 ## ECE298-ComplexLinearAlg-F20

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### ECE 298 ComplexLinearAlg-F20 Schedule (Fall 2020)

Part I: Lectures, Reading assignments and videos: Complex algebra (6 Lecs)
WeekMWF
(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 DueHomework 2 AE-1): Problems: ... pdf, Due on Lec 7 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)
WeekMWF
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 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 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)
WeekMWF
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)
48
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 cutsHomework 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 requestDE-2 (pdf), Due on Lec 15; DE2-sol.pdfDE-1 due on Lec15 13 M 11/16 Lecture: Multi-valued functions; Riemann Sheets, Branch cuts & points; Domain coloringVisualizing 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 & FormulaHomework 6: DE-3: Inverse LT; Impedance; Transmission lines; Problems: ... (pdf), Due on Lec 19; DE3-sol.pdfDE-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 $$Z(s)=N(s)/D(s)$$; ReviewDE-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