Summary and Outline

Circulant and Elliptic Feedback Delay Networks for Artificial Reverberation

This section reviews FDNs along the lines indicated by Jot [9,8] with some modifications.

As depicted in Fig. Fig. 1, an FDN is built using N delay lines,
each having a length in seconds given by
, where
is the sampling period. The complete FDN is given by the
following relations:

where , are the delay-line outputs at time sample . If for each , we obtain the conventional state-variable description of a discrete-time linear system [10]. In the present case, are normally large integers, so the variables form only a small subset of the system state at time , with the remaining state variables being the samples contained within the delay lines at time . Using the transform, assuming zero initial conditions, we can rewrite (2) in the frequency domain as

where , , and . The diagonal matrix is called the ``delay matrix'' and is called the ``feedback matrix.''

The state variables of the FDN can be collected into a vector as follows: List the variables contained in the first delay line from the cell to the second cell, then those contained in the second delay line from the cell to the second cell, and so on for the other delay lines; then attach the first cell of all the delay lines in increasing order, and finally the output variables to .

By assuming that each delay line is longer than two samples, the
state-variable description corresponding to this variable format for the
system (2) can be found to be

where

(6) | |||

(7) |

The state-transition matrix is

(9) |

(10) |

(11) |

(12) | |||

The order of the system (5) is equal to the sum of the delay-line lengths:

From (4), the transfer function is easily found to be

Note that when for all , the FDN specializes to a fully
general state-space description [10]. This implies
*any* linear, time-invariant, discrete-time system can be
formulated as a special case of an FDN since every state-space
description is a special case. This suggests that a wide variety of
stable FDNs can be generated by starting with any stable LTI system
whatsoever and performing the substitution
on each delay element, or any other conformal mapping which
takes the unit circle to itself (another example being the
Schroeder-allpass transformation
). Stability is preserved even when the
unit-sample delays of the original state space description are mapped
using different conformal maps. This can be seen from the matrix
power series expansion

As long as , by making use of the triangle inequality and the Cauchy-Schwarz (or mutual consistency) inequality [20], we can write

(15) |

Therefore, as long as decays exponentially with , stability is assured. The above derivation extends immediately to time-varying feedback matrices , provided for some worst-case .

The poles of the FDN are the solutions of either

The matrix
is not uniquely determined by . In
fact, our ordering of the state variables differs from that used by
Jot [8]. Our ordering gives

By application of the matrix inversion lemma [10] to the transfer
function (13), the system zeros are found as the solutions
of

The formulation of (2) represents a prototype structure in
the sense that, with the appropriate choice of feedback matrix, it is a
lossless structure. In practice, we must insert attenuation coefficients
and filters in the feedback loop. For example, one may insert a
gain [9]

In a practical realization, we normally need to introduce
frequency-dependent losses such that higher frequencies decay faster. We
can do this by introducing lowpass filters after each delay line in place
of the gains . In this case, local uniformity of mode decay is still
achieved by condition (20), where and are made
frequency dependent:

Notice that uniform decay of all the modes, albeit arguably desirable in artificial reverberators for a smooth late time response, is not found in actual rooms. Normal modes are associated with standing waves which have an absorption that depends on their orientation. For example, in a rectangular enclosure, waves traveling in a direction normal to a wall are less absorbed than oblique waves [17, p. 392], so that the corresponding standing waves (expressible as the superposition of traveling waves in opposite directions) have different reverberation times. The room-acoustics interpretation of FDNs provided in Section V points to ways of modeling such uneven decays.

Digital Waveguide Networks

Summary and Outline

Circulant and Elliptic Feedback Delay Networks for Artificial Reverberation

``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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