Acoustics of Music Semester 2

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Acoustics of MusicSemester 2

• Dr Ian Drumm

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Waveforms

• Aims
• An introduction to Fourier Analysis and Synthesis
  Techniques
• Learning Outcomes
• Explain the mathematical basis for additive
  synthesis
• The study of complex musical timbres
• The synthesising of complex musical timbres from
  simpler components

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Fourier Series

• Linear Superimposition of Sinusoids to build
  complex waveforms
• If periodic (repeating)

Jean Baptiste Joseph Fourier 1768-1830
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Fourier Series

• Decompose our complex periodic waveform into a
  series of simple sinusoids
• Where

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Square Wave Example

• Consider
• Clearly period T2 hence ?p
• When we integrate we need to do so over sections
• t 0 to 1 and t1 to 2

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• So to find series calculate coefficients a0,an
  and bn

When evaluating an note the sin function is 0
when angle is every multiple of p
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• Knowing
• We need to consider the cos function to determine
  values of bn for n1,2,3,. etc

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• We found coefficients to be
• Hence Fourier Series for a square wave is

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• Lets try implementing series in Matlab
• We can see its not quite a square wave (Gibbs
  phenomenon) given series should be infinite.
  However with a for loop we can add more to the
  series

w500 tlinspace(0,2pi,100) y14/pi(sin(wt)1
/3sin(3wt)1/5sin(5wt)) plot(t,y1)
To n10
To n1000
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Saw tooth example

• Evaluating Fourier series for saw tooth produces

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Triangle Example

• Evaluating Fourier series for triangle wave

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Fourier Transforms

• The Fourier Transform
• The Inverse Fourier Transform

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Discrete Fourier Transforms

• The Discrete Fourier Transform
• The Discrete Inverse Fourier Transform

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Implementing a DFT

• Essentially we summate the products of each
  signal element multiplied by the exponential term
  for a given value of m
• Hence using these sums we build a frequency
  spectrum
• Note each element in the new spectrum is a
  complex number, so we need to find its magnitude
  to plot

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Applying DFT to earlier Square, Saw and Triangle
Series
xysquare fRange2000 fNyquistfRange/2 for
m0(fRange-1) sum0 for k0(n-1)
sumsumx(k1)exp(-j2pimk/n) end
X(m1)1/nsum end fAxislinspace(0,fRange,fRange
) plot(fAxis(1fNyquist),abs(X(1fNyquist)))
Note this application is computationally
expensive hence the development of optimised
algorithms known as Fast Fourier Transforms (FFTs)
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Partials

• Fourier series pertains to complex waveforms that
  are periodic, being the summing of harmonics
  f,2f,3f, etc
• Real timbres quasi periodic thus yield
  An-harmonic components.
• Harmonic and An-harmonic components are
  collectively referred to as partials
• DFTs (and FFTs) can reveal an-harmonic content in
  signals that arent periodic though a succession
  of short time Fourier transforms is required to
  faithfully capture their behaviour.
• Given real musical notes are time varying
  emulating partials is important for realistic
  sounds