fourier’s theorem beats????. fourier series – periodic functions
TRANSCRIPT
Fourier’s Theorem
Beats????
Fourier Series – Periodic Functions
T
n
0
T
n
0
for a function f t where:
2 the coefficients are calculated by:
T
2A f t cos n t dt for n 0, 1, 2, 3, ....
T
2B f t sin n t dt for n 1, 2, 3, ....
T
0n n
n 1
0 1 2 n
1 2 n
Af t A cos n t B sin n t or,
2
1f t A A cos t A cos 2 t A cos n t ....
2 B sin t B sin 2 t B sin n t ....
Why this works!• Fourier’s Hammer – say you wanted to find A2
• Multiple each term by cos(2t) and time average.
0 1 2 n
1 2 n
1f t A A cos t A cos 2 t A cos n t ....
2 B sin t B sin 2 t B sin n t ....
20 1 2 n
1 2 n
1f t cos 2 t A cos 2 t A cos t cos 2 t A cos 2 t A cos n t cos 2 t ....
2 B sin t cos 2 t B sin 2 t cos 2 t B sin n t cos 2 t ....
T T T T T21 2 n
0
0 0 0 0 0
T T T1 2 n
0 0 0
A A A1 1f t cos 2 t dt A cos 2 t dt cos t cos 2 t dt cos 2 t dt cos n t cos 2 t dt ....
T 2T T T T
B B B sin t cos 2 t dt sin 2 t cos 2 t dt sin n t cos 2 t dt ....
T T T
T T
222
0 0
A1 1f t cos 2 t dt cos 2 t dt A
T T 2
Example
0 when 0 tf t
1 when t 2
f(t)
1
-2
t
2
2 radT 2 sec 1
T sec
Note:
Coefficients
2 2
n
0 0
22
n
n
1 1 1A f t cos nt dt 1cos nt dt 0 cos nt dt
1 1A cos nt dt sin nx
n
1A sin n 2 sin n 0
n
2 2
n
0 0
22
n
n
n n
n
1 1 1B f t sin nt dt 1sin nt dt 0 sin nt dt
1 1B sin nt dt cos nx
n
1B cos n 2 cos n
n1
B 1 cos n for n odd numbers otherwise b 0n2
B for n odd numbersn
Example
f(t)
1
-2
t
2
...
5
t5
3
t3
1
t12
2
1tf
sinsinsin
Fourier Transform of a Square Wave
0
0.2
0.4
0.6
0.8
1
0 7 15 23 31 39
Frequency (rad/s)A
mp
litu
de
Time Domain Frequency Domain
0 when 0 tf t
1 when t 2
Demos
• Mathematica• Logger Pro
-6 -4 -2 2 4 6
-1
-0.5
0.5
1
1.5
Odd and Even Functions tft-f :Function Odd
tft-f :FunctionEven
Even Odd
Odd and Even Functions tft-f :Function Odd
tft-f :FunctionEven
Even Odd
.... 3, 2, 1,nfor dttntfT
2bb
0a then odd is xf If
T
0nn
n
sin
0b
.... 3, 2, 1, 0nfor dttntfT
2a
even then is tf Ifn
T
0n ,cos
Fourier Transforms
jwtf t g w e dw
jwt1g w f t e dt
2
Spectral Density
Dirac Delta Function
t 0 t 0
t dt 1
0 t
2t1
t2
Spectral Density of a Delta Function
jwt1g w f t e dt
2
/ 2
jwt
/ 2
1 1 1 1g w t e dt 1 dt
2 2 2
What if Spectral Density is a Delta Function
g w w
jwt jwt j tf t g w e dw w e dw e
Heavyside Step Function
t 0 t 0
1 t t dt1 t 0
jwt jwt1 1g w f t e dt 1 t e dt
2 2
jwt jwt
0 0
1 1 1 1 1 1 1e dt e 0 1
2 2 jw 2 jw 2 jw
Table of Fourier Transforms (1.15.1)
So What????
jwtF t G w e dw
jwtf w, t G w e
jwtf w, t G wu w, t e
Z w Z w
jwtG wU t u w, t dw e dw
Z w
Example 1.15.6
• A simple oscillator at rest is struck with a force F(t) = (F) 1(t) where F = 1 N. Find the displacement and speed of the oscillator using section 1.15.