Finite Difference

The finite difference approximation (FDA) amounts to replacing derivatives by finite differences, or

$\displaystyle \frac{d}{dt} x(t) \buildrel$$\displaystyle \mbox{\tiny$\Delta$}$$\displaystyle \over=\lim_{\delta\to 0} \frac{x(t) - x(t-\delta)}{\delta} \approx \frac{x(n T)-x[(n-1)T]}{T}.
$

The z transform of the first-order difference operator is $ (1-z^{-1})/T$. Thus, in the frequency domain, the finite-difference approximation may be performed by making the substitution

$\displaystyle s \rightarrow \frac{1-z^{-1}}{T}
$

The first-order difference is first-order error accurate in $ T$. Better performance can be obtained using the bilinear transform, defined by the substitution

$\displaystyle s \longrightarrow c\left(\frac{1-z^{-1}}{1+z^{-1}}\right)$   where$\displaystyle \quad c=\frac{2}{T}.
$


``MUS 206: Modeling Acoustic Tubes and Wind Instrument Bores/Bells'' by Tamara Smyth, Department of Music, University of California, San Diego (UCSD).
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Copyright © 2019-05-22 by Tamara Smyth.
Please email errata, comments, and suggestions to Tamara Smyth<trsmyth@ucsd.edu>