Sound%20Synthesis%20With%20Digital%20Waveguides

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Sound Synthesis With Digital Waveguides

• Jeff Feasel
• Comp 259
• March 24 2003

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The Wave Equation (1D)

• Ky eÿ
• y(t,x) string displacement
• y ?2/?x2 y(t,x)
• ÿ ?2/?t2 y(t,x)
• Restorative Force Inertial Force

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The Wave Equation (1D)

• Same wave equation applies to other media.
• E.g., Air column of clarinet
• Displacement -gt Air pressure deviation
• Transverse Velocity -gt Longitudinal volume
  velocity of air in the bore.

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Numerical Solution

• Brute Force FEM.
• At least one operation per grid point.
• Spacing must be lt ½ smallest audio wavelength.
• Too expensive. Not used in modern synth devices.

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Traveling Wave Solution

• Linear and time-invariant.
• Assume K and e are fixed.
• Class of solutions
• y(x,t) yR(x-ct) yL(xct)
• c sqrt(K / e)
• yR and yL are arbitrary smooth functions.
• yR right-going, yL left-going.

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Traveling Wave Solution

• E.g., plucked string

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Digital Waveguide Solution

• Digital Waveguide (Smith 1987).
• Constructs the solution using DSP.
• Sampled solution is
• y(nT,mX) y(n-m) y-(nm)
• y(n) yR(nT)
• y-(n) yL(nT)
• T, X time, space sample size

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Waveguide DSP Model

• Two-rail model
• Signal is sum of rails at a point.

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More Compact Representation

• Only need to evaluate it at certain points.
• Lump delay filters together between these points.

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Lossy Wave Equation

• Lossy wave equation
• Ky eÿ µ ?y/?t
• Travelling wave solution
• y(nT,mX) gm y(n-m) g-m y-(nm)
• g e-µT/2e

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Lossy Wave Equation

• DSP model
• Group losses and delays.

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Freq-Dependent Losses

• Losses increase with frequency.
• Air drag, body resonance, internal losses in the
  string.
• Scale factors g become FIR filters G(?).

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Dispersion

• Stiffness of the string introduces another
  restorative force.
• Makes speed a function of frequency.
• High frequencies propagate faster than low
  frequencies.

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Terminations

• Rigid terminations
• Ideal reflection.
• Lossy terminations
• Reflection plus frequency-dependent attenuation.

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Excitation

• Excitation
• Initial contents of the delay lines.
• Signal that is fed in.
• E.g., Pluck

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Commuted Waveguide

• Karjalainen, Välimäki, Tolonen (1998) streamline
  the model.
• Use LTI properties of the system, and
  Commutativity of filters.
• Create Single Delay Loop model, which is more
  computationally efficient.

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Commuted Waveguide

• Start with bridge output model.

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Commuted Waveguide

• Find single excitation point equivalent.

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Commuted Waveguide

• Obtain waveform at the bridge.

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Commuted Waveguide

• Force ImpedanceVelocity Diff

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Commuted Waveguide

• Loop and calculate bridge output.

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Extensions To The Model

• Certain components have negligible effect on
  sound. Can be removed.
• Dual polarization.
• Sympathetic coupling.
• Tension-modulation nonlinearity.

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Finding Parameter Values

• Parameters for the filters must be estimated.
• Use real recordings.
• Iterative methods to determine parameters.

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DSP Simulation

• Have a DSP model. How do we implement it?
• Hardware DSP chips.
• Software
• PWSynth
• STK http//ccrma-www.stanford.edu/software/stk/
• Microsoft DirectSound?

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References

• Karjalainen, Välimäki, Tolonen. Plucked-String
  Models From the Karplus-Strong Algorithm to
  Digital Waveguides and Beyond. Computer Music
  Journal, 1998.
• Laurson, Erkut, Välimäki. Methods for Modeling
  Realistic Playing in Plucked-String Synthesis
  Analysis, Control and Synthesis. Presentation
  DAFX00, December 2000.
• http//www.acoustics.hut.fi/vpv/publications/daf
  x00-synth-slides.pdf
• Smith, J. O. Music Applications of Digital
  Waveguides. Technical Report STAN-M-39, CCRMA,
  Dept of Music, Stanford University.
• Smith, J. O. Physical Modeling using Digital
  Waveguides. Computer Music Journal. Vol 16,
  no. 4. 1992.