## Allpass Digital Waveguide Networks

We now describe the class of multi-input, multi-output (MIMO) allpass
filters which can be made using *closed waveguide networks*. We
will see that feedback delay networks can be obtained as a special
case.

### Signal Scattering

The digital waveguide was introduced in §2.4. A basic fact from
acoustics is that traveling waves only happen in a *uniform
medium*. For a medium to be uniform, its *wave impedance*^{3.17}must be *constant*. When a traveling wave
encounters a *change* in the wave impedance, it will
*reflect*, at least partially. If the reflection is not total,
it will also partially *transmit* into the new impedance. This
is called *scattering* of the traveling wave.

Let denote the constant impedance in some waveguide, such as a stretched steel string or acoustic bore. Then signal scattering is caused by a change in wave impedance from to . We can depict the partial reflection and transmission as shown in Fig.2.33.

The computation of reflection and transmission in both directions, as
shown in Fig.2.33 is called a
*scattering junction*.

As derived in Appendix C, for force or pressure waves, the
*reflection coefficient* is given by

That is, the coefficient of reflection for a traveling pressure wave leaving impedance and entering impedance is given by the

*impedance step over the impedance sum*. The

*reflection coefficient*fully characterizes the scattering junction.

For *velocity* traveling waves, the reflection coefficient is
just the negative of that for force/pressure waves, or (see
Appendix C).

Signal scattering is *lossless*, *i.e.*, wave energy is neither
created nor destroyed. An implication of this is that the
*transmission coefficient*
for a traveling pressure wave leaving impedance and entering
impedance is given by

### Digital Waveguide Networks

A *Digital Waveguide Network* (DWN) consists of any number of
digital waveguides interconnected by scattering junctions. For
example, when two digital waveguides are connected together at their
endpoints, we obtain a two-port scattering junction as shown in
Fig.2.33. When three or more waveguides are
connected at a point, we obtain a *multiport scattering
junction*, as discussed in §C.8. In other words, a digital
waveguide network is formed whenever digital waveguides having
arbitrary wave impedances are interconnected. Since DWNs are
lossless, they provide a systematic means of building a very large
class of MIMO allpass filters.

Consider the following question:

In other words, how do we addUnder what conditions may I feed a signal from one point inside a given allpass filter to some other point (adding them) without altering signal energy at any frequency?

*feedback paths*anywhere and everywhere, thereby maximizing the richness of the recursive feedback structure, while maintaining an overall allpass structure?

The *digital waveguide* approach to allpass design
[430] answers this question by maintaining a *physical
interpretation* for all delay elements in the system. Allpass filters
are made out of *lossless digital waveguides* arranged in
*closed, energy conserving networks*. See Appendix C for further
discussion.

**Next Section:**

The Reverberation Problem

**Previous Section:**

Allpass Filters