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Name change: was Complex linear algebra now Circuit & System Analysis
Course Explorer;
University calendar; Academic calendar
Week | M | W | F |
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12 | L1: 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) |
13 | L4: 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) |
14 | L7: 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) 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 |
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 |
Week | M | W | F |
---|---|---|---|
15 | L10: 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); |
16 | L13: 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) |
17 | L16: 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 |
18 | L19: 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) |
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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