# Modal Decomposition of Feedback Delay Networks

## TL;DR

This paper introduces an efficient numerical method for modal decomposition of feedback delay networks, enhancing sound synthesis and reverberation analysis by revealing their modal behavior.

## Contribution

It presents a novel pole finding algorithm based on Ehrlich-Aberth iteration, significantly improving computational efficiency for FDN modal analysis.

## Key findings

- The new method achieves up to three orders of magnitude faster computation.
- Few modes dominate the impulse response energy in FDNs.
- Modal analysis aids in optimizing artificial reverberation systems.

## Abstract

Feedback delay networks (FDNs) belong to a general class of recursive filters which are widely used in sound synthesis and physical modeling applications. We present a numerical technique to compute the modal decomposition of the FDN transfer function. The proposed pole finding algorithm is based on the Ehrlich-Aberth iteration for matrix polynomials and has improved computational performance of up to three orders of magnitude compared to a scalar polynomial root finder. We demonstrate how explicit knowledge of the FDN's modal behavior facilitates analysis and improvements for artificial reverberation. The statistical distribution of mode frequency and residue magnitudes demonstrate that relatively few modes contribute a large portion of impulse response energy.