the university of north carolina at chapel hill sound synthesis with digital waveguides jeff feasel...
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Sound Synthesis With Digital WaveguidesJeff FeaselComp 259March 24 2003
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
The Wave Equation (1D)
• Ky’’ = εÿ♦ 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 UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
The Wave Equation (1D)
• Same wave equation applies to other media.
• E.g., Air column of clarinet:♦ Displacement -> Air pressure
deviation♦ Transverse Velocity -> Longitudinal
volume velocity of air in the bore.
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Numerical Solution
• Brute Force FEM.• At least one operation per
grid point.• Spacing must be < ½
smallest audio wavelength.• Too expensive. Not used in
modern synth devices.
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Traveling Wave Solution
• Linear and time-invariant.♦ Assume K and ε are fixed.
• Class of solutionsy(x,t) = yR(x-ct) + yL(x+ct)
c = sqrt(K / ε)yR and yL are arbitrary smooth functions.yR right-going, yL left-going.
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Traveling Wave Solution
• E.g., plucked string:
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Digital Waveguide Solution• Digital Waveguide (Smith
1987).• Constructs the solution using
DSP.• Sampled solution is:
y(nT,mX) = y+(n-m) + y-(n+m)y+(n) = yR(nT)
y-(n) = yL(nT)
T, X = time, space sample size
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Waveguide DSP Model
• Two-rail model
• Signal is sum of rails at a point.
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
More Compact Representation
• Only need to evaluate it at certain points.
• Lump delay filters together between these points.
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Lossy Wave Equation
• Lossy wave equationKy’’ = εÿ + μ ∂y/∂t
• Travelling wave solutiony(nT,mX) = gm y+(n-m) + g-m y-(n+m)g = e-μT/2ε
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Lossy Wave Equation
• DSP model
• Group losses and delays.
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Freq-Dependent Losses
• Losses increase with frequency.
• Air drag, body resonance, internal losses in the string.
• Scale factors g become FIR filters G(ω).
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
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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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Terminations
• Rigid terminations♦ Ideal reflection.
• Lossy terminations♦ Reflection plus frequency-dependent
attenuation.
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Excitation
• Excitation♦ Initial contents of the delay lines.♦ Signal that is “fed in”.
• E.g., Pluck:
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
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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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Commuted Waveguide
• Start with bridge output model.
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Commuted Waveguide
• Find single excitation point equivalent.
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Commuted Waveguide
• Obtain waveform at the bridge.
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Commuted Waveguide
• Force = Impedance*Velocity Diff
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Commuted Waveguide
• Loop and calculate bridge output.
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
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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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Finding Parameter Values• Parameters for the filters
must be estimated.• Use real recordings.• Iterative methods to
determine parameters.
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
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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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
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: DAFX’00, December 2000.http://www.acoustics.hut.fi/~vpv/publications/dafx00-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.