Upload
axel-workman
View
53
Download
2
Embed Size (px)
DESCRIPTION
Sound Synthesis With Digital Waveguides. Jeff Feasel Comp 259 March 24 2003. The Wave Equation (1D). Ky’’ = εÿ y(t,x) = string displacement y’’ = ∂ 2 /∂x 2 y(t,x) ÿ = ∂ 2 /∂t 2 y(t,x) Restorative Force = Inertial Force. The Wave Equation (1D). - PowerPoint PPT Presentation
Citation preview
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.