Sound Synthesis With Digital Waveguides

1. Sound Synthesis With Digital Waveguides Jeff Feasel Comp 259 March 24 2003

2. 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

3. 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. 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. 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.

6. Traveling Wave Solution • E.g., plucked string:

7. 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

8. Waveguide DSP Model • Two-rail model • Signal is sum of rails at a point.

9. More Compact Representation • Only need to evaluate it at certain points. • Lump delay filters together between these points.

10. 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

11. Lossy Wave Equation • DSP model • Group losses and delays.

12. 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. Dispersion • Stiffness of the string introduces another restorative force. • Makes speed a function of frequency. • High frequencies propagate faster than low frequencies.

14. Terminations • Rigid terminations • Ideal reflection. • Lossy terminations • Reflection plus frequency-dependent attenuation.

15. Excitation • Excitation • Initial contents of the delay lines. • Signal that is “fed in”. • E.g., Pluck:

16. 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. Commuted Waveguide • Start with bridge output model.

18. Commuted Waveguide • Find single excitation point equivalent.

19. Commuted Waveguide • Obtain waveform at the bridge.

20. Commuted Waveguide • Force = Impedance*Velocity Diff

21. Commuted Waveguide • Loop and calculate bridge output.

22. Extensions To The Model • Certain components have negligible effect on sound. Can be removed. • Dual polarization. • Sympathetic coupling. • Tension-modulation nonlinearity.

23. Finding Parameter Values • Parameters for the filters must be estimated. • Use real recordings. • Iterative methods to determine parameters.

24. 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. 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.