Allpass Feedback Delay Networks

TL;DR

This paper extends the theory of allpass feedback delay networks (FDNs), characterizes uniallpass FDNs, and provides methods to design and analyze allpass properties for various configurations, broadening their application scope.

Contribution

It introduces a comprehensive characterization of uniallpass FDNs and offers solutions for the completion problem, expanding the design space of allpass delay networks.

Findings

Uniallpass FDNs can be characterized for arbitrary delay connections.

A method to determine gain parameters for allpass FDNs given a feedback matrix.

Extended the design space for classical allpass and reverberator systems.

Abstract

In the 1960s, Schroeder and Logan introduced delay line-based allpass filters, which are still popular due to their computational efficiency and versatile applicability in artificial reverberation, decorrelation, and dispersive system design. In this work, we extend the theory of allpass systems to any arbitrary connection of delay lines, namely feedback delay networks (FDNs). We present a characterization of uniallpass FDNs, i.e., FDNs, which are allpass for an arbitrary choice of delays. Further, we develop a solution to the completion problem, i.e., given an FDN feedback matrix to determine the remaining gain parameters such that the FDN is allpass. Particularly useful for the completion problem are feedback matrices, which yield a homogeneous decay of all system modes. Finally, we apply the uniallpass characterization to previous FDN designs, namely, Schroeder's series allpass and Gardner's nested allpass for single-input, single-output systems, and, Poletti's unitary reverberator for multi-input, multi-output systems and demonstrate the significant extension of the design space.