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Linear complex algebra: ECE Webpage ECE298-F19; ECE-298-F19; UIUC Course Explorer: ECE-298-F19; Time: TBD MWF; Location: TBD ECEB (official); Register

ECE 298JA Schedule (Fall 2019)

L W D Date Lecture and Assignment

Part I: Number systems (10 Lectures)
1 1 M 8/28 Introduction & Historical Overview; Lecture 0: pdf;

The Pythagorean Theorem & the Three streams: 1) Number systems (Integers, rationals) 2) Geometry 3) {$\infty$} {$\rightarrow$} Set theory {$\rightarrow$} Calculus symbols
Read: Class-notes
Homework 0: Matlab/Octave tutorial: pdf

2 W 9/1 Lecture: The role of physics in Mathematics, Eigen analysis; The Fundamental theorems of Mathematics:
Read: Class-notes
3 F 9/18 Lecture: Analytic geometry as physics (Stream 2) Polynomials, Analytic functions, {$\infty$} Series Taylor series; ROC; expansion point
Read: Class-notes
Homework 1 (NS-1): Basic Matlab commands: pdf (v. 1.06), Due 9/6 (1 week); help
4 M 9/22 Lecture: Polynomial root classification by convolution; Summarize Lec 3: Series representations of analytic functions, ROC
Historical notes on complex numbers: Solution of the quadratic (Brahmagupta, 628), cubic (c1545), quartic (c1535), quintic cannot be solved (Abel, 1826)
and much more
Fundamental Thm of Algebra (pdf) &
Read: Class-notes
5 2 W 9/25 Lecture: Residue expansions of rational functions, Impedance {$Z(s) = \frac{P_m(s)}{P_n(s)}$} and its utility in Engineering applications
Read: Class-notes
6 F 9/27 Lecture: Analytic Geometry; Scalar and vector products
Read: Class-notes
NS-1 Due
Homework 2 (AE-1) Polynomials & Analytic functions and their inverse, Convolution, Newton's method
(pdf, 1 week)
7 M 9/29 Lecture: Gaussian elimination (intersection); Pivot matrices {$(\Pi_n)$}: {$U = \Pi_n^N P_n A$} gives upper-diagional {$U$}
Read: Class-notes
8 6 W 10/2 Lecture: Transmission matrix method (composition of polynomials, bilinear transformation)
Read: Class-notes
9 F 10/4 Lecture: The Riemann sphere (1851); (the extended plane) pdf
Mobius Transformation (youtube, HiRes), pdf description
Mobius composition transformations, as matrices
Software: Octave: zviz.zip, python
Read: Class-notes
AE-1 due
Homework 3 (AE-2): Linear systems of equations; Gaussian elimination; ABCD method; (pdf Due 1 week)
10 M 10/6 Lecture: Visualizing complex valued functions Colorized plots of rational functions

Read: Class-notes

11 7 W 10/9 Lecture:Read: Class-notes
Fourier Transforms (signals) Fourier Transform (wikipedia), Notes on the Fourier series and transform from ECE 310
(including tables of transforms and derivations of transform properties)
12 F 10/11 Lecture: AE-2 Due
Laplace transforms (systems); The importance of Causality
Convolution of the step function: {$u(t) \leftrightarrow 1/s$} vs. {$2\tilde{u}(t) \equiv 1+ \mbox{sgn}(t) \leftrightarrow 2\pi \delta(\omega) + 2/j\omega$}

13 M 10/13 Lecture:Read: Class-notes; Laplace Transform, Types of Fourier transforms
The 10 postulates of Systems (aka, Networks) pdf
The important role of the Laplace transform re impedance: {$z(t) \leftrightarrow Z(s)$}
A.E. Kennelly introduces complex impedance, 1893 pdf

Fundamental limits of the Fourier re the Laplace Transform: {$\tilde{u}(t)$} vs. {$u(t)$}

14 8 W 10/16 Lecture: 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)
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) Analytic properties
Euler's standard circular-function package (Logs, exp, sin/cos);
Inversion of analytic functions: Example: {$\tan^{-1}(z) = \frac{1}{2i}\ln \frac{i-z}{i+z}$}, the inverse of Euler's formula (1728)
Read: Class-notes
15 F 10/20 Lecture:

AE-2 Due
Homework 4 (AE-3): Complex algebra; visualizing complex functions; Mobius transformations; (pdf due 1 week)
Cauchy-Riemann (CR) conditions
Cauchy-Riemann conditions and differentiation wrt {$s$}: {$Z^\prime(s) \equiv \frac{dZ(s)}{ds} = \frac{dZ(s)}{d\sigma} = \frac{dZ(s)}{dj\omega}$}
Differentiation independent of direction in {$s$} plane: {$Z(s)$} results in CR conditions: {$\frac{\partial R(\sigma,\omega)}{\partial\sigma} = \frac{\partial X(\sigma,\omega)}{\partial\omega}$} & {$\frac{\partial R(\sigma,\omega)}{\partial\omega} = -\frac{\partial X(\sigma,\omega)}{\partial\sigma}$}
Cauchy-Riemann conditions require Laplace's Equation: {$\nabla^2 R=0$} & {$\nabla^2 X=0$}.
Discussion: Laplace's equation requires conservative vector fields: (1, 2)
Read: Class-notes & Boas pages 13-26; Derivatives; Convergence and Power series

16 9 M 10/23 Lecture: Complex analytic functions and Brune impedance
Complex impedance functions {$Z(s)$}, {$\Re Z(\sigma>0) \ge 0$}, Simple poles and zeros & 9 Postulates
Time-domain impedance {$z(t) \leftrightarrow Z(s) \Rightarrow v(t) = z(t) \star i(t) $} defines power
Read: Class-notes
17 W 10/25 Lecture: Time out: Come with questions: Review session on: multi-valued functions, complex integration,
Riemann sheets, colorized plots, branch cuts, Review of Fundamental Theorems of complex analytic functions.
Laplace's equation and its role in Engineering Physics. Impedance. What is the difference between a mass and an inductor?
Nonlinear elements; Examples of systems and the 10 postulates of systems.
18 F 10/27 Lecture: Three complex integration Theorems: Part I
1) Cauchy's Integral Theorem: {$\oint f(z) dz =0$} (Boas p. 45) vs. 2D Green's Thm (p. 49); Stokes (Thm, Bio)
AE-3 due
Homework 5 (DE-1): Series, differentiation, CR conditions, Bi-Harmonic functions: pdf, Due Oct 30
19 10
M 10/30 Lecture: Three complex integration Theorems: Part II
2) Cauchy's Integral Formula: {$\frac{1}{2\pi j} \displaystyle \oint_{{\partial}_{\gamma}} \frac{f(z)}{z-z_0}dz = f(z_0) \, U(\gamma) \equiv 0$} if {$z_0 \notin \gamma^\circ$}
3) Cauchy's Residue Theorem; Example by brute force integration: {$\oint_{|s|=1} \frac{ds}{s}= 2\pi j$}
Read: Class-notes & Boas p. 33-43 Complex Integration; Cauchy's Theorem
20 W 11/1 Lecture: The Inverse Laplace Transform (ILT); poles and the Residue expansion: The case for causality {$t<0$}
Cauchy's Residue theorem {$\Leftrightarrow$} 2D Green's Thm (in {$\mathbb C$})
Read: Class-notes
21 F 11/3 Lecture: Inverse Laplace Transform: Use of the Residue theorem {$t>0$}
Case for causality: Closing the contour: ROC as a function of {$e^{st}$}.
Examples: {$F(s)=1 \leftrightarrow \delta(t)$} and {$u(t) \leftrightarrow 1/s$}
Case of RC impedance {$ z(t) = R\delta(t)+u(t)/C \leftrightarrow R+1/sC $}
RC admittance {$ y(t) = e^{-t}u(t) \leftrightarrow 1/(s+1) $}
Semi-capacitor: {$ u(t)/\sqrt{t} \leftrightarrow \sqrt{\pi/s} $}

DE-1 due
Homework 6 (DE-2): Inverse Laplace Transforms; Residue integration: pdf, Due Nov 6

22 11
M 11/6 Lecture: General properties of Laplace Transforms:
Modulation, Translation, Convolution, periodic functions, etc. (png)
Table of common LT pairs (png)
Read: Class-notes
23 W 11/8 Lecture: Review of Laplace Transforms, Integral theorems, etc
Sol to DE-3 handout
Read: Class-notes
24 F 11/10 Lecture: General properties of Impedance (Z) and Transmission (ABCD) functions:
Impedance {$Z(s) = V(s)/I(s) \rightarrow $} Minimum phase impedance {$\rightarrow$} Simple poles & zeros in LHP ({$\sigma \le 0$})
Transfer {$H(s)=V_2/V_1, I_2/I_1 \rightarrow $} Allpass: {$|e^{-\jmath\phi(\omega)}|=1 \rightarrow$} poles in LHP, zeros in RHP
Wiener's factorization theorem: {$H(s) = M(s)A(s)$} with factors Minimum phase {$M(s)$} & Allpass {$A(s)$}
DE-2 Due
Homework 7 (DE-3): pdf, Due Nov 10
25 13
M 11/27 Lecture:Read: Class-notes
26 15
W 12/11 Lecture: The low-frequency quasi-static approximation: i.e., {$a < \lambda=c/f$} or {$f < c/a$}) are used for:
Brune's Impedance ({$a \ll \lambda$}), Kirchhoff's Laws, the telegraph wave equation starting from Maxwell's equations.
Impedance boundary conditions and generalized impedance:
{$Z(s)\equiv \frac{\cal P}{\cal V} = r_0 \frac{1+\Gamma(s)}{1-\Gamma(s)}$} where {$ \Gamma(s) \equiv {\cal P}_-/{\cal P}_+ $} and {$r_0 = {\cal P_+}/{\cal V_+}$}, with {${\cal P}= {\cal P}_+ +{\cal P}_-$} and {${\cal V}= {\cal V}_+ -{\cal V}_-$}.

Read: Class-notes

- F 12/18 Final ExamDE-3 Due

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