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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 
f ( x)  a0 
an cos 


 L 
n1


 2p nx 
bn sin 

 L 
n1


Notation issues:

n1
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 = nfundamental
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The trick: finding the “Fourier coefficients”, an and bn
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compare to: f ( x)  a0 
an x n
Applications (a short list)
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“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 
f ( x)  a0 
an cos 


 L 
n1

a0 
1
L
L

an 
f ( x )dx
0
bn 
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What does
What does
1
a0 
L
2
a1 
L
L

0
2
L
2
L

0

0
 2p nx 
bn sin 

L


n1

L
L

 2p nx 
f ( x) cos 
dx

 L 
 2p nx 
f ( x)sin 
dx

 L 
L
 f ( x)dx
mean?
0
 2p x 
f ( x) cos 
dx

 L 
mean?
Let’s wait a
minute for
derivation.
Example: square wave

 2p nx 
f ( x)  a0 
an cos 


 L 
n1

a0 
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1
L
L

0
f ( x )dx
an 
2
L
L

0

 2p nx 
bn sin 

L


n1

 2p nx 
f ( x) cos 
 dx
L


bn 
2
L
L

0
 2p nx 
f ( x)sin 
dx

 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.

f ( x) 

n 1
(odd only)
 4
 np

  2p nx 
 sin  L 
 

 4   2p x   4   6p x   4   10p x 
f ( x)    sin 
   sin 
 
sin 
 ...




 p   L   3p   L   5p   L 
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Plots with Mathematica:
Deriving the coefficient equations

 2p nx 
f ( x)  a0 
an cos 


 L 
n1

a0 
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1
L
L

0
f ( x )dx
an 
2
L
L

0

 2p nx 
bn sin 

L


n1

 2p nx 
f ( x) cos 
 dx
L


bn 
2
L
L

0
 2p nx 
f ( x)sin 
dx

 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
1

2

1
 2p nx 
sin 

np
 L 
(The next few slides from Dr. Durfee)
N 0
N 1
N 2
N 3
N  10
N  500