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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).
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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/∂x2 y(t,x) • ÿ = ∂2/∂t2 y(t,x) • Restorative Force = Inertial Force
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.
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.
Traveling Wave Solution • Linear and time-invariant. • Assume K and ε are fixed. • Class of solutions y(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.
Traveling Wave Solution • E.g., plucked string:
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
Waveguide DSP Model • Two-rail model • Signal is sum of rails at a point.
More Compact Representation • Only need to evaluate it at certain points. • Lump delay filters together between these points.
Lossy Wave Equation • Lossy wave equation Ky’’ = εÿ + μ ∂y/∂t • Travelling wave solution y(nT,mX) = gm y+(n-m) + g-m y-(n+m) g = e-μT/2ε
Lossy Wave Equation • DSP model • Group losses and delays.
Freq-Dependent Losses • Losses increase with frequency. • Air drag, body resonance, internal losses in the string. • Scale factors g become FIR filters G(ω).
Dispersion • Stiffness of the string introduces another restorative force. • Makes speed a function of frequency. • High frequencies propagate faster than low frequencies.
Terminations • Rigid terminations • Ideal reflection. • Lossy terminations • Reflection plus frequency-dependent attenuation.
Excitation • Excitation • Initial contents of the delay lines. • Signal that is “fed in”. • E.g., Pluck:
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.
Commuted Waveguide • Start with bridge output model.
Commuted Waveguide • Find single excitation point equivalent.
Commuted Waveguide • Obtain waveform at the bridge.
Commuted Waveguide • Force = Impedance*Velocity Diff
Commuted Waveguide • Loop and calculate bridge output.
Extensions To The Model • Certain components have negligible effect on sound. Can be removed. • Dual polarization. • Sympathetic coupling. • Tension-modulation nonlinearity.
Finding Parameter Values • Parameters for the filters must be estimated. • Use real recordings. • Iterative methods to determine parameters.
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?
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.