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Summary of Backward Euler


\begin{eqnarray*}
v_n &=& v_{n-1} + T\,\hat{\dot v}_n\\
&\Longleftrightarrow& \zbox{\hat{\dot v}_n \;=\;\frac{v_n-v_{n-1}}{T}}\\ [10pt]
\updownarrow &=& \qquad \updownarrow\\ [10pt]
V(z) &=& z^{-1}V(z) + T\,\hat{\dot V}(z)\\ [10pt]
\Rightarrow\;&& \zbox{\hat{\dot V}(z) \;=\;\frac{1-z^{-1}}{T}V(z)}
\end{eqnarray*}

Expressing BE as a conformal map from $ s$ to $ z$ :

$\displaystyle \zbox{s\;\leftarrow\; \frac{1-z^{-1}}{T}}
$

The ideal differentiator $ H(s)=s$ , which is a first-order continuous-time LTI filter, is mapped to a first-order discrete-time LTI filter $ H(z)=(1-z^{-1})/T$ .


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``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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