Optimization for Colorless Reverberation

Method

The feedback matrix A and the input and output gains b_i and c_i of a lossless FDN are optimized to minimize the coloration of the resulting reverberation. The loss combines a spectral flatness and a temporal sparsity term, and is minimized with the Adam optimizer using gradients approximated by central finite differences. The delay lengths m_i are kept constant, so each pair of examples below shares the same decay characteristics and differs only in coloration. The N = 16 example uses the circulant matrix, which requires N parameters instead of N².

Full details are given in the paper and, at greater length, in the thesis:

FDN Sandbox: Real-Time Experimentation and Analysis of FDNs Alexandre St-Onge and Gary Scavone. Proc. 29th Int. Conf. on Digital Audio Effects (DAFx26), Cambridge, MA, USA, 1–4 September 2026.

Optimization of Feedback Delay Networks for Room Acoustics Reproduction Alexandre St-Onge. Master's thesis, Music Technology, Schulich School of Music, McGill University, 2026.

Each example is a run of optim_tool from fdn_opt, driven by scripts/run_demo.py; see the README to reproduce them.

Results at a glance

Values are given as initial → optimized. The spectral flatness targets 0.5575; a lower temporal sparsity and a lower ℒ_color are better.

FDN	Parameterization	Spectral flatness	Temporal sparsity	Echo density @100 ms	ℒ_color	Time

Notes

Within each pair, both impulse responses were normalized by the same factor, so that what is compared is the coloration and not the loudness.
The metrics are computed over the first second of each impulse response, using the same analysis window and FFT size as the optimization itself. The values of ℒ_color are reported by optim_tool at its initial and final parameters and are not recomputed here.
The optimization is seeded, but the finite-difference gradients are evaluated in parallel, so the reported losses may differ in the last digits from one machine to another.
Echo density builds up slowly for small FDNs, so some profiles do not reach the fully diffuse threshold within the one-second analysis window. This is reported as such rather than extrapolated.