(Register here) 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) | University Holiday | L2: Roots of Polynomials (Lec2-360-S21 @1:25; Lec2-3600-F20 @3min) |
13 | L3: 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) |
14 | L6: 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;roots (eigen values) assuming Fractional\(\mathbb{F}\) coefficients Read: 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) Next time merge Lec 4 & 5 Read: 3.2,.1,.2, B1, B3 Eigen-analysis, (pp. 88-93) NS-1 Due Homework 2 AE-1: Problems: pdf, Due on Lec 8; AE1-sol.pdf |
5 | F | 4/2 | Lecture: Use Companion Matrix to solve Pell and Fibinocci Eqs. (p. 57-61, 65-7) Lec 3 video @44 mins: Pell & Fibonocci companion Matrix + solutions; Read: 3.2.3 & Eigen-analysis (Appendix B.3); |
6 | M | 4/5 | Lecture: 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, 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 |
Week | M | W | F |
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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) |
15 | L9: 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 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 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) |
Week | M | W | F |
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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) | |
16 | L12: Exam I (NS1, AE1, Schwarz inequality (p. 118, 124); probs AE3 (#9, #10) HW: AllSol.zip | 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) |
17 | 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) | 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: \(t<0\))) |
18 | L18: LT (t>0) Lec18 360-S21; Lec18 360-F20 | L19: 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) |
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 request Homework 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.pdf DE-1 due on Lec15 |
13 | W | 4/21 | Lecture: Multi-valued functions; Riemann Sheets, Branch cuts & points; 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, 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 DE-3 Due, DE3-sol.pdf, Full Solution to train problem. |
- | 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 |
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