1 Introduction to Math Methods in Electrical Engineering An EE100 Lecture Dr. Bob Berinato Fall 2014

2 Good Engineers are Good Problem Solvers … and the best problem solvers have a large toolbox of math methods and techniques Several tools we will consider today  Complex Number Notation and Applications  Properties of Curves  Derivatives  Integrals

3 EE (and Math) at Work: Synthetic Aperture Radar (SAR) Radar can measure range (round-trip time delay) and Doppler shift A moving radar can use the data it collects to create an “image” Math methods are central to understanding and engineering SAR systems  Complex Math and Fourier Transforms are essential tools

4 What are the 5 most important “numbers” in mathematics? 2 integers: 0, 1 2 irrationals: , e 1 complex: Could these possibly be related in one equation?

5 Leonhard Euler – An Amazing Individual! Euler’s Formula: expresses sines and cosines in complex number notation

6 Application: The Phasor Consider the following complex signal, called a phasor This is the complex notation for a sinusoid with phase  and amplitude A Note that at time t = 0, the phasor has value Aexp(j  ) Then the phase increases (and repeats every 2  radians, i.e., every integer multiple of 1/f 0 ) Grapically, this is shown as a rotating vector of length A that makes a complete revolution every 1/f 0 seconds

7 Application: Signal Notation Real Signal Notation: Complex Signal Notation: Magnitude Phase Center Frequency Complex Envelope, or Amplitude MagnitudePhase Re[z(t)] = s(t), the “real” signal

8 The Fourier Series and Fourier Transform The Fourier Series is a representation of a periodic signal (repeats over and over again with period T) as the weighted sum of phasors at discrete frequencies Similarly, the (inverse) Fourier Transform is a representation of a signal as the weighted sum (integral) of phasors at a continuum of frequencies

9 What Do You See?

10 Some Things You Might See Maximum value at t = 0, where v(0) = 1 Zero-crossings at t =  1,  2,  3, … Many local maxima and minima  There are actually an infinite number  Positive and negative values  The area under the curve has both positive and negative contributors The curve is “even”, i.e., symmetric about t = 0

11 The Name of This Curve This function appears in many contexts in EE  The output of an ideal low-pass filter  The diffraction pattern from a rectangular optical or microwave aperture  The spectrum of a radar pulse  Sampling theory  Fourier series and Fourier transforms Note that this curve has a “name” because it is important in many applications – don’t be intimidated by fancy names of curves  Other examples of interesting curves with names are “sin”, “cos”, “log”, “bessel” functions, “legendre” polynomials, …

12 Finding the Maxima and Minima of a Curve A very common problem in many fields is to find the maxima and minima of a function The solution relates to the idea of the slope of a curve For a line y = mx + b, the slope is equal to m (and is constant for all x)

13 The Derivative For a general curve, the “derivative” of the function gives the slope – this is the first major part of Calculus  The derivative is a function of x, i.e., it changes as x changes  It can be thought of as the slope of the tangent line

14 So Where are the Maxima and Minima? The maxima and minima occur where the derivative equals zero! If the slope decreases as x increases, then it is a maximum If the slope increases as x increases, then it is a minimum

15 The Maxima and Minima of v(t) = sinc(t)? zero crossings of dv/dt yield maxima and minima

16 How Did We Find The Value of sinc(0)? At t = 0, the value of sinc(0) is 0/0 – This is undefined! The derivative comes to the rescue L’Hopital’s Rule: When the function equals 0/0, the value is obtained by taking the derivative of the numerator and denominator From calculus we learn Thus

17 Suppose we consider a simple pulse function The area under this curve is clearly equal to 1 But what if we have a curve like v(t) = sinc(t)? This motivates the “integral” of a function – this is the second major part of Calculus Concept: Partition the t-axis into a large number of approximately rectangular slices For the pulse example above, the t-axis could be divided into slices that are each 1-  sec long  Each slice has an area of  The sum of the 10 6 slices within the pulse is equal to 1 The Area Under a Curve (t)(t) t½ –½ 1

18 Let’s find the integral of a function between t = a and t = b It can be approximated by the 46 slices shown below In the limit, we write The Integral green – positive red – negative

19 The Area Under the Sinc Curve Interestingly enough, with an infinite number of positive and negative contributors to the area under the curve, the total area is 1 sinc(t) is an “even” function so the total area is twice the area of the positive-time area It isn’t easy to derive this so don’t worry about how this is found

20 A Great New Learning Opportunity Join the MOOC revolution … Massive Open Online Courses  A great way to add to your toolbox and grow your problem-solving skills and offer a wide array of math, science, and engineering courseswww.coursera.orgwww.edx.orgwww.udacity.com These are the same courses being taught at major universities, and require students to do homework and take exams In the last few years, I’ve taken the following Coursera courses  Cryptography I from Stanford  Game Theory from Stanford  Intro to Philosophy from University of Edinburgh  Leading Strategic Innovation in Organizations from Vanderbilt  Computational Neuroscience from University of Washington  Introduction to Functional Analysis from Ecole Centrale Paris  Future of (Mostly) Higher Education from Duke University UAH hosted its 1 st MOOC this year, Intro to Chemical Engineering by Dr. Chittur The courses can be quite challenging, but real learning occurs!

21 Summary Electrical Engineering is full of challenging mathematics The best EE students (and professionals) are those who do well in math Important concepts for you to master during your undergraduate program include  Calculus – Limits, Derivatives, Integrals  Differential Equations – Ordinary & Partial  Linear Algebra and Linear Operators  Complex Analysis (where “j ” lives)  Fourier Series, Fourier Transforms, and Laplace Transforms  Probability and Statistics Don’t just memorize formulas, know how to derive them and what they mean!!!  Dirac: “I understand what an equation means if I have a way of figuring out the characteristics of its solutions without actually solving it.” Learn to Love Math – It Can Be Contagious What are you doing with your 2,000,000 minutes???