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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;)L2: Roots of Polynomials Lec2-360-F21@6:30 min (Lec2-360-S21 @1:25; Lec2-3600-F20 @3 min)L3: Companion Matrix + Examples Lec3-360-F21 @6:20 min (Lec3-360-S21-NoAudio @5:30; Lec3-360-F20 @2 min)
43L4: Eigen-analysis, analytic solution (Lec4-360-S21 @1:43; Lec4-360-F20)L5: Eigen-analysis (Lec5-360-S21; Lec5-360-F20 @2 min)L6: Eigen-analysis; Taylor series & Analytic functions (Lec6-360-F20 @2, Lec6-360-F20)
44L7: 3.9,.1 \({\cal FT}\) of signals vs. systems 360-L7 S21 (NO Audio from 2-6min);
(F20, Lec6-360; ECE-493: F20, L11-360 @10 min); STFT (Example: Remote Control
L8: Impedance (L8-360 S21, F20: L8-360)L9: Exam I (NS1, AE1, Schwarz inequality (p. 118, 124); probs AE3 (#9, #10) HW: AllSol.zip
L D Date Lecture and Assignment

Part I: Introduction to 2x2 matricies (8 Lectures)
1 M 10/18 Lecture: Introduction & Overview: (Read Ch. 1 p. 1-17), Intro + history; \(S\)3.1 Read p.69-73,
Homework 1: NS-1: pdf, NS-3: Prob #3, #4 pdf, Due on Lec 4; NS1-sol.pdf, NS3-sol.pdf
2 W 10/20 Lecture: Roots of polynomials; Matlab Examples; Allm.zip;
Read: 3.1 (p. 73-80) Roots of polynomials+monics; Newton's method.
3 F 10/22 Lecture: Companion Matrix;
Pell's equation: \(m^2-Nn^2=1\) with \(m,n,N\in{\mathbb N}\), (p. 57-68) Fibonacci Series \(f_{n+1} = f_n + f_{n-1}\), with \(n,f_n \in{\mathbb N}\) Companion matrix; @39 mins: Pell & Fibonocci companion Matrix + solutions; roots (eigen values) assuming Fractional \(\in\mathbb{F}\) coefficients
Read: 3.1.3,.4, (p.84-8) More on Monic roots; @23 mins:
4 M 10/25 Lecture: Eigen-analysis; Examples + analytic solution (Appendix B.3, p. 310)
Read: 3.2,.1,.2, B1, B3 Eigen-analysis, (p. 80-4, 88-93)
NS-1 Due
Homework 2 AE-1: Problems: pdf, Due on Lec 8; AE1-sol.pdf
5 W 10/27 Lecture: Eigen-analysis (Appendix B)
Read: 3.2.3 Eigen-analysis: Solution of Pell's and Fibinocci's Eqs., (p. 57-61, 65-7)
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: 3.2,.3,.4, Eigen-analysis: Taylor series (\(\S3.2.3\), p. 93-8) & Analytic Functions (p. 98-100)
Homework 3: AE-3: Problems: 2x2 complex matrices; scalar products (AE3.pdf), Due by Lec 10, AE3-sol.pdf
7 M 11/1 Lecture: Fourier transforms for signals vs. Laplace transforms for systems:
Read: 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: Class-notes \(\S\) 3.10
8 W 11/3 Lecture: The important role of the Laplace transform re impedance: \(z(t) \leftrightarrow Z(s)\);
Read: 3.2.5,.3.10, Impedance (p. 100-1) & \(\cal LT \);
Read: Hamming Digital filters: The idea of an Eigenfunction, \(\S\) 2.4, 2.6 p. (pdf-pages 38, 46); 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 extended from Lec 7
9 F 11/5 Exam 1: 1-3 PM: Zoom or 2015-ECEB; Submit to Gradescope; Paper copy upon request
Homework 5: DE-2 Problems: Integration, differentiation wrt \(s\); Cauchy theorems; LT; Residues; power series, RoC; LT; Problems:
DE-2 (pdf), Due on Lec 15; DE2-sol.pdf
DE-1 due on Lec15
Part II: Lecture + videos: Complex algebra (12 Lecs)
WeekMWF
45L10: Integration in complex plane F21: L9-360 (@2m), F20: L9-360L11: 3.2,.4,.5 Complex Taylor series, Residues, Convolution; FTCC: (L10-360, S21, L10-360, F20 @4:00)L12: 3.10,.1-.3 Complex analytic functions (L11-360 @5m, S21; L11-360, F20)
46L13: 3.11,.1,.2 Multi-valued functions; Domain coloring (L13-360, S21; L13-360, F20)L14: 3.5.5, 3.6,.1-.5 1) multivalued functions; 2) Schwarz inequality; 3) Triangle inequality; 4) Riemann's extended plane (L14-360,F20)L15: Cauchy's intergral thms CT-1,2,3; DE-3, Due on L19 360 video, S21 (no audio until @18:00m); (L15-360, F20)
47L16: Transmission line problem (Lec16-360, S21, Lec16-360, F20)L17: Wave function \(\kappa(s)\) when sound speed depends on frequency; (Lec17-III-360, S21, Lec17-360, F20 @ 4min (Inv LT: \(t<0\)))L18: LT (t>0) Lec18 360-S21; Lec18 360-F20
48L19: S21: Review for final Exam (360-S21);
F20: LT Properties: (Lec19-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)
9 F 11/5 Lecture: Integration in the complex plane: FTC vs. FTCC;
Read: 3.2.6 (p. 101-3) Complex analytic functions, e.g.: \(Z(s) \leftrightarrow z(t)\); 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 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

9 F 11/8 Exam 1: 1-3 PM: Zoom or 2015-ECEB; Submit to Gradescope; Paper copy upon request
Homework 5: DE-2 Problems: Integration, differentiation wrt \(s\); Cauchy theorems; LT; Residues; power series, RoC; LT; Problems:
DE-2 (pdf), Due on Lec 15; DE2-sol.pdf
DE-1 due on Lec15
10 M 11/8 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 AE3-sol.pdf

11 W 11/10 Lecture: Multi-valued complex functions; Riemann sheets; Branch cuts (not on Exam1)
13 M 11/15 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 11/17 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
15 F 11/19 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
Thanksgiving Break
16 M 11/29 Lecture: Train-mission problem (ABCD matrix method); More on the Cauchy Residue theorem;
17 W 12/1 Lecture: Analysis of the wave propagation function \(\kappa(s)\in\mathbb{C}\) when speed of sound depends on frequency \(s=\sigma + \jmath\omega\).
18 F 12/3 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 \(2\tilde{u}(t) \equiv 1+ \mbox{sgn}(t) \leftrightarrow 2\pi \delta(\omega) + 2/j\omega\)
19 M 12/6 Lecture: Properties of the Laplace Transform: Modulation, convolution; impedance/admittance, poles and zeros \(Z(s)=N(s)/D(s)\); Review
DE-3 Due, DE3-sol.pdf, Full Solution to train problem.
20 W 12/8 Lecture: Review
- R 12/9 Reading Day Optional student Q&A session 9-11AM, 1-2PM

- M ?? Final: TBD
Offical UIUC exam schedule:
- TBD Letter grade statistics, Letter grade statistics, Letter grade statistics

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