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ECE 298 ComplexLinearAlg-F21 Schedule (Spring 2021)

Part I: Lecture + videos: Complex algebra (Calendar week 12-14; 6 Lecs)
12L1: Intro + Overview (Lec1-360-F20 Starts @ 6:30 min; S21: no 360 recording) University HolidayL2: Roots of Polynomials (Lec2-360-S21 @1:25; Lec2-3600-F20 @3min)
13L3: Companion Matrix + Examples (Lec3-360-S21-NoAudio @5:30; Lec3-360-F20 @2min)L4: Eigen-analysis, analytic solution (Lec4-360-S21 @1:43; Lec4-360-F20)L5: Eigen-analysis (Lec5-360-S21; Lec5-360-F20 @2min)
14L6: Eigen-analysis; Taylor series & Analytic functions (Lec6-360-F20 @2, Lec6-360-F20)Next time mv L6 to Part II 
L D Date Lecture and Assignment

Part I: Introduction to 2x2 matricies (6 Lectures)
1 M 3/22 Lecture: Introduction & Overview: (Read Ch. 1 p. 1-17), Intro + history; \(S\)3.1 Read p.69-73,
Homework 1 (NS-1): Problems: pdf, Due on Lec 4; NS1-sol.pdf
W 3/24 University Holiday
2 F 3/26 Lecture: Roots of polynomials; Matlab Examples; Allm.zip;
Read: 3.1 (p. 73-80) Roots of polynomials+monics; Newton's method.
3 M 3/29 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;roots (eigen values) assuming Fractional\(\mathbb{F}\) coefficients
Read: 3.1.3,.4, (p.84-8) More on Monic roots; @23 mins:
4 W 3/31 Lecture: Eigen-analysis; Examples + analytic solution (Appendix B.3, p. 310)
Next time merge Lec 4 & 5
Read: 3.2,.1,.2, B1, B3 Eigen-analysis, (pp. 88-93)
NS-1 Due
Homework 2 AE-1: Problems: pdf, Due on Lec 8; AE1-sol.pdf
5 F 4/2 Lecture: Use Companion Matrix to solve Pell and Fibinocci Eqs. (p. 57-61, 65-7)
Lec 3 video @44 mins: Pell & Fibonocci companion Matrix + solutions;

Read: 3.2.3 & Eigen-analysis (Appendix B.3);

6 M 4/5 Lecture: 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, 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
Part II: Lecture + videos: Transforms (3 Lecs)
14 Merge II and III?? L7: 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)
L8: Impedance (L8-360 S21, F20: L8-360)
15L9: Integration in complex plane S21: L9-360 (@2m), F20: L9-360  
L D Date Lecture and Assignment
Part II: Fourier and Laplace Transforms (3 Lectures)
7 W 4/7 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 F 4/9 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 M 4/12 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

Part III: Lecture + videos: Complex algebra (12 Lecs)
15 L10: 3.2,.4,.5 Complex Taylor series, Residues, Convolution; FTCC: (L10-360, S21, L10-360, F20 @4:00)L11: 3.10,.1-.3 Complex analytic functions (L11-360 @5m, S21; L11-360, F20)
16L12: Exam I (NS1, AE1, Schwarz inequality (p. 118, 124); probs AE3 (#9, #10) HW: AllSol.zipL13: 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)
17L15: Cauchy's intergral thms CT-1,2,3; DE-3, Due on L19 360 video, S21 (no audio until @18:00m); (L15-360, F20)L16: 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: \(t<0\)))
18L18: LT (t>0) Lec18 360-S21; Lec18 360-F20L19: 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
(360-Review, F20)

L D Date Lecture and Assignment
Part III: Complex analytic analysis (6 Lectures)
10 W 4/14 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 F 4/16 Lecture: Multi-valued complex functions; Riemann sheets; Branch cuts (not on Exam1)
12 M 4/19 Exam 1: 1-3 PM: Zoom or 3017-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
13 W 4/21 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 F 4/23 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 M 4/26 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
16 W 4/28 Lecture: Train-mission problem (ABCD matrix method); More on the Cauchy Residue theorem;
17 F 4/30 Lecture: Analysis of the wave propagation function \(\kappa(s)\in\mathbb{C}\) when speed of sound depends on frequency \(s=\sigma + \jmath\omega\).
18 M 5/3 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 W 5/5 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.
- R 5/6 Reading Day Optional student Q&A session 9-11AM, 1-2PM

- M 5/7 Final: 1:30-4:30 PM via zoom + in person on paper 3017 ECEB;
Offical UIUC exam schedule:
- TBD Letter grade statistics

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