 ## ECE298-ComplexLinearAlg-F20

### ECE 298 ComplexLinearAlg-F20 Schedule (Fall 2020)

Part I: Reading assignments and videos: Complex algebra (5 Lecs)
WeekMWF
(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)
L6:
 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 BNS-1 DueHomework 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)
WeekMWF
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 DueHomework 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 matrixComplex-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)
WeekMWF
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
6L15:L16:L17:
718:L19:L20:
 L D Date Lecture and Assignment Part III: Complex analytic analysis (6 Lectures) 9 M 4/13 AE-1 DueExam 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 applicationsHomework 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 cutsHomework 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); pdfMobius Transformation (youtube, HiRes), pdf description Mobius composition transformations, as matrices 15 M 4/27 Lecture: Cauchy’s Integral theorem & FormulaHomework 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; ReviewDE-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