Verfahrenstechnische ProduktionStudienarbeit Angewandte Informationstechnologie

WS 2008 / 2009

Fourier Series and the Fourier Transform

Karl Kellermayr

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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 (time-signal)?

• What are fourier-series good for?

• The fourier-serie of a periodical function.

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3

Introduction Time and Frequency DomainFourier 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)

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Example of periodic function: Sinusoid

What is the amplitude, period, frequency, and angular (radian) frequency of this sinusoid?

-8

-6

-4

-2

0

2

4

6

8

0 0,01 0,02 0,03 0,04 0,05

Time (sec)

Period: T = 1/f =0,02 secFrequency:f = 1/T = 50 HzAmplitude: A = 7

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Which parameters characterice a Sinusoid

Signal: x(t)=A.sin(t)Period: Time necessary to go through one cycle

T = 2/ω = 1/f

Frequency: Cycles per second (Hertz, Hz)f = 1/T

Angular frequency (Kreisfrequenz): Radians per second (radian…Winkel im Bogenmaß)ω = 2 f

Amplitude: A for example, could be 5 volts or 5 amps

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=

3 sin(x) A

+ 1 sin(3x) B A+B

+ 0.8 sin(5x) CA+B+C

+ 0.4 sin(7x) DA+B+C+D

Example: A sum of sines and cosines

sin(x) A

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Periodical functions

-1,5

-1

-0,5

0

0,5

1

1,5

0 2 4 6 8 10 12 14 16

x

y

Periodische Funktion

-0,5

-0,3

-0,1

0,1

0,3

0,5

0,7

0,9

1,1

1,3

1,5

0 2 4 6 8 10 12 14 16

Zeit t

Wert y

(t)

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Periodical functions

Sägezahn

-0,2

0

0,2

0,4

0,6

0,8

1

1,2

0 2 4 6 8 10 12 14 16

Zeit t

Wert y

(t)y

y(t)=0,5+0,5sin(t)+0,2sin/ 3t)

-0,2

0

0,2

0,4

0,6

0,8

1

1,2

1,4

0 2 4 6 8 10 12 14 16

Zeit t

Wert

y(t

)

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Fourier-series of an arbitrary periodical function

A periodical function y = f(x) with period p = 2 can be in some situations developed to an infinite trigonometric series:

f (x) a0

2 an cos(nx) bn sin(nx)

n1

-1,5000

-1,0000

-0,5000

0,0000

0,5000

1,0000

1,5000

x

0,5

1,1

1,7

2,3

2,9

3,5

4,1

4,7

5,3

5,9

6,5

f(f)=sin(x)

g(x)=sin(3x) /3

h(x)=sin(5x)/5

i(x)=sin(7x)/7

(4/PI)*(f+g+h+i)(x)

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Calculation of Fourier-Coefficients

a0 1

f (x)dx

0

2

an 1

f (x)cos(nx)dx (n N)

0

2

bn 1

f (x)sin(nx)dx (n N)

0

2