Loaded Scattering Implementation Notes

As noted in [25] et al., it is natural in
practice to implement the junction load impedance (or admittance) as
bank of *parallel biquads*:

where is the number of biquads, and we define

A hasty implementation of Eq.(C.97) might overlook the fact that all junction transfer-functions (transmittances, reflectances, etc.) must have the

A basic trick for implementing the *reciprocal* of a transfer
function without altering its denominator(s) is to place it in
a *feedback loop*. If
denotes the starting
transfer function, then placing it in a feedback loop gives
. Now split
into its instantaneous and delayed
components
, where
. If
(otherwise pull out unit-sample delays until it is),
then we can realize
as

which can be implemented as a feedback loop containing as the feedback filter, and the scale factor in the forward path, as shown in Fig.C.30. In the case of a parallel biquad realization, structure is preserved with altered numerators obtained by extracting the instantaneous gain from each section, as we'll see the following examples.

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