Normalized Scattering Junctions

Using (C.53) to convert to *normalized waves*
, the
Kelly-Lochbaum junction (C.60) becomes

as diagrammed in Fig.C.22. This is called the

It is interesting to define
, always
possible for passive junctions since
, and note that
the normalized scattering junction is equivalent to a *2D rotation*:

where, for conciseness of notation, the time-invariant case is written.

While it appears that scattering of normalized waves at a two-port junction requires four multiplies and two additions, it is possible to convert this to three multiplies and three additions using a two-multiply ``transformer'' to power-normalize an ordinary one-multiply junction [436].

The *transformer* is a lossless two-port defined by [137]

The transformer can be thought of as a device which steps the wave impedance to a new value without scattering; instead, the traveling signal power is redistributed among the force and velocity wave variables to satisfy the fundamental relations (C.57) at the new impedance. An impedance change from on the left to on the right is accomplished using

as can be quickly derived by requiring . The parameter can be interpreted as the ``turns ratio'' since it is the factor by which force is stepped (and the inverse of the velocity step factor).

Figure C.23 illustrates a *three-multiply*
normalized-wave scattering junction [436]. The impedance of
all waveguides (bidirectional delay lines) may be taken to be
.
Scattering junctions may then be implemented as a denormalizing
transformer
, a one-multiply scattering junction
, and a renormalizing transformer
. Either
transformer may be commuted with the junction and combined with the
other transformer to give a three-multiply normalized-wave scattering
junction. (The transformers are combined on the left in
Fig.C.23).

In slightly more detail, a transformer steps the wave impedance (left-to-right) from to . Equivalently, the normalized force-wave is converted unnormalized form . Next there is a physical scattering from impedance to (reflection coefficient ). The outgoing wave to the right is then normalized by transformer to return the wave impedance back to for wave propagation within a normalized-wave delay line to the right. Finally, the right transformer is commuted left and combined with the left transformer to reduce total computational complexity to one multiply and three adds.

It is important to notice that transformer-normalized junctions may
have a large dynamic range in practice. For example, if
, then Eq.(C.69) shows that the
transformer coefficients may become as large as
. If
is the ``machine epsilon,'' *i.e.*,
for typical
-bit two's complement arithmetic normalized
to lie in
, then the dynamic range of the transformer
coefficients is bounded by
. Thus, while
transformer-normalized junctions trade a multiply for an add, they
require up to
% more bits of dynamic range within the junction
adders. On the other hand, it is very nice to have normalized waves
(unit wave impedance) throughout the digital waveguide network,
thereby limiting the required dynamic range to root physical power in
all propagation paths.

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