 ## ECE298-CircuitSystemAnalysis-S22

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Name change: was Complex linear algebra now Circuit & System Analysis
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### ECE 298 Complex Systems Analysis Schedule (Spring 2022)

Part I: Lecture + videos: Complex linear algebra (Calendar week 12-14; 6 Lecs)
WeekMWF
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 ECE293-Sep29.20.pdf Part I: Introduction to 2x2 complex matricies (9 Lectures)AUDITORY-request@LISTS.MCGILL.CA 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 matrixComplex-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)
WeekMWF
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.pdfHomework: 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 classHomework 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 matricesDE-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 - 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  -