A Physical Derivation of Wave Digital Filters

• To each element, such as a capacitor or inductor, attach a length of transmission line at impedance $R_0$, and make it infinitesimally long. (Take the limit as the length of the transmission line goes to zero.)
  • The infinitesimal transmission line is terminated by the element.

  • The line impedance is arbitrary because it has been physically introduced.

  • If two such line-augmented elements are connected together by their transmission lines, scattering will clearly be induced at the junction in the usual way.

• Calculate the reflectance of the terminated line. That is, find the Laplace transform of the return wave divided by the Laplace transform of the input wave going into the line:
  • For a capacitor $C$ (impedence $R_C(s)=1/(Cs)$), we get the reflectance $S_C(s) = (R_C(s)-R_0)/(R_C(s)+R_0)$, which simplifies to
    $$S_C(s) = \frac{1 - R_0 C s}{1 + R_0 C s}$$

  • For an inductor $L$, we get $(Ls-R_0)/(Ls+R_0)$, or
    $$S_L(s) = \frac{ s - R_0/L }{ s + R_0/L}$$

  • For a resistor $R$, we get $(R-R_0)/(R+R_0)$, or
    $$S_R(s) = \frac{1 - R_0/R }{ 1 + R_0/R }$$

  • Note that both the capacitor and inductor reflectances are stable allpass filters, as they must be. Also, the resistor reflectance is always less than 1, no matter what line impedance $R_0$ we choose.

• Observe that there is a natural choice for each transmission-line impedance which will give us a normalized, universal reflectance for each element:
  • For the capacitor, $R_0 = 1/C \Rightarrow$
    $$S_C(s) = \frac{1 - s}{1 + s}$$

  • For the inductor, $R_0=L \Rightarrow$
    $$S_L(s) = - \frac{1 - s}{1 + s}$$

  • And for the resistor, $R_0 = R \Rightarrow$
    $$S_R(s) = 0$$

• Going to discrete time via the bilinear transform means making the substitution
  $$s = c \frac{z-1}{z+1}$$
  where c is some arbitrary positive constant, usually taken to be $c=2/T$.

• Solving for $z$ gives
  $$z = \frac{1+s/c}{1-s/c}$$

• 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)
    $$S_C\left(\frac{z-1}{z+1}\right) = z^{-1}$$

  • For the ``wave digital inductor'' (or mass)
    $$S_L\left(\frac{z-1}{z+1}\right) = - z^{-1}$$

  • And for the ``wave digital resistor'' (or dashpot)
    $$S_R\left(\frac{z-1}{z+1}\right) = 0$$
    as before in the continuous-time case.

Equivalently, we may obtain the same results by setting $c=2/T$ in the bilinear transform (which defines a frequency-scaling) and take the transmission-line (port) impedances to be instead $R_L = Lc = 2L/T$ for the inductor, and $R_C = T/(2C)$ for the capacitor (thereby compensating the frequency scaling).