Feedback Delay Network (FDN)

• Feedback Delay Network (FDN) is a recursive filter topology that uses multiple delay lines and a structured feedback matrix to generate high echo density reverberation.
• It incorporates frequency-dependent attenuation via configurable filters, enabling precise control over spectral decay and T60 matching.
• Recent developments integrate FDNs into differentiable optimization frameworks for real-time adaptive reverberation and dynamic spatial audio synthesis.

A Feedback Delay Network (FDN) is a parametric, recursive filter topology extensively used in artificial reverberation, spatial audio rendering, sound synthesis, and physical modeling. An FDN consists of multiple parallel delay lines whose outputs are recursively mixed and fed back via a structured feedback matrix, typically with additional frequency-dependent attenuation and configurable input/output coupling. Distinguished by their high echo density, tunable spectral and temporal decay, and efficient implementation, FDNs underpin state-of-the-art real-time reverberation algorithms and modern differentiable audio synthesis pipelines.

1. Mathematical Formulation and Core Structure

An N-line single-input–single-output (SISO) FDN is described in state-space form as:

$$s(n+1) = A\,s(n) + b\,x(n) \qquad y(n) = c^T\,s(n) + d\,x(n)$$

where:

• $s(n)$: N-dimensional state vector, with each $s_i(n)$ representing a tapped delay of length $m_i$,
• $A \in \mathbb{C}^{N \times N}$: feedback matrix,
• $b, c \in \mathbb{C}^N$: input and output gain vectors,
• $d \in \mathbb{C}$: direct feedthrough gain,
• $x(n)$: input,
• $y(n)$: output (Schlecht et al., 2019).

The corresponding transfer function in the $z$-domain is:

$H(z) = c^T \left(D(z)^{-1} - A\right)^{-1} b + d,$

with $D(z) = \mathrm{diag}(z^{-m_1}, ..., z^{-m_N})$.

In implementations with frequency-dependent attenuation, each delay is followed by an absorption filter $\alpha_i(z)$, yielding $D_\alpha(z) = \mathrm{diag}(z^{-m_i} \alpha_i(z))$.

A modal decomposition is possible:

$$H(z) = d + \sum_{k=1}^M \frac{R_k}{1 - \lambda_k z^{-1}},$$

with $M = \sum m_i$, $\lambda_k$ FDN poles, and $R_k$ residues. The time-domain impulse response is aggregated as $h(n) = \sum_k h_k(n)$, where each $h_k(n)$ contributes an exponentially decaying, frequency-specific resonance.

2. Feedback Matrix Topologies and Losslessness

The feedback matrix $A$ governs energy preservation and decorrelation. The FDN is lossless if, for all choices of delays, all eigenvalues (the roots of $p_{A, \ell}(z) = \det[D(z^{-1}) - A]$) lie on the unit circle. The full class of unilossless (delay-independent lossless) matrices consists of those which, possibly after permutation, decompose into blocks each diagonally similar to a unitary matrix (Schlecht et al., 2016):

$D^{-1} A D = U \quad (U^* U = I)$

Classic choices include:

• Unitary $A$: $A^* A = I$ (e.g., Hadamard, DFT, Householder matrices).
• Triangular $A$ with unimodular diagonals.
• More generally, block-triangular matrices after decompositions into irreducible blocks.

This lossless property allows arbitrary delay choices and per-line attenuation scaling, underpinning stable artificial reverberation.

3. Frequency-Dependent Decay and Attenuation Filtering

Realistic artificial reverberation demands frequency-dependent decay. This is achieved by inserting attenuation filters $H_k(z)$ after each delay line within the feedback loop. Modern approaches use differentiable parametric equalizers (PEQ), each as a cascade of second-order sections (SOS). The PEQ structure enables independent tuning of center frequency ($\omega_0$), gain ($G$), and quality factor ($Q$) per band (Ibnyahya et al., 25 Nov 2025).

A central finding is that center frequencies and $Q$ can be shared across all lines, while gain is scaled by the delay length $m_k$ to match the desired $T_{60}(f)$ (band-dependent reverberation time):

$$20 \log_{10} |H_\mathrm{PEQ, k}(f)| = -60 \cdot m_k / [T_{60}(f)\, f_s]$$

This enables state-of-the-art $T_{60}$ curve matching with far fewer parameters and lower computational cost than traditional per-line graphic EQs.

4. Differentiable FDNs and Optimization Frameworks

Recent research has focused on embedding the entire FDN—including delay lengths, feedback matrix, gain vectors, and attenuation filters—within end-to-end differentiable pipelines. Parameters are optimized by backpropagation to minimize loss functions encoding energy decay, echo density, spectral flatness, or psychoacoustic metrics (Mezza et al., 29 Mar 2024, Santo et al., 17 Feb 2024, Gerami et al., 30 Sep 2025, Götz et al., 27 Oct 2025).

Key differentiable parameterizations include:

• Orthogonal feedback matrices via exponentials of skew-symmetric matrices.
• Per-band or per-line attenuation implemented as differentiable PEQ cascades.
• Learnable delay lengths using frequency-domain fractional delay interpolation.
• Losses computed on energy decay curves (EDC), echo density profiles (EDP), modal energy distributions, and perceptual metrics such as Fréchet Audio Distance.

These methods permit real-time fitting of compact FDNs to measured RIRs or perceptual targets, surpassing analytic and heuristic tuning in accuracy and flexibility.

5. FDN Modal Analysis, Energy Distribution, and Echo Density

Modal decomposition reveals that the poles of the FDN correspond to the system's resonant modes, with residues quantifying excitation energy per mode. In random orthogonal FDNs, the mode frequencies are nearly equidistributed, minimizing modal gaps, while residue magnitudes follow a log-Rayleigh distribution. Empirical studies show only about 10–20% of modes contribute approximately 90% of the impulse response energy, suggesting aggressive mode-pruning is possible without perceptible degradation (Schlecht et al., 2019).

Echo density, defined as the rate at which the IR "fills in" to a Gaussian envelope, is crucial to natural-sounding reverberation. Echo density and mixing time are optimized via careful selection of feedback matrix structure and delay lengths; paraunitary filter feedback matrices and the "velvet" feedback matrix achieve a high echo density at lower computational cost than conventional scalar FDNs (Schlecht et al., 2019).

6. Architectural Extensions: Grouped, Multi-band, and Scattering FDNs

FDN architectures have been generalized to model complex acoustic phenomena:

• Grouped FDNs (GFDN) partition delay lines into groups, each with unique decay rates, enabling multi-slope energy decay as in coupled volumes (Das et al., 8 Aug 2025).
• Multi-band FDNs process audio in parallel across octave bands, yielding per-band control of $T_{60}$ and spatial variation (Das et al., 8 Aug 2025).
• Filter feedback matrices generalize the feedback operation to paraunitary FIR/IIR matrices, supporting advanced scattering behavior and decorrelation (Schlecht et al., 2019).

Table: Key FDN Architectures

Architecture	Key Feature	Reference
Scalar FDN	Lossless matrix, per-line delays	(Schlecht et al., 2016)
Differentiable FDN	Gradient-trained parameters	(Mezza et al., 29 Mar 2024)
GFDN	Multi-slope decay, spatialization	(Das et al., 8 Aug 2025)
Velvet FFM	High echo density, sparse multiply	(Schlecht et al., 2019)
Parametric PEQ filters	Shared-band attenuation	(Ibnyahya et al., 25 Nov 2025)

7. Applications and Practical Considerations

FDNs are the foundation of modern artificial reverberation in game audio, AR/VR, spatial audio rendering, automatic room acoustics synthesis, and perceptually-guided sound matching (Götz et al., 27 Oct 2025, Gerami et al., 30 Sep 2025). Differentiable, data-driven FDNs support dynamic adaptation to changing room geometry and source/receiver motion, with low-latency parameter re-optimization suitable for embedded and wearable devices.

Design best practices include:

• Using unitary or unilossless feedback matrices for stability and high echo density.
• Choosing delay lengths that are coprime or "Lundeby-spaced" to avoid periodicities and maximize modal diversity (Schlecht, 2020).
• Employing band-limited attenuation filters for perceptually-accurate decay time control.
• Modulating input/output gains and delay structure to achieve desired early-to-late energy ratios, definition, and clarity.

Limitations include the challenge of fitting highly frequency-dependent decays with a single FDN and the upper bound on echo density imposed by small N; grouped or filter-matrix extensions provide practical solutions.

In summary, FDNs provide a mathematically principled, highly efficient, and tunable platform for artificial reverberation and room acoustic modeling, with recent advances enabling joint spectral/temporal shaping, differentiable parameter learning, and scalable dynamic spatial audio synthesis (Schlecht et al., 2019, Ibnyahya et al., 25 Nov 2025, Santo et al., 17 Feb 2024, Das et al., 8 Aug 2025).