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Complex Linear Algebra: Time:MWF 1:00-1:50 PM; Location: 3081 ECEB ECE298-CLA; (Register here)

ECE 298 ComplexLinearAlg-S19 Schedule (Spring 2019)

L W D Date Lecture and Assignment

Part I: Introduction to complex 2x2 matricies (6 Lectures)
1 11 M 3/11 Lecture: Introduction & Overview:
1) Integers, fractionals, rationals, real vs. complex, vectors and matrices;
2) Common Math Notation symbols
3) Matlab tutorial: pdf
4) Polynomials and Newton's complex root finding method; Polynomial root classification by convolution;
5) Fundamental Thm of Algebra (pdf);
6) Series representations of analytic functions, ROC
7) Historical notes on complex numbers: Solution of the quadratic (Brahmagupta, 628), cubic (c1545), quartic (Tartaglia et al..., 1535),
quintic cannot be solved (Abel, 1826) and much much more
Read: Class-notes
Homework 1 (NS-1): Basic Matlab commands: pdf, Due on Lec 3; help.m
2 W 3/13 Lecture: Complex analytic functions, geometry; vector scalar (i.e., dot) products
Read: Class-notes
Homework 2 (NS-2): pdf, Due on Lec 6
3 F 3/15 Lecture: Pyth triplets (Lec 7); Inverse matrix via Gaussian Elimination (Lec 15)

Read: Class-notes
NS-1 Due

12 Spring Break
4 13 M 3/25 Lecture: Analysis of simple LRC circuits by matrix composition: ABCD (transmisison matrix method, composition of polynomials)
Read: Class-notes: Lec 16 page 120-125
'''Homework 3 (NS-3):Pell's equation, Fibonacci sequence; pdf, Due on Lec 8
5 W 3/27 Lecture: Pell's equation: \(M^2-Nn^2=1 (m,n,N\in{\mathbb N})\) & Fibonacci Series \(f_{n+1} = f_n + f_{n-1}, (n,f_n \in{\mathbb N})\)
Companion matrix and eigen-analysis (eigenvalues, eigenvectors)
Read: Class-notes Lec 8 and Lec 9
L W D Date Lecture and Assignment
Part II: Complex analytic analysis (9 Lectures)
6 F 3/29 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: Lec 19
NS-2 Due

7 14 M 4/1 Lecture: Laplace transforms and Causality; Residue expansions

Read: Class-notes

8 W 4/3 Lecture: The 10 system 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 vs. the Laplace Transform: \(\tilde{u}(t)\) vs. \(u(t)\)
Read: Class-notes
NS-3 Due

9 F 4/5 Exam I: 7:00-9:30 PM; 3081 ECEB room confirmed; NS1-NS3, Lec 11; 2x2 complex matrix analysis;
Analysis of Exam 1 pdf Ver 1.7, Apr 18, 2019
10 15 M 4/8 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) Continued fractions; 5) Analytic properties
History: The amazing Bernoulli family; Fluid mechanics; airplane wings; natural logarithms
Beginnings of modern mathematics: Euler and Bernoulli, 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
Homework 4 (DE1): Series, differentiation, CR conditions, Branch cuts: pdf, Due on Lec 13
11 W 4/10 Lecture: Differentiation in the complex plane: Complex Taylor series;
Cauchy-Riemann (CR) 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:
\(\frac{\partial R(\sigma,\omega)}{\partial\sigma} = \frac{\partial X(\sigma,\omega)}{\partial\omega}\) and \(\frac{\partial R(\sigma,\omega)}{\partial\omega} = -\frac{\partial X(\sigma,\omega)}{\partial\sigma}\)
Cauchy-Riemann conditions require that Real and Imag parts of \(Z(s) = R(\sigma,\omega) + j X(\sigma,\omega)\) obey Laplace's Equation:
\(\nabla^2 R=0\), namely: \(\frac{\partial^2R(\sigma,\omega)}{\partial \sigma^2} + \frac{\partial^2 R(\sigma,\omega)}{\partial \omega^2} =0 \) and \(\nabla^2 X=0\), namely: \(\frac{\partial^2 X(\sigma,\omega)}{\partial \sigma^2} + \frac{\partial^2 X(\sigma,\omega)}{\partial \omega^2} =0\),
Biharmonic grid (zviz.m)
Discussion: Laplace's equation means conservative vector fields: (1, 2)
Read: Class-notes & Boas pages 13-26; Derivatives; Convergence and Power series
12 F 4/12 Lecture: Complex analytic functions; Brune Impedance \(Z(s) = {P_m(s)}/{P_n(s)}\) and its utility in Engineering applications
Read: Class-notes

13 16 M 4/15 Lecture: Multi-valued complex functions; Riemann sheets; Branch cuts
Read: Class-notes
DE-2: Integration, differentiation wrt \(s\); Cauchy theorems; LT; Residues; power series, RoC; LT; (pdf), due on Lec 16;
DE-1 due
14 W 4/17 Lecture: Complex analytic mapping (Domain coloring)
Visualizing complex valued functions Colorized plots of rational functions

Software: Matlab: zviz.zip, python
Read: Class-notes

15 F 4/19 Lecture: Riemann’s extended plane: The Riemann sphere (1851); pdf
Mobius Transformation (youtube, HiRes), pdf description
Mobius composition transformations, as matrices
Read: Class-notes
16 17 M 4/22 Lecture: Cauchy’s Integral theorem & Formula
Read: Class-notes
DE-3: Inverse LT; Impedance; Transmission lines; (pdf); due on Lec 20; DE-2 due
17 W 4/24 Lecture: Train-mission problem (ABCD matrix method); More on the Cauchy Residue theorem;

Read: Class-notes (ABCD matrix method)

18 F 4/26 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. \(2\tilde{u}(t) \equiv 1+ \mbox{sgn}(t) \leftrightarrow 2\pi \delta(\omega) + 2/j\omega\)
Read: Class-notes;
19 18 M 4/29 Lecture: Inverse Laplace transform via the Residue theorem \(t > 0\)

Read: Class-notes;

20 W 5/1 Lecture: Properties of the Laplace Transform: Modulation, convolution; Review
DE-3 Due
- R 5/2 Reading Day

- M 5/6/2019 Time and place confirmed: Official: Final Exam 7-10PM ECE 3081

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