Modal Decomposition of Feedback Delay Networks

TL;DR

This work tackles the challenging problem of obtaining a full modal decomposition for large feedback delay networks (FDNs) used in artificial reverberation. It introduces a polynomial matrix formulation of the generalized characteristic polynomial and solves for all poles simultaneously using the Ehrlich-Aberth Iteration, augmented with an efficient approximate deflation scheme and residue computation via adjugates. The approach delivers dramatic speedups over traditional matrix-eigenvalue or scalar-root methods, enabling reliable modal synthesis and extensive statistical analyses of pole-residue distributions under various attenuation schemes. The results reveal that, in lossless random FDNs, mode frequencies are nearly equidistributed while a small subset of high-energy modes dominates late reverberation, offering practical guidance for FDN design and optimization.

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.

Figures (7)

• Figure 1: Conceptual overview of modal decomposition and synthesis of a feedback delay network (FDN). Top left: FDN block diagram with a set of delay lines $\bm{D}_{\bm{m}}(z)$, connected via a feedback matrix $\bm{A}$, and input and output gains $\bm{b}$ and $\bm{c}$ for input and output signals $x$ and $y$, respectively. Thick lines indicate multiple signals. Top right: FDN modes with four parameters each: frequency, decay rate, initial amplitude and phase (not depicted). Bottom right: Time-domain impulse responses of the resonators corresponding to the FDN modes. Bottom left: Time-domain impulse response of the FDN.
• Figure 2: Average number of full iterations in the EAI for 500 random FDNs with total delay $\mathfrak{{N}}$ between 50 and $10^4$ samples and a random orthogonal feedback matrix. The average number of full iterations indicate the average number of Newton steps each pole requires to converge. For low matrix size $N$, the sign of matrix determinant $\det\lparen*\rparen{\bm{A}}$ and parity of $N$ plays a significant role.
• Figure 3: Computation time comparison of EAI with MATLAB build-in functions eig and roots. For system order $\mathfrak{{N}} > 5 \cdot 10^4$, the memory requirements of eig and roots become prohibitive on a personal computer configuration. The results are identical to a maximum error less than $\epsilon < 10^{-10}$.
• Figure 4: Modal Decomposition of 8-FDN with target reverberation time $T_{60}(0) = 2$ seconds and $T_{60}(\pi) = 0.4$ seconds using one-pole attenuation filters Jot:1991tq. Delays are $\bm{m} = [*]{2300,499,1255,866,729,964,1363,1491}$ and $\bm{A}$ is a random orthogonal matrix. \ref{['fig:plot_onePoleAbsorptionPol']} Pole magnitudes converted to reverberation time. Minimum and maximum bounds are computed from \ref{['eq:magnitudeBoundsDiagonal']}. \ref{['fig:plot_onePoleAbsorptionRes']} Residue magnitudes with and without attenuation. The mean difference between the residue magnitudes is 0.48 dB.
• Figure 5: Modal Decomposition of 8-FDN with inhomogeneous attenuation $\bm{\alpha}(z)$ according to an average delay length $\overline{m} = 1074$ for all one-pole attenuation filters in \ref{['eq:delayProportional']}. Identical delays, feedback matrix and target reverberation time as in Fig. \ref{['fig:OnePole']} were used. Minimum and maximum bounds are computed from \ref{['eq:magnitudeBoundsDiagonal']}.

Theorems & Definitions (1)

• Theorem 1: see Monden:1980bo