1 Prof. Nizamettin AYDIN Digital Signal Processing

2 Lecture 7 Fourier Series & Spectrum Digital Signal Processing

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

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

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

7 Harmonic Signal PERIOD/FREQUENCY of COMPLEX EXPONENTIAL:

8 Example

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

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

12 FS: Rectified Sine Wave {a k }

13 SQUARE WAVE EXAMPLE 0 – t x(t).01

14 FS for a SQUARE WAVE {a k }

15 DC Coefficient: a 0

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 Spectrum from Fourier Series

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

19 Fourier Series Synthesis

20 Synthesis: 1st & 3rd Harmonics

21 Synthesis: up to 7th Harmonic

22 Fourier Synthesis

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

24 Fourier Series Demos Fourier Series Java Applet –Greg Slabaugh Interactive – MATLAB GUI: fseriesdemo –

25 fseriesdemo GUI

26 Fourier Series Java Applet

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