 ## ECE298-ComplexLinearAlg-S21

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### ECE 298 ComplexLinearAlg-S21 Schedule (Spring 2021)

Part I: Lecture + videos: Complex algebra (Calendar week 12-14; 6 Lecs)
WeekMWF
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; @39 mins: Pell & Fibonocci companion Matrix + solutions; roots (eigen values) assuming Fractional$$\mathbb{F}$$ coefficientsRead: 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)Read: 3.2,.1,.2, B1, B3 Eigen-analysis, (p. 80-4, 88-93)NS-1 DueHomework 2 AE-1: Problems: pdf, Due on Lec 8; AE1-sol.pdf 5 F 4/2 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 M 4/5 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
Part II: Lecture + videos: Transforms (3 Lecs)
WeekMWF
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 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 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)
WeekMWF
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 requestHomework 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.pdfDE-1 due on Lec15 13 W 4/21 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 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 - 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