1. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Sound Synthesis With Digital Waveguides Jeff Feasel Comp 259 March 24 2003

2. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 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

3. 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.

4. 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.

5. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Traveling Wave Solution Linear and time-invariant. ♦ Assume K and ε are fixed. Class of solutions y(x,t) = y R (x-ct) + y L (x+ct) c = sqrt(K / ε) y R and y L are arbitrary smooth functions. y R right-going, y L left-going.

6. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Traveling Wave Solution E.g., plucked string:

7. 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) = y R (nT) y - (n) = y L (nT) T, X = time, space sample size

8. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Waveguide DSP Model Two-rail model Signal is sum of rails at a point.

9. 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.

10. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Lossy Wave Equation Lossy wave equation Ky’’ = εÿ + μ ∂y/∂t Travelling wave solution y(nT,mX) = g m y + (n-m) + g -m y - (n+m) g = e -μT/2ε

11. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Lossy Wave Equation DSP model Group losses and delays.

12. 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).

13. 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.

14. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Terminations Rigid terminations ♦ Ideal reflection. Lossy terminations ♦ Reflection plus frequency-dependent attenuation.

15. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Excitation ♦ Initial contents of the delay lines. ♦ Signal that is “fed in”. E.g., Pluck:

16. 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.

17. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Commuted Waveguide Start with bridge output model.

18. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Commuted Waveguide Find single excitation point equivalent.

19. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Commuted Waveguide Obtain waveform at the bridge.

20. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Commuted Waveguide Force = Impedance*Velocity Diff

21. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Commuted Waveguide Loop and calculate bridge output.

22. 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.

23. 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.

24. 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?

25. 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.