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Advanced Digital Signal Processing
Prof. Nizamettin AYDIN
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Fourier Series Coefficients
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Work with the Fourier Series Integral
ANALYSIS via Fourier Series For PERIODIC signals: x(t+T0) = x(t) Spectrum from the Fourier Series
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HISTORY Jean Baptiste Joseph Fourier 1807 thesis (memoir) Heat !
On the Propagation of Heat in Solid Bodies Heat ! Napoleonic era
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SPECTRUM DIAGRAM Recall Complex Amplitude vs. Freq 100 250 –100 –250
100 250 –100 –250 f (in Hz)
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Harmonic Signal PERIOD/FREQUENCY of COMPLEX EXPONENTIAL:
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Fourier Series Synthesis
COMPLEX AMPLITUDE
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Harmonic Signal (3 Freqs)
T = 0.1
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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
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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
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INTEGRAL Property of exp(j)
INTEGRATE over ONE PERIOD
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ORTHOGONALITY of exp(j)
PRODUCT of exp(+j ) and exp(-j )
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Isolate One FS Coefficient
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SQUARE WAVE EXAMPLE –.02 .02 0.04 1 t x(t) .01
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FS for a SQUARE WAVE {ak}
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DC Coefficient: a0
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Fourier Coefficients ak
ak is a function of k Complex Amplitude for k-th Harmonic This one doesn’t depend on the period, T0
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Spectrum from Fourier Series
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Fourier Series Integral
HOW do you determine ak from x(t) ?
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Fourier Series & Spectrum
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Example
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Example In this case, analysis just requires picking
off the coefficients.
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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
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FS: Rectified Sine Wave {ak}
Half-Wave Rectified Sine
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FS: Rectified Sine Wave {ak}
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Fourier Coefficients ak
ak is a function of k Complex Amplitude for k-th Harmonic This one doesn’t depend on the period, T0
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Fourier Series Synthesis
HOW do you APPROXIMATE x(t) ? Use FINITE number of coefficients
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Fourier Series Synthesis
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Synthesis: 1st & 3rd Harmonics
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Synthesis: up to 7th Harmonic
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Fourier Synthesis
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Gibbs’ Phenomenon Convergence at DISCONTINUITY of x(t)
There is always an overshoot 9% for the Square Wave case
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Fourier Series Demos Fourier Series Java Applet
Greg Slabaugh Interactive MATLAB GUI: fseriesdemo
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