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Beta Parameters
It is customary in the wave digital filter literature to define the
beta parameters
as
![$\displaystyle \fbox{$\displaystyle \beta_i \isdef \frac{2R_i}{\sum_{j=1}^N R_j}$} \qquad\mbox{(Beta Parameters)} \protect$](img4968.png) |
(F.33) |
where
are the port impedances (attached element reference
impedances). In terms of the beta parameters, the force-wave series
adaptor performs the following computations:
![$\displaystyle v_J(n)$](img4014.png) |
![$\displaystyle =$](img241.png) |
(Common Junction Velocity)![$\displaystyle \protect$](img2254.png) |
(F.34) |
![$\displaystyle v^{-}_i(n)$](img4970.png) |
![$\displaystyle =$](img241.png) |
(Outgoing Velocity Waves)![$\displaystyle \protect$](img2254.png) |
(F.35) |
However, we normally employ a mixture of parallel and series adaptors,
while keeping a force-wave simulation. Since
, we obtain, after a small amount of algebra, the following
recipe for the series force-wave adaptor:
![$\displaystyle f^{{+}}_J(n)$](img4973.png) |
![$\displaystyle =$](img241.png) |
(Total Incoming Force)![$\displaystyle \protect$](img2254.png) |
(F.36) |
![$\displaystyle f^{{-}}_i(n)$](img4900.png) |
![$\displaystyle =$](img241.png) |
(Outgoing Force Waves)![$\displaystyle \protect$](img2254.png) |
(F.37) |
We see that we have
multiplies and
additions as in the
parallel-adaptor case.
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