- To each element, such as a capacitor or inductor, attach a
length of transmission line at impedance
, 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.

- The infinitesimal transmission line is
- 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
(impedence
), we get the reflectance
, which simplifies to
- For an inductor
, we get
, or
- For a resistor
, we get
, or
- 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 we choose.

- For a capacitor
(impedence
), we get the reflectance
, which simplifies to
- 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,
- For the inductor,
- And for the resistor,

- For the capacitor,
- Going to discrete time via the bilinear transform means
making the substitution
- Solving for
gives
- In this case, we see that setting
further simplifies our
universal reflectances in the digital domain:
- For the ``wave digital capacitor'' (or spring)
- For the ``wave digital inductor'' (or mass)
- And for the ``wave digital resistor'' (or dashpot)

- For the ``wave digital capacitor'' (or spring)

Equivalently, we may obtain the same results by setting in the bilinear transform (which defines a frequency-scaling) and take the transmission-line (port) impedances to be instead for the inductor, and for the capacitor (thereby compensating the frequency scaling).

Download WaveDigitalFilters.pdf

Download WaveDigitalFilters_2up.pdf

Download WaveDigitalFilters_4up.pdf

Copyright ©

Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

[Automatic-links disclaimer]