Introduction

Introduction

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 , with 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 inputs and outputs by replacing the -sample delay line with an order ``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
by (1) replacing the single -sample
delay line by a diagonal matrix of delay lines of different lengths,
and (2) replacing the feedback gain by the matrix , where
is any unitary matrix,^{2} and is any diagonal matrix having all
elements less than in magnitude, where
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.

Lossless Reveberator Prototypes

Introduction

Introduction

``Circulant and Elliptic Feedback Delay Networks for Artificial Reverberation'', by Davide Rocchesso and Julius O. Smith III, preprint of version in

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