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Announcements 10/19/11 Prayer Chris: today: 3-5 pm, Fri: no office hours Labs 4-5 due Saturday night Term project proposals due Sat night…

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Announcements 10/19/11 Prayer Chris: today: 3-5 pm, Fri: no office hours Labs 4-5 due Saturday night Term project proposals due Sat night (emailed to me) – One proposal per group; CC your partner(s) – See website for guidelines, grading, ideas, and examples of past projects. HW 22 due MONDAY instead of Friday. (HW 23 also due Monday) We’re half-way done with semester! Exam 2 starts a week from tomorrow! a. Review session: either Monday, Tues, or Wed. Please vote by tomorrow night so I can schedule the room on Friday. Anyone need my “Fourier series summary” handout? Pearls Before Swine Reading Quiz In the Fourier transform of a periodic function, which frequency components will be present? a. Just the fundamental frequency, f0 = 1/period b. f0 and potentially all integer multiples of f0 c. A finite number of discrete frequencies centered on f0 d. An infinite number of frequencies near f0, spaced infinitely close together e. 1320 KFAN (1320 kHz), home of the Utah Jazz… if there’s a season Fourier Theorem Any function periodic on a distance L can be written as a sum of sines and cosines like this: 2p nx 2p nx f ( x) a0 n1 an cos L n1 bn sin L Notation issues: compare to: f ( x) a0 n1 an x n a. a0, an, bn = how “much” at that frequency b. Time vs distance c. a0 vs a0/2 d. 2p/L = k (or k0) 2p/T = w (or w0 ) e. 2pn/L = nfundamental The trick: finding the “Fourier coefficients”, an and bn Applications (a short list) “What are some applications of Fourier transforms?” a. Electronics: circuit response to non-sinusoidal signals b. Data compression (as mentioned in PpP) c. Acoustics: guitar string vibrations (PpP, next lecture) d. Acoustics: sound wave propagation through dispersive medium e. Optics: spreading out of pulsed laser in dispersive medium f. Optics: frequency components of pulsed laser can excite electrons into otherwise forbidden energy levels g. Quantum: wavefunction of an electron in “particle in a box” situations, aka “infinite square well” How to find the coefficients 2p nx 2p nx f ( x) a0 n1 an cos L n1 bn sin L L L 2p nx 1 2 a0 f ( x )dx an f ( x) cos dx Let’s wait a L 0 L 0 L minute for L 2p nx derivation. 2 bn f ( x)sin dx L 0 L L f ( x)dx 1 a0 What does L mean? 0 L 2p x 2 What does a1 L f ( x) cos L dx mean? 0 Example: square wave 2p nx 2p nx f ( x) a0 n1 an cos L n1 bn sin L L L L 2p nx 2p nx 2 1 2 a0 f ( x )dx an f ( x) cos dx bn f ( x)sin dx L 0 L 0 L L 0 L f(x) = 1, from 0 to L/2 f(x) = -1, from L/2 to L (then repeats) a0 = ? 0 an = ? 0 b1 = ? 4/p b2 = ? Could work out each bn individually, but why? bn = ? 4/(np), only odd terms Square wave, cont. 4 2p nx f ( x) n 1 np sin L (odd only) 4 2p x 4 6p x 4 10p x f ( x) sin sin sin ... p L 3p L 5p L Plots with Mathematica: Deriving the coefficient equations 2p nx 2p nx f ( x) a0 n1 an cos L n1 bn sin L L L L 2p nx 2p nx 2 1 2 a0 f ( x )dx an f ( x) cos dx bn f ( x)sin dx L 0 L 0 L L 0 L To derive equation for a0, just integrate LHS and RHS from 0 to L. To derive equation for an, multiply LHS and RHS by cos(2pmx/L), then integrate from 0 to L. (To derive equation for bn, multiply LHS and RHS by sin(2pmx/L), then integrate from 0 to L.) Recognize that when n and m are different, cos(2pmx/L)cos(2pnx/L) integrates to 0. (Same for sines.) Graphical “proof” with Mathematica Otherwise, if m=n, then integrates to (1/2)L (Same for sines.) Recognize that sin(2pmx/L)cos(2pnx/L) always integrates to 0. Sawtooth Wave, like HW 22-2 2p nx 1 1 sin 2 np L (The next few slides from Dr. Durfee) N 0 N 1 N 2 N 3 N 10 N 500

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