Bilinear Transformation

To digitize via the bilinear transform, we make the substitution

$$s = c \frac{1-z^{-1}}{1+z^{-1}}$$

where $c$ is any positive real constant (typically $2/T$ ).

For the ideal mass reflectance

$$S_m(s) = \frac{ms-R_0}{ms+R_0}$$

the bilinear transform yields

$$\tilde{S}_m(z) = \frac{p_m-z^{-1}}{1-p_mz^{-1}}$$

with

$$p_m \isdef \frac{mc-R_0}{mc+R_0}$$

Note that $\vert p_m\vert<1$ and $\vert\tilde{S}_m(e^{j\omega T})\vert=1$ . The stable allpass nature of the digitized mass reflectance is preserved by the bilinear transform, as always.

Important Observation:

If we choose $R_0=mc$ , then $p_m=0$ and $\tilde{S}_m(z) = -z^{-1}\,\,\Rightarrow\,\,$ no delay-free path through the mass reflectance