laplace Transform: Modulation, convolution; impedance/admittance, poles and zeros \(Z(s)=N(s)/D(s)\); Review
DE3-sol.pdf, Full Solution to train problem.
Read: \(\S\)5.7 (p. 220-222)
- R 12/9 Reading Day Optional student 'Review Q&A session 9-11AM, 1-2PM

- M Final: Dec 14, 8-11 AM
Offical UIUC exam schedule:
- Letter grade statistics: F21, F20, F16

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