(Finite Difference Schemes)

- There are many methods for converting ODEs to difference equations
- For white-box modeling, we'll use a very simple,
order-preserving methods which
*replaces each derivative or integral with a first-order finite difference:*

for sufficiently small (the sampling interval)

- This is formally known as the
*Backward Euler*(BE), or*backward difference*method for differentiation approximation - In addition to BE, we'll look at Forward Euler (FE), BiLinear
Transform (BLT), and a few others
- For a more advanced treatment of finite difference schemes, see
**Numerical Sound Synthesis**by Stefan Bilbao (2009, Wiley)

- Backward Euler Finite-Difference Equation for a Force-Driven Mass
- Accuracy of Backward Euler
- Summary of Backward Euler
- Delay-Free Loops
- Forward-Euler (FE)
- Centered Finite Difference
- Trapezoidal Rule for Numerical Integration
- Filter Design Approach
- Ideal Differentiator Frequency Response
- Explicit and Implicit Finite Difference Schemes
- Semi-Implicit Finite Difference Schemes

Download DigitizingNewton.pdf

Download DigitizingNewton_2up.pdf

Download DigitizingNewton_4up.pdf

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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