Delay Network-based Artificial Reverberators

A curated list of papers and code on delay-network-based artificial reverberation. This repository collects references and implementations for Feedback Delay Networks (FDN) and Scattering Delay Networks (SDN). It is a living resource and will be updated as new work appears.

Made public on Friday, November 14th, 2025.
Last edit to the list on December 30th, 2025.

Contents

• Toolboxes and libraries
• Feedback Delay Networks
  • ML optimization
• Feedback Delay Networks for Spatial Audio
  • ML optimization
• Scattering Delay Networks
  • ML optimization
• Early Reverbs and FDN Theory
• Other resources
• Acknowledgements
• Contributing

🔧 Toolboxes and libraries

Reference	Description	language	Repository
S. J. Schlecht. "FDNTB: The feedback delay network toolbox." International Conference on Digital Audio Effects (DAFx), 2020.	Comprehensive FDN toolbox: special feedback matrices, topologies, attenuation filters, modal decomposition and examples.	Matlab	fdnToolbox
G. Dal Santo, G. M. De Bortoli, K. A. Prawda, S. Schlecht, & V. Välimäki. "FLAMO: An Open-Source Library for Frequency-Domain Differentiable Audio Processing." (ICASSP 2025).	Frequency-domain differentiable audio processing. Contains differentiable implementations of common LTI audio modules with learnable parameters.	PyTorch	flamo
G. Dal Santo, K. A. Prawda, S. Schlecht, & V. Välimäki. "FLARE: An Open-Source Library for RIR Synthesis and Analysis in PyTorch." (AES AI/ML for Audio, 2025).	Room Impulse Response synthesis and Analysis in PyTorch (based on FLAMO). Contains classes for differentiable FDN and grouped FDN implementations.	PyTorch	flare

📝 Feedback Delay Networks

Reference	Content Type	Main Contributions	Notes
M. Chemistruck, K. Marcolini, & W. Pirkle. "Generating matrix coefficients for feedback delay networks using genetic algorithm." Proceedings of the 133rd Audio Engineering Society Convention, 2012.	Parameter estimation	Uses a genetic algorithm to generate or optimize feedback matrix coefficients for FDNs to improve reverberation characteristics.	..
S. J. Schlecht and E. A. P. Habets. "Time-varying feedback matrices in feedback delay networks and their application in artificial reverberation." The Journal of the Acoustical Society of America, 2015.	Informed design	Introduce time-varying feedback matrices to modulate pole locations to break temporal patterns, and model air movement. Derives the condition for FDN pole angles to be uniformely distributed.	fdnToolbox Matlab. Practical considerations in this paper
J. Coggin & W. Pirkle. "Automatic design of feedback delay network reverb parameters for impulse response matching." Proceedings of the 141st Audio Engineering Society International Convention (AES), 2016.	Parameter estimation	Proposes an automatic optimization method based on a genetic algorithm for analysis–synthesis matching.	..
S. J. Schlecht and E. A. P. Habets. "Feedback Delay Networks: Echo Density and Mixing Time." IEEE/ACM Transactions on Audio, Speech, and Language Processing, 2017.	Theory and Analysis	Explains echo density and mixing time and how these depend on FDN parameters, with emphasis on delay lengths	fdnToolbox Matlab
S. J. Schlecht and E. A. P. Habets. "Accurate Reverberation Time Control in Feedback Delay Networks." Proceedings of the International Conference on DAFx, 2017.	Attenuation filters optimization	Constrained nonlinear least-squares optimization to align attenuation filters with a target reverberation time across frequencies	fdnToolbox Matlab. Requires target RT
K. Prawda, V. Välimäki, and S. Schlecht. "Improved reverberation time control for feedback delay networks." Proceedings of the International Conference on DAFx, 2019.	Attenuation filters optimization	Presents a GEQ design that incorporates an additional high-shelf filter and an optimization method based on a frequency-dependent weighting matrix.	Requires target RT
S. J. Schlecht and E. A. P. Habets. "Modal Decomposition of Feedback Delay Networks." IEEE/ACM Transactions on Audio, Speech, and Language Processing, 2019.	Theory and Analysis	Discusses the connection between the FDN's modal representation and its parameters. Presents a pole finding algorithm	fdnToolbox Matlab
J. Shen & R. Duraiswami. "Data-driven feedback delay network construction for real-time virtual room acoustics." PervasiveHealth: Pervasive Computing Technologies for Healthcare, 2020.	Parameter estimation	Propose a data-driven approach to automatically generate a pre-tuned FDN for any given room described by a set of room parameters.	..
S. J. Schlecht and E. A. P. Habets. "Scattering in Feedback Delay Networks." IEEE/ACM Transactions on Audio, Speech, and Language Processing, 2020.	Informed design	Generalizes the feedback matrix to arbitrary lossless filter-feedback matrices to increase echo density. Introduces the "velvet" feedback matrix for very dense IRs at low cost.	fdnToolbox I and fdnToolbox II Matlab. Presents the velvet feedback matrix which can create ultra-dense impulse responses at a minimal computational cost
O. Das & J. S. Abel. "Grouped feedback delay networks for modeling of coupled spaces." J. Audio Eng. Soc., 2021.	Informed design	Architecture where groups of delay lines sharing different target decay rates. Used to model multi-stage decay	GFDN C++ Plugin
Ibnyahya, I., & Reiss, J. D. A method for matching room impulse responses with feedback delay networks. In Audio Engineering Society Convention 153, 2022.	Informed design + Parameter estimation	Uses genetic algorithm to find frequency-independent parameters. Present a method to extract early reflection. Informed design for frequency-dependent filters.	MatchReverb Matlab
O. Das, S. J. Schlecht & E. De Sena. "Grouped feedback delay networks with frequency-dependent coupling." IEEE/ACM Transactions on Audio, Speech, and Language Processing, 2023.	Informed design	Expands previous work on GFDN by introducing frequency-dependent coupling among the various FDN groups.	GFDN C++ Plugin
S. J. Schlecht, J. Fagerström, and V. Välimäki. "Decorrelation in Feedback Delay Networks." IEEE/ACM Trans. Audio Speech Lang. Process., 2023.	Theory and Analysis	Analysis of multichannel correlation induced by FDNs	..
S. J. Schlecht, M. Scerbo, E. De Sena and V. Välimäki, "Modal Excitation in Feedback Delay Networks," in IEEE Signal Processing Letters, vol. 31, pp. 2690-2694, 2024.	Analysis	Presents a method for computing modal shapes of an FDN having a large order and a moderate number of delay lines.	..
V. Välimäki, K. Prawda, and S. J. Schlecht. "Two-Stage Attenuation Filter for Artificial Reverberation." IEEE Signal Processing Letters, 2024.	Informed design of attenuation filters	SOTA design consisting of a first-order low-shelf filter and one-third-octave GEQ	Two_stage_filter Matlab. Uses the J. Liski's GEQ
Machine Learning Optimization
S. Lee, H. S. Choi, Lee K., "Differentiable artificial reverberation." IEEE/ACM Transactions on Audio, Speech, and Language Processing 30: 2541-2556, 2022.	Parameter estimation	Presents a differentiable FDN, along with differentiable Filtered Velvet Noise. Introduces a parameter estimation network for analysis-synthesis and blind estimation task in an end-to-end manner.	Unofficial implementation Python. First of its kind. Online audio examples available.
G. Dal Santo, K. Prawda, S. J. Schlecht, and V. Välimäki. "Differentiable Feedback Delay Network for colorless reverberation." International Conference on Digital Audio Effects (DAFx23), Copenhagen, Denmark, Sept. 4-7, 2023.	Parameter optimization	Optimize the gain parameters of the FDN to reduce metallic artifacts and increase temporal density. Using this optimization allows the reduction of the number of channels needed for a smooth response.	diff-fdn-colorless (dafx23 branch) Python. Online audio examples available.
Dal Santo, G., Alary, B., Prawda, K., Schlecht, S., & Välimäki, V. RIR2FDN: An improved room impulse response analysis and synthesis. In International Conference on Digital Audio Effects (pp. 230-237), 2024. University of Surrey.	Informed design + Parameter optimization	Present a pipeline to design a smooth-sounding FDN to match a given RIR. Evaluates the results with a perceptual study.	rir2fdn Python + Matlab filter design. Online audio examples available.
A. I. Mezza, R. Giampiccolo, E. De Sena, and A. Bernardini. Data-driven room acoustic modeling via differentiable feedback delay networks with learnable delay lines. EURASIP Journal on Audio, Speech, and Music Processing, 2024(1), 51.	Parameter optimization	Presents a method for optimizing FDN parameters in the time domain. Introduces the echo density profile loss and the optimization of delay lines along with the other FDN parameters.	..
A. I. Mezza, R. Giampiccolo, and A. Bernardini, Modeling the frequency-dependent sound energy decay of acoustic environments with differentiable feedback delay networks. In Proceedings of the 27th International Conference on Digital Audio Effects (DAFx24) (pp. 238-245), 2024.	Parameter optimization	Extends the author's previous work to frequency-dependent FDNs.	..
Dal Santo, G., Prawda, K., Schlecht, S. J., and Välimäki, V. "Optimizing tiny colorless feedback delay networks," EURASIP Journal on Audio, Speech, and Music Processing, 2025(1), 13.	Parameter optimization	Improved the previous colorless FDN work by making the optimization iFFT-free.	diff-fdn-colorless Python. Online audio examples available (GitHub)
I. Ibnyahya, and J. Reiss. "Differentiable Attenuation Filters for Feedback Delay Networks." 28th International Conference on Digital Audio Effects (DAFx25), 2025.	Attenuation filters optimization	Optimization of a parametric equalizer to reduce the number of required second-order sections.	irr_match Python
Götz, P., Dal Santo, G., Schlecht, S. J., Välimäki, V., & Habets, E. A. (2025). Matching Reverberant Speech Through Learned Acoustic Embeddings and Feedback Delay Networks. arXiv preprint arXiv:2510.23158 (Under review).	Parameter estimation	Present a parameter estimation network to solve the reverberant signal matching task with a differentiable FDN.	Online audio examples available

📝 FDNs for Spatial Audio

Reference	Content Type	Main Contributions	Notes
J. Anderson and S. Costello. "Adapting artificial reverberation architectures for B‑format signal processing." Proc. Ambisonics Symposium (Graz, Austria), 2009.	Perceptually motivated design	B-format spatial reverb. Early reflections are modeled using multiple cascaded delay stages, while late reverb is modeled using a delay network and a steering gain.	Core algorithm works in A-format, requiring B to A conversion at input. Has a lot of practical considerations.
S. J. Schlecht and E. A. P. Habets. "Sign-Agnostic Matrix Design for Spatial Artificial Reverberation with Feedback Delay Networks." AES Conf. on Spatial Reproduction, 2018.	Informed design, optimization	Method to design unilossless and spatially-aware feedback matrix. Has direct connections with SDN design.	Applications in aesthetic reverbs, coupled rooms, and SDN. Online audio examples and animations available.
B. Alary, A. Politis, S. J. Schlecht & V. Välimäki. "Directional feedback delay network." AES: Journal of the Audio Engineering Society, 2019.	Informed design	Uses multichannel delay lines in the Ambisonics domain with directional attenuation. A directional weighting matrix transform modifies the energy distribution over time.	..
B. Alary, & A. Politis, "Frequency-dependent Directional feedback delay network." In: IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2020.	Informed design	Uses multichannel delay lines encoded for a set of directions, together with direction and frequency dependent attenuation.	Simplifies the processing presented in the 2019 paper above.
Machine Learning Optimization
R. Giampiccolo, A. I. Mezza, and A. Bernardini, "Differentiable MIMO Feedback Delay Networks for Multichannel Room Impulse Response Modeling," In Proceedings of the 27th International Conference on Digital Audio Effects (DAFx24) (pp. 278-285), 2024.	..	Extends the author's previous work to MIMO FDNs.	..
R. Giampiccolo, A. I. Mezza, M. Pezzoli, S. Koyama, A. Bernardini, and F. Antonacci, "Modeling the Impulse Response of Higher-Order Microphone Arrays using Differentiable Feedback Delay Networks", In Proceedings of the International Conference on Digital Audio Effects (DAFx25) (pp. 180-187), 2025.	..	Presents a novel loss function to optimize FDNs and learn the energy distribution in space, as well as in the time-frequency domain.	Online audio examples available
..	..	..	..

📝 Scattering Delay Networks

Reference	Content Type	Application	Notes
E. De Sena, H. Hacıhabiboğlu, and Z. Cvetković, "Scattering Delay Network: An Interactive Reverberator for Computer Games," in Proc. 41st AES International Conference: Audio for Games, London, UK, February 2011.	Informed	..	..
E. De Sena, H. Hacıhabiboğlu, Z. Cvetković & J. O. Smith. "Efficient synthesis of room acoustics via scattering delay networks." IEEE/ACM Transactions on Audio, Speech, and Language Processing, 2015.	Informed	..	Code available here
M. Scerbo, O. Das, P. Friend & E. De Sena. "Higher-Order Scattering Delay Networks for Artificial Reverberation." 25th International Conference on Digital Audio Effects (DAFx).	..	..	..
L. Vinceslas, M. Scerbo, H. Hacıhabiboğlu, Z. Cvetković & E. De Sena. "Low-Complexity Higher Order Scattering Delay Networks." IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA).	..	..	..
..	..	..	..
..	..	..	..
Machine Learning Optimization
A. I. Mezza, R. Giampiccolo, E. De Sena, and A. Bernardini, "Differentiable Scattering Delay Networks for Artificial Reverberation." Proceedings of the International Conference on Digital Audio Effects (DAFx25), 2025.	Informed and optimized	..	Code available here. Presents a methodology for the optimization of SDN parameters to account for variability in geometry and/or floor plan and match the time-frequency decay of sound energy.
..	..	..	..

📝 Early Reverbs and FDN Theory

Reference	Year	Notes
M. R. Schroeder and B. F. Logan. "Colorless artificial reverberation." J. Audio Eng. Soc.	1961	Cascade of allpass filters
M. R. Schroeder. "Natural-sounding artificial reverberation." Journal of the Audio Engineering Society.	1962	Combination of parallel comb filters with series allpass filters. Known as the Schroeder reverb
M. A. Gerzon. "Synthetic stereo reverberation, parts I and II." Studio Sound.	1971(I), 1972(II)	Feedback delay network - multichannel allpass reverberator
J. A. Moorer. "About this reverberation business." Computer Music Journal.	1979	Introduces lowpass filters within comb filters to model high frequency damping. Uses a sparse FIR filter to model early reflections
J. Stautner and M. Puckette. "Designing Multi-Channel Reverberators." Computer Music Journal.	1982	"Consolidates" Gerzon's reverb
W. G. Gardner. "A real-time multichannel room simulator." J. Acoustical Society of America.	1992	Nested allpass filters
W. G. Gardner. "The virtual acoustic room." Master's thesis, MIT.	1992	Diffuse reverberators for small/medium/large rooms, based on nested allpass filters
J.-M. Jot and A. Chaigne. "Digital delay networks for designing artificial reverberators." Proc. 90th Conv. Audio Eng. Soc.	1991	Introduces the concept of delay-proportional attenuation filters
J.-M. Jot. "An Analysis/synthesis approach to real-time artificial reverberation." Proc. ICASSP.	1992	Explains how to design attenuation and tone corrector filters from a target RIR
J. Dattorro. "Effect design, part 1: Reverberator and other filters." J. Audio Engineering Society.	1997	Tutorial-like paper with complete design, coefficient values and practical insights
D. Rocchesso and J. O. Smith III, "Circulant and elliptic feedback delay networks for artificial reverberation," IEEE Trans. Speech, Audio Process., vol. 5, no. 1, pp. 51–63, 1997.	1997	Shows that lossless FDNs can be achieved by any feedback matrix having unit-modulus eigenvalues and linearly independent eigenvectors. Presents the circulant matrix, for efficient implementations. Note that eq (29) is flawed, as noted in the Appendix of S. J. Schlecht's PhD thesis
D. Rocchesso, "Maximally diffusive yet efficient feedback delay networks for artificial reverberation," IEEE Signal Process. Lett., vol. 4, no. 9, pp. 252 – 255, 1997.	1997	Uses Galois sequences arranged in a circulant matrix to produce a maximum echo density in the time response.
..	..	..

📚 Other resources

• "Artificial Reverberation" chapter of Miller Puckette's Theory and Techniques of Electronic Music book (December 2006)
• "Efficient Reverb Rendering for Auditory Scenes" by Jean Marc Jot DAFx17 Tutorial
• Sean Costello's "Getting started with reverb design" ValhallaDSP blog.
• "On reverb design" Talk by Sean Costello about the history of artificial reverberators and how he designs a reverb algorithm. Talk (March 2019)
• "Designing the Make Noise Erbe-Verb - Reverb Design Lecture" by Tom Erbe Lecture at UCSB (2019)
• "Let's Write a Reverb" by Geraint Luff ADC 21
• "Feedback Delay Networks for Artificial Reverberation" by Sebastian J. Schlecht CCRMA seminars 22
• "Odd Challenges of Using Deep Learning in Designing a Feedback Delay Network Reverb" by Wojciech Kacper Werkowicz and Benjamin Whateley ADC 23
• "Building Flexible Audio DDSP Pipelines: A Case Study on Artificial Reverb" by Gloria Dal Santo DAFx25 Tutorial

🤝 Acknowledgements

This list is largely inspired by:

• S. J. Schlecht. "Feedback Delay Networks in Artificial Reverberation and Reverberation Enhancement" (PhD thesis, 2018).
• B. Alary. "Analysis and Synthesis of Directional Reverberation" (PhD thesis, Aalto University, 2021).
• V. Välimäki, J. D. Parker, L. Savioja, J. O. Smith, & J. S. Abel. "Fifty years of artificial reverberation" (IEEE TASLP, 2012).

🫶 Contributing

Contributions are welcome! Thank you for helping keep this list up to date. The easiest flow for adding references is:

1. Fork the repository.
2. Create a branch with a descriptive name.
3. Edit README.md and add your reference as a new row in the appropriate table.
4. Commit your change with a short message.
5. Push your branch and open a pull request against the main branch.

Notes:

• Avoid promotional or non-technical entries; keep the list a factual resource.
• By contributing, you confirm you have the right to share the bibliographic information and links you provide.
• We don't want this to become a tier list. Please try to be as objective as possible.

If you have questions about contribution format or want help making larger structural edits (e.g., converting tables to a machine-readable format), please open an issue or mention it in your PR. Thanks!