## 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

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Allpass Filters