1. 1 Prof. Nizamettin AYDIN naydin@yildiz.edu.tr http://www.yildiz.edu.tr/~naydin Digital Signal Processing

2. 2 Lecture 7 Fourier Series & Spectrum Digital Signal Processing

3. 3 License Info for SPFirst Slides This work released under a Creative Commons License with the following terms:Creative Commons License Attribution The licensor permits others to copy, distribute, display, and perform the work. In return, licensees must give the original authors credit. Non-Commercial The licensor permits others to copy, distribute, display, and perform the work. In return, licensees may not use the work for commercial purposes—unless they get the licensor's permission. Share Alike The licensor permits others to distribute derivative works only under a license identical to the one that governs the licensor's work. Full Text of the License This (hidden) page should be kept with the presentation

4. 4 READING ASSIGNMENTS This Lecture: –Fourier Series in Ch 3, Sects 3-4, 3-5 & 3-6 Replaces pp. 62-66 in Ch 3 in DSP First Notation: a k for Fourier Series Other Reading: –Next Lecture: Sampling

5. 5 LECTURE OBJECTIVES ANALYSISANALYSIS via Fourier Series –For PERIODIC signals: x(t+T 0 ) = x(t) SPECTRUMSPECTRUM from Fourier Series –a k is Complex Amplitude for k-th Harmonic

6. 6 0100250–100–250 f (in Hz) SPECTRUM DIAGRAM Recall Complex Amplitude vs. Freq

7. 7 Harmonic Signal PERIOD/FREQUENCY of COMPLEX EXPONENTIAL:

8. 8 Example

9. 9 In this case, analysis just requires picking off the coefficients.

10. 10 STRATEGY: x(t)  a k ANALYSIS –Get representation from the signal PERIODIC –Works for PERIODIC Signals Fourier Series –Answer is: an INTEGRAL over one period

11. 11 FS: Rectified Sine Wave {a k } Half-Wave Rectified Sine

12. 12 FS: Rectified Sine Wave {a k }

13. 13 SQUARE WAVE EXAMPLE 0 –.02.020.04 1 t x(t).01

14. 14 FS for a SQUARE WAVE {a k }

15. 15 DC Coefficient: a 0

16. 16 Fourier Coefficients a k a k is a function of k –Complex Amplitude for k-th Harmonic –This one doesn’t depend on the period, T 0

17. 17 Spectrum from Fourier Series

18. 18 Fourier Series Synthesis HOW do you APPROXIMATE x(t) ? Use FINITE number of coefficients 16Kasim2k11

19. 19 Fourier Series Synthesis

20. 20 Synthesis: 1st & 3rd Harmonics

21. 21 Synthesis: up to 7th Harmonic

22. 22 Fourier Synthesis

23. 23 Gibbs’ Phenomenon Convergence at DISCONTINUITY of x(t) –There is always an overshoot –9% for the Square Wave case

24. 24 Fourier Series Demos Fourier Series Java Applet –Greg Slabaugh Interactive –http://users.ece.gatech.edu/mcclella/2025/Fsdemo_Slabaugh/fourier.htmlhttp://users.ece.gatech.edu/mcclella/2025/Fsdemo_Slabaugh/fourier.html MATLAB GUI: fseriesdemo –http://users.ece.gatech.edu/mcclella/matlabGUIs/index.htmlhttp://users.ece.gatech.edu/mcclella/matlabGUIs/index.html

25. 25 fseriesdemo GUI

26. 26 Fourier Series Java Applet

27. 27 Harmonic Signal (3 Freqs) T = 0.1 a3a3 a5a5 a1a1