Transcript Folie 1 - bplaced

Verfahrenstechnische Produktion
Studienarbeit Angewandte Informationstechnologie
WS 2008 / 2009
Fourier Series and the Fourier Transform
Karl Kellermayr
Fourier – Series / Fourier – Reihen / Fourier
Transform
•
Fourier - What is a fourier-serie? What is
meaned by time-domain and fourier-domain
(frequency domain) of a function (timesignal)?
•
What are fourier-series good for?
•
The fourier-serie of a periodical function.
Folie 2
Introduction Time and Frequency Domain
Fourier Series – Fourier Transform
• Jean Baptiste Joseph Fourier
(1768-1830)
• French Mathematician: La Théorie
Analitique de la Chaleur (1822)
• Fourier Series:
Any periodic function can be expressed as
a sum of sine and/or cosine functions Fourier Series
• Fourier Transform:
Even functions that are not periodic but
have a finite area under curve can be
expressed as an integral of sines and
cosines multiplied by a weighing function.
• Both the Fourier Series and the Fourier
Transform have an inverse operation, that
means functions can be described in 2
different domains:
Original Domain (Time Domain)
Fourier Domain (Frequency Domain)
(from: http://en.wikipedia.org/wiki/Joseph_Fourier)
Folie 3
3
Example of periodic function: Sinusoid
What is the amplitude, period, frequency, and angular
(radian) frequency of this sinusoid?
8
6
4
Period:
T = 1/f =0,02 sec
Frequency:
f = 1/T = 50 Hz
Amplitude:
A=7
2
0
-2 0
0,01
0,02
0,03
0,04
0,05
-4
-6
-8
Time (sec)
Folie 4
Which parameters characterice a Sinusoid
Signal: x(t)=A.sin(t)
Period:
Time necessary to go through one cycle
T = 2p/ω = 1/f
Frequency: Cycles per second (Hertz, Hz)
f = 1/T
Angular frequency (Kreisfrequenz):
Radians per second
(radian…Winkel im Bogenmaß)
ω = 2p f
Amplitude: A for example, could be 5 volts or 5 amps
Folie 5
Example: A sum of sines and cosines
=
3 sin(x)
A
+ 1 sin(3x)
B
+ 0.8 sin(5x)
C
+ 0.4 sin(7x)
D
A+B
A+B+C
A+B+C+D
Folie 6
Periodical functions
1,5
1
0
0
2
4
6
8
10
12
14
16
-0,5
-1
Periodische Funktion
-1,5
1,5
x
1,3
1,1
Wert y(t)
y
0,5
0,9
0,7
0,5
0,3
0,1
0
2
4
6
8
10
12
14
16
-0,1
Zeit t
-0,3
-0,5
Folie 7
Periodical functions
Sägezahn
1,2
1
Wert y(t)y
0,8
0,6
0,4
0,2
0
0
2
4
6
8
10
12
14
16
Zeit t
-0,2
y(t)=0,5+0,5sin(t)+0,2sin/3t)
1,4
1,2
Wert y(t)
1
0,8
0,6
0,4
0,2
0
0
-0,2
2
4
6
8
10
12
14
16
Zeit t
Folie 8
Fourier-series of an arbitrary
periodical function
A periodical function y = f(x) with period p = 2p can be in some
situations developed to an infinite trigonometric series:

0
n
n
n1
a
f (x)   a  cos(nx)  b  sin(nx)
2
1,5000
1,0000
0,5000
f(f)=sin(x)
g(x)=sin(3x) /3
6,5
5,9
5,3
4,7
4,1
3,5
2,9
2,3
1,7
1,1
0,5
h(x)=sin(5x)/5
x
0,0000
i(x)=sin(7x)/7
(4/PI)*(f+g+h+i)(x)
-0,5000
-1,0000
-1,5000
Folie 9
Calculation of
Fourier-Coefficients
1
a0  
p
2p
1
an  
p
2p
1
bn  
p
2p
 f (x)dx
0
 f (x)  cos(nx)dx
(n  N)
0
 f (x)  sin(nx)dx
(n  N)
0
Folie 10