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# Fourier series expansion

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Fourier series expansion

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### Fourier series expansion

1. 1. Fourier series expansion
2. 2. Spectral analysisMost part of signals involved in systemsworking, are time-varying quantities.Although a signal physically exists intime domain, we can represent it in theso called frequency domain, in which itconsists of a series of sinusoidalcomponents at various frequencies.The frequency domain description iscalled spectral analysis.24/05/2012 2Fourier series expansion
3. 3. LTI systemsThe spectral analysis of signals, coupled withfrequency response characteristics of systems,allows us to have a good approach in designwork.In fact, when we can study the behavior of alinear time-invariant (LTI) system in presence ofa particular sinusoidal signal, we can alsostudy the behavior in presence of all thesinusoidal signals, and therefore in presence ofall the signals which we can considercomposed by a series of sinusoidal signals.24/05/2012Fourier series expansion 3
4. 4. Jean Baptiste Joseph FourierThe mathematical methodswhich help us in this workare based on the studies ofa French physicist andmathematician, Jean Bap-tiste Joseph Fourier wholived between XVIII and XIXcentury.24/05/2012Fourier series expansion 4
5. 5. MethodsThere are two kinds of methods:• Series expansion of periodic signals• Transform of non-periodic signals24/05/2012Fourier series expansion 5
6. 6. Series expansionEvery periodic function can be represented asthe expansion of a series of sinusoidal functions:24/05/2012Fourier series expansion 6where
7. 7. Example 1 - 124/05/2012Fourier series expansion 7
8. 8. Example 1 - 224/05/2012Fourier series expansion 8periodic odd function
9. 9. Example 1 - 324/05/2012Fourier series expansion 9in the end:If b=0, the function represents the restriction of function sgn(t)into the ] interval, periodically extended outside. In thefigure below (left) are represented the first five Fourierpolynomials of this function.The amplitude spectrum is a line spectrum (right figure).
10. 10. Example 224/05/2012Fourier series expansion 10periodic even function
11. 11. Example 324/05/2012Fourier series expansion 11
12. 12. Example 4 - 124/05/2012Fourier series expansion 12  
13. 13. Example 4 - 224/05/2012Fourier series expansion 13
14. 14. Example 4 - 324/05/2012Fourier series expansion 14 
15. 15. Example 4 - 424/05/2012Fourier series expansion 15
16. 16. Example 524/05/2012Fourier series expansion 16
17. 17. Example 624/05/2012Fourier series expansion 17