1. Advanced Digital Signal Processing
Prof. Nizamettin AYDIN

2. Fourier Series Coefficients

3. Work with the Fourier Series Integral
ANALYSIS via Fourier Series
For PERIODIC signals: x(t+T0) = x(t)
Spectrum from the Fourier Series

4. HISTORY Jean Baptiste Joseph Fourier 1807 thesis (memoir) Heat !
On the Propagation of Heat in Solid Bodies
Heat !
Napoleonic era

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

7. Harmonic Signal
PERIOD/FREQUENCY of COMPLEX EXPONENTIAL:

8. Fourier Series Synthesis
COMPLEX
AMPLITUDE

9. Harmonic Signal (3 Freqs)
T = 0.1

10. SYNTHESIS vs. ANALYSIS SYNTHESIS Synthesis can be HARD ANALYSIS Easy
Given (wk,Ak,fk) create x(t)
Synthesis can be HARD
Synthesize Speech so that it sounds good
ANALYSIS
Hard
Given x(t), extract (wk,Ak,fk)
How many?
Need algorithm for computer

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

12. INTEGRAL Property of exp(j)
INTEGRATE over ONE PERIOD

13. ORTHOGONALITY of exp(j)
PRODUCT of exp(+j ) and exp(-j )

14. Isolate One FS Coefficient

15. SQUARE WAVE EXAMPLE
–.02
.02
0.04 1 t
x(t)
.01

16. FS for a SQUARE WAVE {ak}

17. DC Coefficient: a0

18. Fourier Coefficients ak
ak is a function of k
Complex Amplitude for k-th Harmonic
This one doesn’t depend on the period, T0

19. Spectrum from Fourier Series

20. Fourier Series Integral
HOW do you determine ak from x(t) ?

21. Fourier Series & Spectrum

22. Example

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

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

25. FS: Rectified Sine Wave {ak}
Half-Wave Rectified Sine

26. FS: Rectified Sine Wave {ak}

27. Fourier Coefficients ak
ak is a function of k
Complex Amplitude for k-th Harmonic
This one doesn’t depend on the period, T0

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

29. Fourier Series Synthesis

30. Synthesis: 1st & 3rd Harmonics

31. Synthesis: up to 7th Harmonic

32. Fourier Synthesis

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

34. Fourier Series Demos Fourier Series Java Applet
Greg Slabaugh
Interactive
MATLAB GUI: fseriesdemo