Complexity Reduction: A Physical Approach

The scattering matrix of a physical junction of waveguides (38) can
be expressed in terms of the alpha parameters as

The vector of outgoing pressure waves can be expressed as

where can be interpreted as the ``junction pressure.'' Thus, the matrix-vector multiplication can be seen to require only multiplications and additions.

Equation (64) can be rewritten as

This is the canonical form of the scattering relations for a physical -way scattering junction parametrized in terms of the alpha parameters.

Whenever possible, it is advisable to set the branch impedances so
that for some , and therefore only
multiplications are needed. This is achieved when the th
waveguide admittance equals the sum of all the other
admittances. According to Fettweis' terminology for wave digital
filters, the th junction port is called a *reflection-free
port*, and the th branch is said to be *adapted* to the other
branches. Physically, a wave traveling into the junction on a
reflection-free port will be fully scattered into the other branches
meeting at the junction, with no reflection back along branch .

In time-varying applications, it is also convenient to maintain
scaling such that
so that no
divisions or renormalizations are necessary in the alpha parameters.
This is easily
accomplished by setting the admittance of branch , say, to

(65) |

Since the alpha parameters sum to in the lossless case,
the first element, say, of the vector
can be expressed as

which implies

In Fettweis' terminology, the port corresponding to the first waveguide is called the

An important advantage provided by the organization of computations (68) and (69) is that they force the junction to be row allpass complementary (see section 5) even under coefficient quantization. The following theorem establishes the connection between allpass complementarity and structural losslessness of the scattering matrix.

(69) |

Thus, we must find positive such that

Let where is any positive constant. Since is positive, so is . (While is restricted to certain values, may be any positive real value). The condition (71) becomes

which holds if and only if

(72) |

We see that exact losslessness of a physical waveguide junction is
assured provided only that the actual alpha parameters are positive
and sum precisely to 2 *after* quantization. The proof extends
readily to nonnegative
by assigning
to any waveguide
branch corresponding to a port for which
. (Note that
corresponds to a branch which contributes no signal to the junction,
and a branch with zero admittance can convey no signal power. Such a
situation is physically degenerate so that if quantization forces
to zero for some , the corresponding branch may be
removed from the junction.)

The above theorem is quite simple from a physical point of view: All
we require for exact losslessness is that the quantized scattering
matrix correspond to *some* valid lossless scattering matrix.
From the physical correspondence (63) between the alpha parameters and the
branch wave admittances it is clear that losslessness holds when
. For
more general passivity, we can add a positive real ``load'' admittance
to the sum over above to obtain that scattering
passivity holds if and only if the alpha parameters are positive and
sum to a value *not exceeding* .

The above discussion is only concerned with passivity of the scattering matrix itself. The application of the matrix is assumed to be exact. Since rounding of the final outgoing waves is necessary in practice, we add

Since extended intermediate precision is required to ensure passivity, it is of interest to examine the additional bits required.

Hence (a) one additional bit is sufficient for the inner-product accumulator for . That the extra bit is necessary is shown by taking for all in which case for any set of lossless alpha parameters.

The outgoing wave variables are computed as , and a single bit is clearly sufficient to provide the needed headroom for a single subtraction in general. That the second headroom bit is necessary is shown by considering the example , , , , , in which case . If is not allowed, then an example is obtained with , , , , for which . For this to exceed , we must have . With or more, we can choose , and with or more, we can take , and the inequality is satisfied. Thus, (b) two extra bits are necessary and sufficient for the computation of each outgoing wave variable .

It is perhaps surprising at first that only one extra bit suffices for the junction pressure computation, no matter how many branches are impinging on the junction. Ordinarily, an inner product of length requires headroom bits. The single headroom bit requirement is a direct result of the alpha parameters being restricted to positive partitions of in the lossless case, or a positive partitioning of a positive number less than in the lossy, passive case.

Since an out-of-range outgoing traveling wave typically must be ``clipped'' in practice (i.e., replaced by if negative and if positive), the two extra carry bits are normally fed along with the next lower bit (the in-range sign bit) to saturation logic which implements clipping as needed.

The computations (68) or (69) can also be
used to implement precisely lossless or passive *time-varying*
junctions. In this case, the passivity constraints become

where is time in samples, since implicitly obeys (67) in formulas (68) and (69). (Here we allow which may be a convenient way to eliminate waveguide branches momentarily in the time-varying case.)

A junction of *normalized* waveguides can be obtained
from (68) or (69) by transformer-coupling
each branch other than the first with a waveguide having wave
admittance . The ideal transformer introduces two additional
multiplications for each branch except one. Therefore, the normalized
junction for arbitrary impedances can be implemented using
multiplications and additions.

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

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