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Announcements 10/17/12

Announcements 10/17/12. Prayer TA: on Fridays Clement can now only have office hours 3-4 pm. Available 12-1 pm by request. Term project proposals due Sat night (emailed to me) “Learn Smart” trial – I’ll send info by email Exam 2 starts a week from tomorrow!

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Announcements 10/17/12

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  1. Announcements 10/17/12 • Prayer • TA: on Fridays Clement can now only have office hours 3-4 pm. Available 12-1 pm by request. • Term project proposals due Sat night (emailed to me) • “Learn Smart” trial – I’ll send info by email • Exam 2 starts a week from tomorrow! • 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

  2. From warmup Extra time on? (nothing in particular) Other comments? this stuff is so awesome! This is very interesting stuff to me, especially how Dr. Durfee related it to the real world. I remember doing Fourier Transforms in Physics 145 and I didn't understand them and thus absolutely hated them! I work in the math lab and I was able to answer a question from someone not in a math class about fourier transforms. It was why the integral is zero if they have different frequencies. My mom doesn't know anything about fourier transforms but I still wanted someone to be proud of me. In case you haven't already seen it: http://xkcd.com/26/

  3. From warmup The most interesting functions that you created

  4. Warmup, cont.

  5. Warmup, cont.

  6. Warmup, cont.

  7. Transforms Transform: one-to-one mapping from function to list of coefficients (or to a function if “spacing” between coefficients becomes infinitely small) Example: Taylor’s series for ex = 1 + x + x2/2! + x3/3! + … ex (1, 1, 1/2!, 1/3!, …) The coefficients tell you the amplitudes of the polynomials that are present. Fourier Transform: The coefficients tell you the amplitudes of the frequencies that are present. Example: 

  8. Fourier Transform Answer from computer: “There are several components at different values of k; all are multiples of k = 0.01. k = 0.01: amplitude = 0 k = 0.02: amplitude = 0 … … k = 0.90: amplitude = 1 k = 0.91: amplitude = 1 k = 0.92: amplitude = 1 …” Do the transform (or have a computer do it)  How did the computer know the k values are all multiples of k = 0.01?

  9. From warmup  Fig. 6.2--what the two plots are showing? The top plot is a Fourier transform of Dr. Durfee saying the word "hello." [It can be represented by a sum of sine waves with different frequencies.] The bottom plot is the sine functions' amplitudes plotted as a function of frequency.

  10. Fourier Theorem • Any function periodic on a distance L can be written as a sum of sines and cosines like this: • Notation issues: • a0, an, bn = how “much” at that frequency • Time vs distance • a0 vs a0/2 • 2p/L = k (or k0) 2p/T = w (or w0 ) • 2pn/L = nfundamental • The trick: finding the “Fourier coefficients”, an and bn

  11. Applications (a short list) • “What are some applications of Fourier transforms?” • Electronics: circuit response to non-sinusoidal signals • Data compression (as mentioned in PpP) • Acoustics: guitar string vibrations (PpP, next lecture) • Acoustics: sound wave propagation through dispersive medium • Medical imaging: constructing 3D image from 2D pictures • Optics: spreading out of pulsed laser in dispersive medium • Optics: frequency components of pulsed laser can excite electrons into otherwise forbidden energy levels • Quantum: wavefunction of an electron in a periodic structure (e.g. atoms in a solid)

  12. How to find the coefficients • What does mean? • What does mean? Let’s wait a minute for derivation.

  13. Example: square wave • f(x) = 1, from 0 to L/2 • f(x) = -1, from L/2 to L (then repeats) • a0 = ? • an = ? • b1 = ? • b2 = ? • bn = ? 0 0 4/p Could work out each bn individually, but why? 4/(np), only odd terms

  14. Square wave, cont. • Plots with Mathematica:

  15. Square wave, cont.

  16. Square wave, cont.

  17. Square wave, cont. nmax = 500 Why is it still not quite getting the corners right?

  18. Deriving the coefficient equations • From Warmup: There are really only two main steps in PpP 6.4: 1) multiply by another cosine (but of a different frequency), and 2) integrate the product over a period. And the important thing to realize is that the product of two cosines of different frequecies, integrated over an integer number of periods, will give you zero. So all of the terms in the summation except one just integrate to zero.

  19. Deriving the coefficient equations a0: just integrate LHS and RHS from 0 to L. an: multiply LHS and RHS by cos(2pmx/L), then integrate bn: multiply LHS and RHS by sin(2pmx/L), then integrate Details: If n  m, then cos(2pmx/L)cos(2pnx/L) integrates to 0. (Same for sines.) If n = m, then it integrates to (1/2)L (Same for sines.) Either way, sin(2pmx/L)cos(2pnx/L) always integrates to 0. Graphical “proof” with Mathematica

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