On Lossless Feedback Delay Networks

TL;DR

This paper characterizes the most general class of lossless feedback matrices in Feedback Delay Networks, showing that all irreducible components must be diagonally similar to a unitary matrix for the network to be lossless.

Contribution

It introduces a generalized class of lossless feedback matrices for FDNs, extending beyond known unitary and triangular matrices.

Findings

Lossless FDNs require all irreducible components of the feedback matrix to be diagonally similar to a unitary matrix.

Examples demonstrate the necessity of this generalized class in existing FDN designs.

The property holds for any set of delay lengths, broadening design options.

Abstract

Lossless Feedback Delay Networks (FDNs) are commonly used as a design prototype for artificial reverberation algorithms. The lossless property is dependent on the feedback matrix, which connects the output of a set of delays to their inputs, and the lengths of the delays. Both, unitary and triangular feedback matrices are known to constitute lossless FDNs, however, the most general class of lossless feedback matrices has not been identified. In this contribution, it is shown that the FDN is lossless for any set of delays, if all irreducible components of the feedback matrix are diagonally similar to a unitary matrix. The necessity of the generalized class of feedback matrices is demonstrated by examples of FDN designs proposed in literature.