1. Verfahrenstechnische Produktion Studienarbeit Angewandte Informationstechnologie WS 2008 / 2009 Fourier Series and the Fourier Transform Karl Kellermayr

2. Folie 2 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.

3. Folie 3 3 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)

4. Folie 4 Example of periodic function: Sinusoid What is the amplitude, period, frequency, and angular (radian) frequency of this sinusoid? Period: T = 1/f =0,02 sec Frequency: f = 1/T = 50 Hz Amplitude: A = 7

5. Folie 5 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

6. Folie 6 = 3 sin(x) A + 1 sin(3x) B A+B + 0.8 sin(5x) C A+B+C + 0.4 sin(7x)D A+B+C+D Example: A sum of sines and cosines sin(x) A

7. Folie 7 Periodical functions

8. Folie 8 Periodical functions

9. Folie 9 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:

10. Folie 10 Calculation of Fourier-Coefficients