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Difference Equations
(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)
Subsections
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
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Download
DigitizingNewton.pdf
Download
DigitizingNewton_2up.pdf
Download
DigitizingNewton_4up.pdf
``
Introduction to Physical Signal Models
'', by
Julius O. Smith III
, (From Lecture Overheads,
Music 420
).
Copyright ©
2020-06-27
by
Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),
Stanford University
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