### Exact Reverb via Transfer-Function Modeling

Figure 3.1 depicts the general reverberation scenario for three
sources and one listener (two ears). In general, the filters should
also include filtering by the pinnae
of the ears, so that each echo can be perceived as coming from the
correct angle of arrival in 3D space; in other words, at least some
reverberant reflections should be *spatialized* so that they
appear to come from their natural directions in 3D space
[248]. Again, the filters change if anything changes in
the listening space, including source or listener position. The
artificial reverberation problem is then to implement some
approximation of the system in Fig.3.1.

In the frequency domain, it is convenient to express the input-output relationship
in terms of the *transfer-function matrix*:

Denoting the impulse response of the filter from source to ear by , the two output signals in Fig.3.1 are computed by six convolutions:

*sparse*. For greater accuracy, each tap may include a lowpass filter which models

*air absorption*[314] and/or spherical spreading loss (see §2.3). For large , the impulse responses are not sparse, and we must either implement very expensive FIR filters, or approximate the tail of the impulse response using less expensive IIR filters; this subject--``late reverberation'' approximation--is taken up in §3.4.

**Next Section:**

Complexity of Exact Reverberation

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Digital Waveguide Networks