Late Reverberation Approximations
Desired Qualities in Late Reverberation
From a perceptual standpoint, the main qualities desired of a good late-reverberation impulse response are
- a smooth (but not too smooth) decay, and
- a smooth (but not too regular) frequency response.
A smooth frequency response exhibits no large, isolated gaps or boosts. It is generally provided when the mode density is sufficiently large in the frequency domain, with the modes being spread out uniformly, as opposed to piling up in the same place or separated to form gaps. On the other hand, the modes should not be too regularly spaced, since this can produce audible periodicity in the time-domain impulse response.
An interesting experiment by Moorer  was to try exponentially decaying white noise as a late reverberation impulse response. This signal satisfies both smoothness criteria (time domain and frequency domain), and it sounds quite natural. However, since natural reverberation decays faster at high frequencies, it is better to say that the ideal late reverberation impulse response is exponentially decaying ``colored'' noise, with the high-frequency energy decaying faster than the low-frequency energy.
Schroeder's rule of thumb for echo density in the late reverb is 1000 echoes per second or more . However, for impulsive sounds, 10,000 echoes per second or more may be necessary for a smooth response [217,153].
Schroeder Allpass Sections
Manfred Schroeder's original papers on the use of allpass filters for artificial reverberation [417,412,152,153] started a lively thread of research which continues to the present. For many years thereafter, digital reverberation algorithms were designed along the lines suggested by Schroeder using delay lines, comb filters, and allpass filters--elements described in Chapter 2. There was even special-purpose hardware developed to implement these structures efficiently in real time . Today, these elements continue to serve as the basis for commercial devices for artificial reverberation and related effects . They are also still typically used in software for artificial reverberation . We will see some examples starting in §3.5 below.
Schroeder's suggested use of allpass filters was especially brilliant because there is nothing in nature to suggest them. Instead, he recognized the conceptual and practical utility of separating the coloration of reverberation from its duration and density aspects. While Schroeder's 1961 paper is entitled ``Colorless Artificial Reverberation,'' there is no such thing as colorless (exactly allpass) reverberation in the real world. However, it makes sense as an idealization of natural reverb. Colorless reverberation is an idealization only possible in the ``virtual world''.
In Schroeder's original work, and in much work which followed, allpass filters are arranged in series, as shown in Fig.3.4.
Each allpass can be thought of as expanding each nonzero input sample from the previous stage into an entire infinite allpass impulse response. For this reason, Schroeder allpass sections are sometimes called impulse expanders  or impulse diffusers. While not a physical model of diffuse reflection, single reflections are expanded into many reflections, which is qualitatively what is desired.
Another interesting interpretation of a Schroeder allpass section is as a digital waveguide model of the driving-point impedance of an ideal string (or cylindrical acoustic tube) which is reflectively terminated at a real impedance. That is, if the input signal is regarded as samples of a physical velocity, then the output signal is proportional to samples of the corresponding force (or pressure) at the same physical point. The delay line contains traveling-wave samples; the first half corresponds to traveling waves heading toward the far end of the string (or tube), while the second half holds traveling-wave samples coming back the other way toward the driving point. For the mathematical details of this interpretation, see Appendix C.
Nested Allpass Filters
Another common method for increasing the density of an allpass impulse response is to nest two or more allpass filters, as described in §2.8.2 and shown in Fig.2.32 on page . In general, a nested allpass filter is created when one or more of its delay elements is replaced by another allpass filter. As we saw in §2.8.2, first-order nested allpass filters are equivalent to lattice filters. This equivalence implies that any order transfer function (any poles and zeros) may be obtained from a linear combination of the delay elements of nested first-order allpass filters, since this is a known property of the lattice filter .
In general, any delay-element or delay-line inside a stable allpass-filter can be replaced with any stable allpass-filter, and the result will be a stable allpass.