The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL

Sound Synthesis With Digital WaveguidesJeff FeaselComp 259March 24 2003

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

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.

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.

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.

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL

Traveling Wave Solution

• E.g., plucked string:

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

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL

Waveguide DSP Model

• Two-rail model

• Signal is sum of rails at a point.

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.

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

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL

Lossy Wave Equation

• DSP model

• Group losses and delays.

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).

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.

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL

Terminations

• Rigid terminations♦ Ideal reflection.

• Lossy terminations♦ Reflection plus frequency-dependent

attenuation.

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL

Excitation

• Excitation♦ Initial contents of the delay lines.♦ Signal that is “fed in”.

• E.g., Pluck:

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.

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL

Commuted Waveguide

• Start with bridge output model.

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL

Commuted Waveguide

• Find single excitation point equivalent.

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL

Commuted Waveguide

• Obtain waveform at the bridge.

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL

Commuted Waveguide

• Force = Impedance*Velocity Diff

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL

Commuted Waveguide

• Loop and calculate bridge output.

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.

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.

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?

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.