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Physical Audio Signal Processing
    Introduction to Lumped Models
       Bilinear Transformation

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Bilinear Transformation

The bilinear transform is defined by the substitution

$\displaystyle s$ $\displaystyle =$ $\displaystyle c\frac{1-z^{-1}}{1+z^{-1}}, \quad c>0, \; c=2/T\;$   (typically) (L.6)
$\displaystyle \,\,\Rightarrow\,\, z$ $\displaystyle =$ $\displaystyle \frac{1+s/c}{1-s/c} = 1 + 2(s/c) + 2(s/c)^2 + 2(s/c)^3 + \cdots$ (L.7)

where $ c$ is some positive constant [85,332]. That is, given a continuous-time transfer function $ H_a(s)$, we apply the bilinear transform by defining

$\displaystyle H_d(z) = H_a\left(c\frac{1-z^{-1}}{1+z^{-1}}\right) $

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

It can be seen that analog dc ($ s=0$) maps to digital dc ($ z=1$) and the highest analog frequency ($ s=\infty$) maps to the highest digital frequency ($ z=-1$). It is easy to show that the entire $ j\omega $ axis in the