Announcements 10/20/10 Prayer Term project proposals due on Saturday night! to me: proposal in body of , 650 word max. See website for guidelines, grading, ideas, and examples of past projects. a. a.If in a partnership, just one from the two of you Exam 2 starts a week from tomorrow. a. a.Exam 2 optional review session: vote on times by tomorrow evening. Survey link sent out this morning. Anyone not get the Fourier transform handout last lecture?

Quick Writing Ralph doesn’t understand what a transform is. As discussed last lecture and in today’s reading, how would you describe the “transform” of a function to him?

Reading Quiz In the Fourier transform of a periodic function, which frequency components will be present? a. a.Just the fundamental frequency, f 0 = 1/period b. b.f 0 and potentially all integer multiples of f 0 c. c.A finite number of discrete frequencies centered on f 0 d. d.An infinite number of frequencies near f 0, spaced infinitely close together

Fourier Theorem Any function periodic on a distance L can be written as a sum of sines and cosines like this: Notation issues: a. a.a 0, a n, b n = how “much” at that frequency a. a.Time vs distance b. b.a 0 vs a 0 /2 c. c.2  /L = k (or k 0 )… compare 2  /T =  (or  0 ) d. d.Durfee: – – a n and b n reversed – – Uses 0 instead of L The trick: finding the “Fourier coefficients”, a n and b n

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

Example: square wave f(x) = 1, from 0 to L/2 f(x) = -1, from L/2 to L (then repeats) a 0 = ? a n = ? b 1 = ? b 2 = ? b n = ? 0 0 4/  Could work out each b n individually, but why? 4/(n  ), only odd terms

Square wave, cont. Plots with Mathematica: square wave Fourier.nb

Deriving the coefficient equations To derive equation for a 0, just integrate LHS and RHS from 0 to L. To derive equation for a n, multiply LHS and RHS by cos(2  mx/L), then integrate from 0 to L. (To derive equation for b n, multiply LHS and RHS by sin(2  mx/L), then integrate from 0 to L.) Recognize that when n and m are different, cos(2  mx/L)  cos(2  nx/L) integrates to 0. (Same for sines.) Graphical “proof” with Mathematica Otherwise integrates to (1/2)  L (and m=n). (Same for sines.) Recognize that sin(2  mx/L)  cos(2  nx/L) always integrates to 0.

Sawtooth Wave, like HW 22-1 (The next few slides from Dr. Durfee)

The Spectrum of a Saw-tooth Wave

Electronic “Low-pass filter” “Low pass filter” = circuit which preferentially lets lower frequencies through. ? Circuit What comes out? How to solve: (1)Decompose wave into Fourier series (2)Apply filter to each freq. individually (3)Add up results in infinite series again

Low-Pass Filter – before filter

Low-Pass Filter – after filter

Low Pass Filter

Actual Data from Oscilloscope