BerinatoFall 11 - 1 Introduction to Math Methods in...

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Unformatted text preview: 1 Introduction to Math Methods in Electrical Engineering An EE100 Lecture Dr. Bob Berinato [email protected] www.dynetics.com Fall 2011 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 ¾ Complex Number Notation and Applications ¾ Properties of Curves ¾ Derivatives ¾ 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 ¾ Complex Math and Fourier Transforms are essential tools www.sandia.gov/radar/imageryku.html 4 What are the 5 most important “numbers” in mathematics? 2 integers: 0, 1 2 irrationals: π , e 1 complex: Could these possibly be related in one equation? 1 − = j 5 Leonhard Euler – An Amazing Individual! 1 = + π j e 1 , sin cos − = + = j j e j θ θ θ a jb 1 θ cos θ sin θ Euler’s Formula: expresses sines and cosines in complex number notation ∑ ∞ = = 1 2 2 6 1 k k π 6 Application: Signal Notation • Real Signal Notation: • Complex Signal Notation: ( ) ( ) ( ) [ ] t t f t a t s o φ π + = 2 cos ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) { } [ ] ( ) ( ) ( ) [ ] t t f t ja t s t t f j t a t f j t j t a t f j t m t z o o o o φ π φ π π φ π + + = + = = = 2 sin 2 exp 2 exp exp 2 exp Magnitude Phase Center Frequency Complex Envelope, or Amplitude Magnitude Phase Re[ z ( t )] = s ( t ), the “real” signal 7 Application: The Phasor • Consider the following complex signal, called a phasor • This is the complex notation for a sinusoid with phase φ and amplitude A • Note that at time...
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This note was uploaded on 12/14/2011 for the course EE 1001 taught by Professor Hite during the Fall '11 term at University of Alabama - Huntsville.

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BerinatoFall 11 - 1 Introduction to Math Methods in...

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