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Prior Work

The field of digital artificial reverberation was launched by M. Schroeder more than thirty years ago [25]. In his pioneering work, he introduced recursive comb filters and allpass filters as suitable means for inexpensive simulation of multiple echoes. In particular, he introduced use of allpass filters of the form $y(n) = g x(n) + x(n-N) - g y(n-N)$, with $N$ any positive integer, for achieving dense echoes with a flat amplitude response. This structure has since been used extensively in artificial reverberation [16].

In the seventies, M. A. Gerzon [4] generalized the single-input, single-output Schroeder allpass to $M$ inputs and outputs by replacing the $N$-sample delay line with an order $M$ ``unitary network'' (a square matrix transfer function having a frequency response matrix which is a unitary matrix at all frequencies, i.e., it must be a ``paraunitary'' transfer function matrix [31]).

J. Stautner and M. Puckette [30] introduced what we now call feedback delay networks (FDNs) as structures well suited for artificial reverberation. These structures are characterized by a set of delay lines connected in a feedback loop through a ``feedback matrix'' (see Fig. Fig. 1). The FDN was obtained as a generalization of the recursive feedback comb filter $y(n) = x(n-N) + g y(n-N)$ by (1) replacing the single $N$-sample delay line by a diagonal matrix of delay lines of different lengths, and (2) replacing the feedback gain $g$ by the matrix $G=UD$, where $U$ is any unitary matrix,2 and $D$ is any diagonal matrix having all elements less than $1-\epsilon$ in magnitude, where $\epsilon > 0$ determines the stability margin. Specific early reflections were implemented by adding scaled copies of the source signal into selected points along the delay lines, corresponding to use of the transposed form of the FIR filter [19]. Early reflections in artificial reverberation were apparently first implemented by J. A. Moorer using a direct-form FIR filter in series with Schroeder allpass filters and air-absorption comb filters [16].

More recently, J. M. Jot has extensively studied FDNs and developed associated techniques for designing good quality reverberators. He suggested the use of efficient special cases of unitary feedback matrices as well as techniques for pole-placement to obtain a desired decay-time vs. frequency [8], and introduced the valuable design principle that, for smoothest (idealized) late reverberation, all modes in a given frequency band should decay at substantially the same rate in order to avoid isolated ringing modes in the late reverberation which tend to sound ``metallic'' [9].

In 1986, digital waveguide networks (DWN) were proposed as a useful starting point for digital reverberator design [26]. The idea was to build an arbitrary closed network of digital waveguides exhibiting the desired early reflections and late echo density, and then introduce loss filters into the network to achieve the desired decay time vs. frequency. Approaching reverberation via lossless prototypes leads to good numerical and stability properties [27,31]. Like FDNs, DWNs make it easy to construct well-behaved, high-order, nearly lossless systems.


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``Circulant and Elliptic Feedback Delay Networks for Artificial Reverberation'', by Davide Rocchesso and Julius O. Smith III, preprint of version in IEEE Transactions on Speech and Audio, vol. 5, no. 1, pp. 51-60, Jan. 1996.

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Copyright © 2005-03-10 by Davide Rocchesso and Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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