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Schroeder Allpass Sections

The original Schroeder-Logan papers on the use of allpass filters for artificial reverberation [420,421,415,153,154] 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 [396]. Today, these elements continue to serve as the basis for commercial devices for artificial reverberation and related effects [104]. They are also still typically used in software for artificial reverberation [86]. 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.

Figure 3.4: A cascade of three Schroeder allpass sections. A typical value for $ g$ is 0.7. The delay-line lengths $ M_i$ are typically mutually prime and spanning successive orders of magnitude, e.g., $ 1051,337,113$ .
\includegraphics{eps/callpass}

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 [485] 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.


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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