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ECE 298 ComplexLinearAlg-F20 Schedule (Fall 2020)

Part I: Reading assignments and videos: Complex algebra (5 Lecs)
1L1: 1 (Read p. 1-17) Intro + history; 3.1 Read p.69-73
(298CLA, Lec 1 360, Lec 1-zoom)
L2: 3.1,.1,.2, (p. 73-80) Roots of polynomials; Newton's method. (298CLA Lec 2-360, 298-CLA Lec 2-zoom)L3: 3.1.3,.4, (p.84-88) Companion matrix
(493 Lec 3)
2L4: 3.2,.1,.2, B1, B3 Eigenanalysis, (p. 80-84, 88-93)
(493 Lec 4)
L5: 3.2.3 Sol of Fibinocci and Pell's Eqs., (p. 57-61, 65-67)
(493 Lec 5)
L D Date Lecture and Assignment

Part I: Introduction to complex 2x2 matricies (5 Lectures)
1 M 10/19 Lecture: Introduction & Overview:
Homework 1 (NS-1): Problems: ... pdf, Due on Lec 4;
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 3/30 Lecture: Eigen analysis; Appendix B
NS-1 Due
Homework 2 (NS-2): Problems: ... pdf, Due on Lec 7
5 W 4/1 Lecture: Detailed eigen-analysis example for eigenvalues and eigenvectors (Appendix B)
Part II: Reading assignments and videos: Transforms (3 Lecs)
2L4:L5:L6: 3.9,.1 \({\cal FT}\) of signals
(493 Lec 11)
3Labor dayL7: 3.5, Anal Geom, Generalized scalar products
(493 Lec 7)
L8: 3.5.1-.4 \(\cdot, \times, \wedge \) scalar products (493 Lec 8)
L D Date Lecture and Assignment
Part II: Fourier and Laplace Transforms (3 Lectures)
6 F 4/3 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

- M 4/6 Labor day
7 W 4/8 Lecture: The 10 system postulates of Systems (aka, Networks) pdf;
Integration in the complex plane: FTC vs. FTCC; Analytic vs complex analytic functions and Taylor formula
Calculus of the complex \(s=\sigma+j\omega\) plane: \(dF(s)/ds\),
\(\int F(s) ds\) (Boas, see page 8)

The convergent analytic power series: Region of convergence (ROC)
NS-2 Due
Homework 3: AE-1: Polynomials; Problems ... pdf, Due on Lec 10
8 F 4/10

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 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

Part III: Reading assignments and videos: Complex algebra (12 Lecs)
4L9: Review for Exam I
(493 Lec 9)
L10: 3.8,.1-.4 Thevenin parameters; Transmission lines; impedance matrix
(493 Lec 10 No audio!)
L11: 3.2,.4,.5 Analytic Functions, Residues, Convolution (493 Lec 6)
5L12: 3.10,.1-.3 \(\cal LT\) of systems + postulates
(493 Lec 12-zoom, Lec 12-360)\ (493 Lec 6)
L13: 3.11,.1,.2 Complex analytic color maps; Riemann sphere; Bilinear transform
(493 Lec 13-zoom, Lec 13-360)
L14: 3.5.5, 3.6,.1-.5 Gauss Elim; Matrix algebra (systems)
Lec 14-zoom Lec 14-360
L D Date Lecture and Assignment
Part III: Complex analytic analysis (6 Lectures)
9 M 4/13

AE-1 Due
Exam I: 12:00-3:00 PM; via Zoom NS1-NS2, AE1, Lec 11; 2x2 complex matrix analysis; pdf

10 W 4/15 Lecture: Complex analytic functions;
History: Beginnings of modern mathematics: Euler and Bernoulli,
The Bernoulli family; ; natural logarithms Euler's standard circular-function package (Logs, exp, sin/cos);
Brune Impedance \(Z(s) = {P_m(s)}/{P_n(s)}\) and its utility in Engineering applications
Homework 4: DE-1: Problems ... Series, differentiation, CR conditions, Branch cuts: pdf, Due on Lec 13
11 F 4/17 Lecture: Differentiation in the complex plane: Complex Taylor series;
Cauchy-Riemann (CR) conditions and differentiation wrt \(s\)

Discussion of Laplace's equation (1, 2)

12 M 4/20 Lecture: Multi-valued complex functions; Riemann sheets; Branch cuts
Homework 5: DE-2: Integration, differentiation wrt \(s\); Cauchy theorems; LT; Residues; power series, RoC; LT; Problems:DE2 (pdf), due on Lec 16;
DE-1 due
13 W 4/22 Lecture: Complex analytic mapping (Domain coloring)
Visualizing complex valued functions Colorized plots of rational functions

Software: Matlab: zviz.zip, python

14 F 4/23 Lecture: Riemann’s extended plane: The Riemann sphere (1851); pdf
Mobius Transformation (youtube, HiRes), pdf description
Mobius composition transformations, as matrices
15 M 4/27 Lecture: Cauchy’s Integral theorem & Formula
Homework 6: DE-3: Inverse LT; Impedance; Transmission lines; Problems: ... (pdf); due on Lec 20; DE2 due
16 W 4/29 Lecture: Train-mission problem (ABCD matrix method); More on the Cauchy Residue theorem;
17 F 5/1 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 M 5/4 Lecture: Inverse Laplace transform via the Residue theorem \(t > 0\)
19 W 5/6 Lecture: Properties of the Laplace Transform: Modulation, convolution; Review
DE-3 Due
- W 5/6 Last day of instruction.
- R 5/7 Reading Day

- M 12/17 Time and place 7-10, Thurs, Dec 17, via zoom; (official time) Final Exam Thursday, Dec 17, 7-10PM zoom

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