 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 symbolsRead: Class-notesHomework 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 pointRead: Class-notesHomework 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 applicationsRead: Class-notes 6 F 9/27 Lecture: Analytic Geometry; Scalar and vector productsRead: Class-notesNS-1 DueHomework 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) pdfMobius Transformation (youtube, HiRes), pdf description Mobius composition transformations, as matricesSoftware: Octave: zviz.zip, pythonRead: Class-notesAE-1 dueHomework 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 CausalityConvolution 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. FTCCAnalytic vs complex analytic functions and Taylor formulaCalculus 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) conditionsCauchy-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 impedanceComplex impedance functions {$Z(s)$}, {$\Re Z(\sigma>0) \ge 0$}, Simple poles and zeros & 9 PostulatesTime-domain impedance {$z(t) \leftrightarrow Z(s) \Rightarrow v(t) = z(t) \star i(t)$} defines powerRead: 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 I1) Cauchy's Integral Theorem: {$\oint f(z) dz =0$} (Boas p. 45) vs. 2D Green's Thm (p. 49); Stokes (Thm, Bio)AE-3 dueHomework 5 (DE-1): Series, differentiation, CR conditions, Bi-Harmonic functions: pdf, Due Oct 30 19 1044 M 10/30 Lecture: Three complex integration Theorems: Part II2) 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 dueHomework 6 (DE-2): Inverse Laplace Transforms; Residue integration: pdf, Due Nov 6 22 1145 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, etcSol to DE-3 handoutRead: 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 RHPWiener's factorization theorem: {$H(s) = M(s)A(s)$} with factors Minimum phase {$M(s)$} & Allpass {$A(s)$} Exam II TBDDE-2 DueHomework 7 (DE-3): pdf, Due Nov 10 25 1348 M 11/27 Lecture:Read: Class-notes 26 1550 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 TBD