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Bilinear Transformation

The bilinear transform may be defined by

$\displaystyle s$ $\displaystyle =$ $\displaystyle c\frac{1-z^{-1}}{1+z^{-1}}\protect$ (I.9)
$\displaystyle z^{-1}$ $\displaystyle =$ $\displaystyle \frac{1-s/c}{1+s/c}\protect$ (I.10)

where $ c$ is an arbitrary positive constant that we may set to map one analog frequency precisely to one digital frequency. In the case of a lowpass or highpass filter, $ c$ is typically used to set the cut-off frequency to be identical in the analog and digital cases.

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