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

(F.8) |

where is an arbitrary real constant, usually taken to be .

Solving for gives us the inverse bilinear transform:

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

- For the ``wave digital capacitor'' (or spring), Eq.
(F.5) becomes
- For the ``wave digital inductor'' (or mass), Eq.
(F.6) becomes
- And for the ``wave digital resistor'' (or dashpot), Eq.
(F.7) becomes

Note that this choice of
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 (instead of ) in the bilinear transform. From Eq. (F.9), it is clear that changing amounts to a linear frequency scaling of . Such a scaling may be compensated by choosing the waveguide (port) impedances to be (instead of ) for the inductor, and (instead of ) for the capacitor.

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