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Bilinear Transform

The bilinear transform is a one-to-one mapping from the $ s$ plane to the $ z$ plane:

\begin{eqnarray*}
s &=& c\frac{1-z^{-1}}{1+z^{-1}}, \quad c>0, \quad c=\frac{2}{T} \quad \mbox{(typically)} \\ [10pt]
\,\,\Rightarrow\,\,
z &=& \frac{1+s/c}{1-s/c} % \eqsp 1 + 2(s/c) + 2(s/c)^2 + 2(s/c)^3 + \cdots
\end{eqnarray*}

Starting with a continuous-time transfer function $ H_a(s)$ , we obtain the discrete-time transfer function

$\displaystyle \zbox{H_d(z) \;\mathrel{\stackrel{\mathrm{\Delta}}{=}}\;H_a\left(c\frac{1-z^{-1}}{1+z^{-1}}\right)}
$

where ``$ d$ '' denotes ``digital,'' and ``$ a$ '' denotes ``analog.''


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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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