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ECE 298 ComplexLinearAlg-F20 Schedule (Fall 2020)

Part I: Lectures, Reading assignments and videos: Complex algebra (6 Lecs)
43L1: 1 (Read p. 1-17) Intro + history; 3.1 Read p.69-73
(Starts at 7 min: 298-Lec1-360,,
ECE493 Lec1-zoom, Aug 24,2020-zoom)
L2: 3.1,.1,.2, (p. 73-80) Roots of polynomials+Monics; Newton's method.
(Starts @ 3 min: 298 Lec2-360, 298-zoom)
L3: 3.1.3,.4, (p.84-8) More on Monic roots; @23 mins: Companion matrix; @39 mins: Fibonocci Matrix + solution
(Starts @ 2 min: 298Lec3-360, 298-zoom)
44L4: 3.2,.1,.2, B1, B3 Eigen-analysis, (p. 80-4, 88-93)
(Lec4-360, -zoom)
L5: 3.2.3 Eigen-analysis: Solution of Pell's and Fibinocci's Eqs., (p. 57-61, 65-7)
(@ 2min: Lec5-360)
L6: 3.9,.1 \({\cal FT}\) of signals, p. 152-6)
(ECE-298: Lec6-360, ECE-493: L11-360ms)
L D Date Lecture and Assignment

Part I: Introduction to 2x2 matricies (5 Lectures)
1 M 10/19 Lecture: Introduction & Overview:
Homework 1 (NS-1): Problems: pdf, Due on Lec 4; NS1-sol.pdf (Fall 2020: NS-3 to replace NS-1)
2 W 10/21 Lecture: Roots of polynomials; Examples; Allm.zip
3 F 10/23 Lecture: Companion Matrix;
Pell's equation: \(m^2-Nn^2=1\) with \(m,n,N\in{\mathbb N}\), (p. 58, 65-66, 308-310)
Fibonacci Series \(f_{n+1} = f_n + f_{n-1}\), with \(n,f_n \in{\mathbb N}\), (p. 60-61, 66-67)
4 M 10/26 Lecture: Eigen-analysis; Examples + analytic solution (Appendix B.3, p. 310)
NS-1 Due
Homework 2 AE-1): Problems: ... pdf, Due on Lec 7 8; AE1-sol.pdf
5 W 10/28 Lecture: Detailed eigen-analysis example for eigenvalues and eigenvectors (Appendix B)
6 F 10/30 Lecture: Fourier transforms for signals vs. Laplace transforms for systems 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: Class-notes

Part II: Lectures, Reading assignments and videos: Transforms (3 Lecs)
45L7: 3.2,.3,.4, Eigen-analysis: Taylor series (\(\S3.2.3\), p. 93-8) & Analytic Functions (p. 98-100) (L7-360)L8: 3.2.5,.3.10, Impedance (p. 100-1) & \(\cal LT \) (L8-360)L9: 3.2.6 (p. 101-3) Complex analytic functions, e.g.: \(Z(s) \leftrightarrow z(t)\); FTC, FTCC (\(\S\) 4.1, 4.2), (L9-360)
L D Date Lecture and Assignment
Part II: Fourier and Laplace Transforms (3 Lectures)
7 M 11/2 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.
Homework 3: AE-3: Problems 2x2 complex matrices; scalar products pdf, Due on Lec 10, AE3-sol.pdf
8 W 11/4 Lecture: The important role of the Laplace transform re impedance: \(z(t) \leftrightarrow Z(s)\); 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 extended from Lec 7
9 F 11/6 Lecture: Integration in the complex plane: FTC vs. FTCC; 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)
The convergent analytic power series: Region of convergence (ROC)
Homework 4: DE-1: Problems ... Series, differentiation, CR conditions: pdf, DE1-sol.pdf, Due on Lec 12 15

Part III: Lectures, Reading assignments and videos: Complex algebra (12 Lecs)
46L10: 3.2,.4,.5 Complex analytic functions, Residues, Convolution;FTCC: Lec10-360
L11: 3.10,.1-.3 Complex Taylor series; Disc. Exam content (L11-360)
L12: Exam I (NS1, AE1, AE3) HW: AllSol.zip
47L13: 3.11,.1,.2 Multi-valued functions; Domain coloring (L13-360)
L14: 3.5.5, 3.6,.1-.5 Riemann's extended plane (Lec14-360)
L15: Cauchy's intergral thms CT-1,2,3; DE-3, Due on L19 (Lec15-360)
Spring Break
49L16: Transmission line problem (DE-3 Due on L19) (Lec16-360)
L17: Inv LT (t<0) (Lec17-III-360)
L18: LT (t>0) (Lec18-III-360)
50L19: LT Properties (conv, modulation, impedance: \(Z(s)= R(s)+jX(s)\); etc) (Lec19-III-360)
L20: Last day of instruction: Review Assignments; (Lec20-III)
Thur: Reading Day: Optional review for final Student Q&A 9-11, 1-2 PM
L D Date Lecture and Assignment
Part III: Complex analytic analysis (6 Lectures)
10 M 11/9 Lecture: Fundamental theorem of complex calculus; Differentiation in the complex plane: Complex Taylor series;
Cauchy-Riemann (CR) conditions and differentiation wrt \(s\)

Discussion of Laplace's equation and conservative fields: (1, 2)
AE-3 Due

11 W 11/11 Lecture: Multi-valued complex functions; Riemann sheets; Branch cuts
Homework 5: DE-2 Problems: Integration, differentiation wrt \(s\); Cauchy theorems; LT; Residues; power series, RoC; LT; Problems:
12 F 11/13 Exam 1: 8-11 AM: Zoom or 3017-ECEB; Submit to Gradescope; Paper copy upon request
DE-2 (pdf), Due on Lec 15; DE2-sol.pdf
DE-1 due on Lec15
13 M 11/16 Lecture: Multi-valued functions; Riemann Sheets, Branch cuts & points; Domain coloring
Visualizing complex valued functions \(\S 3.11\) (p. 167) Colorized plots of rational functions

Software: Matlab: Working with Octave/Matlab: 3.1.4 (p. 86): zviz.m, zviz.zip, python

14 W 11/18 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
15 F 11/20 Lecture: Cauchy’s Integral theorem & Formula
Homework 6: DE-3: Inverse LT; Impedance; Transmission lines; Problems: ... (pdf), Due on Lec 19;
DE-1 & DE-2 due
- Nov 21-Nov 29 Spring break
16 M 11/30 Lecture: Train-mission problem (ABCD matrix method); More on the Cauchy Residue theorem;
17 W 12/2 Lecture: Inverse Laplace transform \(t \le 0\); Case for causality Laplace Transform,
Cauchy Riemann role in the acceptance of complex functions:
Convolution of the step function: \(u(t) \leftrightarrow 1/s\) vs. \(2\tilde{u}(t) \equiv 1+ \mbox{sgn}(t) \leftrightarrow 2\pi \delta(\omega) + 2/j\omega\)
18 F 12/4 Lecture: Inverse Laplace transform via the Residue theorem \(t > 0\)
19 M 12/7 Lecture: Properties of the Laplace Transform: Modulation, convolution; impedance/admittance, poles and zeros \(Z(s)=N(s)/D(s)\); Review
DE-3 Due
20 W 12/9 Last day of instruction: Review for final
- R 12/10 Reading Day Optional student Q&A session 9-11AM, 1-2PM

- M 12/14 Time and place 7-11:59 AM, Monday, Dec 14, via zoom and live on paper 3017 ECEB;
Offical UIUC exam schedule: If class is scheduled for 1:00PM on Monday then the official exam time is at 7:00-10:00 PM, Thursday Dec. 17
- 12/23 Letter grade statistics

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