Fourier

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Fouriers Theorem
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Fouriers Theorem

• any periodic (or regularly repeating) wave,
  however complicated, can be described in terms of
  an infinite number of sine waves (of various
  amplitudes and phases) added together

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Fouriers Equation
the sine and cosine parts deal with phases of
partials
coefficients (give energies of partials)
represents the fundamental frequency of the
waveform
represents time
number of the harmonic
add all harmonics from n1 to n infinity
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Angular Frequency

• ?1 angular frequency (in radians)

means the same as

• but only works for sine waves
• (one cycle of a sine wave in radians 2p)

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Analysis and Synthesis

• Fourier analysis involves taking a wave and
  breaking it up into its constituent components
• Fourier synthesis involves constructing a wave by
  adding up sine waves

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Fourier Analysis/Synthesis

• Fourier analysis and synthesis are used a lot in
  D.S.P.
• For example, to apply effects to digital audio

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Fourier Synthesis Examples
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Creating a Square Wave
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Coefficient Values for a Square Wave
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Insert Coefficients
insert these co-efficients
into Fouriers equation
to get
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Calculate Terms at Each Time (t)
e.g. at t T1/4
when n1 sin ?1t sin (2?/T1 T1/4) sin
2?/4 sin ?/2 sin 90 1    
 when n3 1/3 sin (3?1t) 1/3 sin (3 2?/T1
T1/4) 1/3 sin 6?/4   1/3 sin 3?/2 1/3
sin 270 1/3 -1 -1/3
when n5 1/5 sin (5?1t) 1/5 sin (5 2?/T1
T1/4) 1/5 sin 10?/4 1/5 sin 5?/2 1/5
sin 450 1/5 1 1/5
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Calculating Terms Cont

• according to the formula multiply each value by
  4/?
• this makes

• first term (n 1) 14/? 1.273 (at t T1/4)
• second term (n3) -1/3 4/? -0.424 (at t
  T1/4)
• third term (n 5) 1/5 4/? 0.255 (at t
  T1/4)

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Plot Values
A plot of the first 5 terms (all t values).
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Add Waves Together
A plot of the first five terms all added together.
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Add More Terms
A plot of the first twenty terms all added
together.
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Square Wave Freq. Domain Plot
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Practicalities

• practical synthesis does not require an infinite
  number of sine waves
• DiscoDSPs Vertigo has 256 partials
• VirSins Cube has 512

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Sawtooth Wave
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Sawtooth - Freq. Domain Plot
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Triangle Wave
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Triangle Freq. Domain Plot
Coefficients