1 / 25

Sound Synthesis With Digital Waveguides

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).

Download Presentation

Sound Synthesis With Digital Waveguides

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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.

More Related