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?

PearlsBefore 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:

Notation issues: a. a0, an, bn = how “much”

at that frequencyb. Time vs distancec. a0 vs a0/2

d. 2/L = k (or k0) 2/T = (or 0 )e. 2n/L = nfundamental

The trick: finding the “Fourier coefficients”, an and bn

01 1

2 2( ) cos sinn n

n n

nx nxf x a a b

L L

01

compare to: ( ) nn

n

f x a a x

Applications (a short list) “What are some applications of Fourier transforms?”

a. Electronics: circuit response to non-sinusoidal signalsb. Data compression (as mentioned in PpP)c. Acoustics: guitar string vibrations (PpP, next lecture)d. Acoustics: sound wave propagation through

dispersive mediume. Optics: spreading out of pulsed laser in dispersive

mediumf. 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

What does mean?

What does mean?

0

0

1( )

L

a f x dxL

0

2 2( )cos

L

nnx

a f x dxL L

0

2 2( )sin

L

nnx

b f x dxL L

01 1

2 2( ) cos sinn n

n n

nx nxf x a a b

L L

0

0

1( )

L

a f x dxL

1

0

2 2( )cos

Lx

a f x dxL L

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) a0 = ? an = ? b1 = ? b2 = ? bn = ?

0

0

1( )

L

a f x dxL

0

2 2( )cos

L

nnx

a f x dxL L

0

2 2( )sin

L

nnx

b f x dxL L

01 1

2 2( ) cos sinn n

n n

nx nxf x a a b

L L

004/Could work out each bn individually, but why?

4/(n), only odd terms

Square wave, cont.

Plots with Mathematica:

1(odd only)

4 2( ) sin

n

nxf x

n L

4 2 4 6 4 10( ) sin sin sin ...

3 5

x x xf x

L L L

Deriving the coefficient equations

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(2mx/L),

then integrate from 0 to L.(To derive equation for bn, multiply LHS and RHS by sin(2mx/L), then integrate from 0 to L.)

Recognize that when n and m are different, cos(2mx/L)cos(2nx/L) integrates to 0. (Same for sines.)

Graphical “proof” with MathematicaOtherwise, if m=n, then integrates to (1/2)L (Same for sines.)

Recognize that sin(2mx/L)cos(2nx/L) always integrates to 0.

0

0

1( )

L

a f x dxL

0

2 2( )cos

L

nnx

a f x dxL L

0

2 2( )sin

L

nnx

b f x dxL L

01 1

2 2( ) cos sinn n

n n

nx nxf x a a b

L L

0N 1N 2N

3N 10N 500N

1 1 2sin

2

nx

n L

Sawtooth Wave, like HW 22-2

(The next few slides from Dr. Durfee)