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Name change: was Complex linear algebra now Circuit & System Analysis
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 University calendar; Academic calendar

ECE 298 Complex Systems Analysis Schedule (Spring 2022)

Part I: Lecture + videos: Complex linear algebra (Calendar week 12-14; 6 Lecs)
12L1: Intro + Overview (Lec1-360-F20 Starts @ 6:30 min; S21: no 360 recording)L2: Roots of Polynomials (Lec2-360-S21 @1:25m; Lec2-360-F20 @3m)L3: Companion Matrix + Examples (Lec3-360-S21 @6:03m (NoAudio); Lec3-360-F20 @2m)
13L4: Eigen-analysis, analytic solution (Lec4-360-S21 @1:43; Lec4-360-F20)L5: Eigen-analysis (Lec5-360-S21; Lec5-360-F20 @2m)L6: Eigen-analysis; Taylor series & Analytic functions (Lec6-360-F20 @2, Lec6-360-F20)
14L7: 3.9,.1 \({\cal FT}\) of signals vs. \({\cal LT}\) of systems 360-L7 S21 (NO Audio from 2-6m);
(F20, Lec6-360, ECE-493: F20, L11-360 @10 m)
L8: Impedance (L8-360 S21, F20: L8-360)L9: Integration in complex plane S21: L9-360 (@2m), F20: L9-360
L D Date Lecture and Assignment


Part I: Introduction to 2x2 complex matricies (9 Lectures)
1 M 3/21 Lecture: Introduction & Overview: (Read Ch. 1 p. 1-17), Intro + history; \(S\)3.1 Read p.69-73,
Homework 1: NS-1 pdf, Due on Lec 4; NS1-sol.pdf
2 W 3/23 Lecture: Roots of polynomials; Matlab Examples; ;
Read: 3.1 (p. 73-80) Roots of polynomials+monics; Newton's method.
3 F 3/25 Lecture: Find Companion Matrix for 1) Pell's equation: \(m^2-Nn^2=1\) with \(m,n,N\in{\mathbb N}\), (p. 57-68),
2) Fibonacci Series \(f_{n+1} = f_n + f_{n-1}\), with \(n,f_n \in{\mathbb N}\).
Read: 3.1.3,.4, (p.84-8) Monic roots; @23 ms 2021 video
4 M 3/28 Lecture: Eigen-analysis; Fibinocci analytic solution (Appendix B.3, p. 310)
Read: 3.2,.1,.2, B1, B3 Eigen-analysis, (pp. 88-93)
Homework 2: AE-1 pdf, AE1-sol.pdf, Due in 1 week, Lec 8;
Homework 3: NS-2: ( Replace with NS-3!!) pdf, NS2-sol.pdf, Due in 2 weeks, Lec 10;
NS-1 Due
5 W 3/30 Lecture: Use Companion Matrix to solve Pell Eq & review Fibinocci Eqs. (p. 57-61, 65-7)
Lec 3 video @44 ms: Pell & Fibonocci companion Matrix + solutions;

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

6 F 4/1 Lecture: Taylor series (\(\S3.2.3\)) & Analytic functions (\(\S3.2.4\)); Echo 360 @5m;
Brief 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; Analytic continuation using Mobius (bilinear) transformation: Pole \(\rightarrow \infty\);

Mobius Transformation: (youtube, HiRes), description pdf
Read: 3,.1,2,.3,.4 Taylor series (\(\S3.2.3\), p. 93-8) & 'Analytic Functions' (p. 98-100)
Homework 4: AE-3 (AE3.pdf), Due by Lec 10, AE3-sol.pdf

7 M 4/4 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 4/6 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: \(\S\) 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
9 F 4/8 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 5: DE-1 pdf, DE1-sol.pdf, Due on Lec 15

Part II: Lecture + videos: Complex algebra (12 Lecs)
15L10: 3.2,.4,.5 Fundamental theorem of complex calculus; Differentiation in the complex plane:
Cauchy-Riemann conditions & differentiation wrt to \(s=\sigma+\jmath\omega\); Residues, Convolution; FTCC:
(L10-360, S21, L10-360, F20 @4:00)
L11: 3.10,.1-.3 Multi-valued complex functions; Riemann sheets; Branch cuts (not on Exam1);
Cauchy's life;
(L11-360 @5m, S21; L11-360, F20)
L12: Exam I (NS2, AE1, AE3, DE1);
16L13: 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)
17L16: Transmission line train problem (Lec16-360, S21, Lec16-360, F20)L17: Wave function \(\kappa(s)\) when sound speed depends on frequency; (Lec17-360, S21), Lec17-360, F20 @ 4m (Inv LT: \(t<0\))L18: LT (t>0) Lec18 360-S21; Lec18 360-F20
18L19: S21: Review for final exam (360-S21);
F20: LT Properties: (Lec19-360-F20)
Thur: Reading Day: Optional review for final Student Q&A 1-2 PM
(360-Review, F20)
L D Date Lecture and Assignment
Part II: Complex analytic analysis (6 Lectures)
10 M 4/11 Lecture: Properties of Impedance/Admittance; Fundamental theorem of complex calculus (Causal, Positive-real, complex analytic);
Cauchy-Riemann (CR) conditions and differentiation wrt the Laplace Frequency \(s=\sigma+\jmath\omega\).

Discussion of Laplace's equation and conservative fields: (1, 2)
AE-3 Due AE3-sol.pdf
Homework: DE-1 due by Lec 15

11 W 4/13 Lecture: Multi-valued complex functions; Riemann sheets; Branch cuts (not on Exam1)
NS-2 Due
12 F 4/15 Exam 1: 9:30-1:30 PM: 3013-ECEB In person only (No zoom); maximum of 3 hour; Topics taken from HWs NS-2, AE-1, AE-3, DE-1, as discussed in class
Homework 6: DE-2 (pdf), Due on Lec 15; DE2-sol.pdf
13 M 4/18 Lecture: Multi-valued functions; Riemann Sheets, Branch cuts & points; colored figures; 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, Allm.zip, python

14 W 4/20 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
DE-2 (pdf)
15 F 4/22 Lecture: Cauchy’s Integral theorem & Formula:
Homework 7: DE-3 (pdf), Due on Lec 19; DE-1 & DE-2 due
16 M 4/25 Lecture: Train-mission problem (ABCD matrix method); More on the Cauchy Residue theorem;
17 W 4/27 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 4/29 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 5/2 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.
- W 5/4 Review, summary and emphasis of the key ideas you have learned in this class (no video)
- R 5/5 Reading Day Optional student Q&A session 10AM-12PM

- W 5/11 Final:In person only on paper: 2017 ECEB; 1:30-4:30 p.m., Wednesday, May 11
 Offical UIUC exam schedule:   HW: AllSol.zip

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