Digitizing Elementary Reflectances by Bilinear Transform

Going to discrete time via the bilinear transform means making the substitution

$$s = c \frac{1-z^{-1}}{1+z^{-1}}$$ (F.8)

where $c>0$ is an arbitrary real constant, usually taken to be $c=2/T$ .

Solving for $z^{-1}$ gives us the inverse bilinear transform:

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

In this case, we see that setting $c=1$ further simplifies our universal reflectances in the digital domain:

• For the ``wave digital capacitor'' (or spring), Eq.(F.5) becomes
  $$\fbox{$\displaystyle \hat{\rho}_C\left(\frac{z-1}{z+1}\right) = z^{-1}$}$$

• For the ``wave digital inductor'' (or mass), Eq.(F.6) becomes
  $$\fbox{$\displaystyle \hat{\rho}_L\left(\frac{z-1}{z+1}\right) = - z^{-1}$}$$

• And for the ``wave digital resistor'' (or dashpot), Eq.(F.7) becomes
  $$\fbox{$\displaystyle \hat{\rho}_R\left(\frac{z-1}{z+1}\right) = 0$}$$
  as before in the continuous-time case.

Note that this choice of $c$ is also the only one that eliminates delay-free paths in the fundamental elements. This allows them to be used as building blocks for explicit finite difference schemes.

We may still obtain the above results using the more typical value $c=2/T$ (instead of $c=1$ ) in the bilinear transform. From Eq.(F.9), it is clear that changing $c$ amounts to a linear frequency scaling of $s=j\omega$ . Such a scaling may be compensated by choosing the waveguide (port) impedances to be $R_L = Lc = 2L/T$ (instead of $R_L=L$ ) for the inductor, and $R_C = T/(2C)$ (instead of $1/C$ ) for the capacitor.