Scattering Delay Network: Modeling Room Acoustics

• Scattering Delay Network (SDN) is a computational model that uses interconnected delay lines and scattering junctions to generate diffuse, physically informed reverberation.
• It maps physical room geometry to delay lengths and wall absorption parameters, ensuring energy conservation and accurate rendering of first-order reflections.
• The integration of filter feedback matrices, including velvet FFMs, enhances micro-diffusion effects and reduces mixing time while maintaining low computational complexity.

A Scattering Delay Network (SDN) is a class of artificial reverberators or acoustic modeling systems that synthesizes diffuse, physically informed room impulse responses through a network of interconnected delay lines and scattering junctions. SDNs generalize feedback delay networks (FDNs) by interpreting them as graphs that physically map to acoustic boundaries and use mathematically constrained scattering operations to ensure energy conservation, realistic modal distributions, and controllable echo densities. SDNs underpin efficient virtual room simulation and are used in applications such as spatial audio rendering, virtual environments, and perceptual decorrelation.

1. Mathematical Structure of the Scattering Delay Network

The SDN models a reverberant space as a directed graph with $K$ nodes positioned on boundary surfaces (typically first-order image source locations) and bidirectional delay lines forming the edges. Each edge between node $i$ and $j$ consists of two signals, $p_{ij}^+[n]$ (incoming to $j$) and $p_{ji}^+[n]$ (incoming to $i$), realized as digital delay lines of length $D_{ij} = \lfloor F_s \ell_{ij} / c \rfloor$ samples, where $\ell_{ij}$ is the physical distance, $F_s$ is the sampling frequency, and $c$ is the speed of sound. At each node $k$, a $K$-port scattering junction transforms the vector of arriving signals $p^+[n] \in \mathbb{C}^{K}$ into outgoing signals $p^-[n]$, mixing and distributing energy across connected edges.

The scattering at each node is governed by

$p^-[n] = S_k(z)\, p^+[n],$

where $S_k(z) = H_k(z) A$ comprises the wall absorption and a unitary (lossless) mixing matrix. $A$ is typically chosen as an isotropic Householder reflector:

$$A = \left( \frac{2}{K} \mathbf{1}\mathbf{1}^T - I \right),$$

which ensures uniform redistribution of energy. For wall absorption $\alpha_k$, the reflection gain is $\beta_k = \sqrt{1-\alpha_k}$.

2. Physical and Perceptual Parameter Mapping

Delay lengths $\ell_{ij}$ are mapped from the physical room geometry, positioning SDN nodes at first-order image points of reflective surfaces. For nodes $k$ and $m$, the delay line implements $D_{km} = \lfloor F_s \| x_k - x_m \| / c \rfloor$. Wall absorption parameters $\alpha_k(f)$ are drawn from measured absorption spectra (e.g., ISO-354) and translated into per-port reflection coefficients. When wall absorption is frequency-dependent, it is modeled as

$$S_k(z) = H_k(z)\, A, \quad |H_k(e^{j\omega})| = \beta_k(\omega).$$

Source directivity and receiver directivity are incorporated as per-port gain vectors:

$$\Gamma_S = [\Gamma_S(\theta_{S,1}), \ldots, \Gamma_S(\theta_{S,K})]^T, \quad \Gamma_M = [\Gamma_M(\theta_{1,M}), \ldots, \Gamma_M(\theta_{K,M})]^T.$$

Perceptually, the SDN accurately renders first-order reflections using geometric-amplitude-correct gain factors, reproducing pathwise delays and amplitudes as in the Image Source Method (Sena et al., 2015).

3. Filter Feedback Matrices and Scattering Generalization

SDNs encompass generalized FDNs using Filter Feedback Matrices (FFMs), which replace scalar feedback elements with entire FIR or IIR filters. An FFM $A(z)$ is

$A(z) = A_0 + z^{-1}A_1 + \cdots + z^{-L}A_L,$

with each $A_\ell \in \mathbb{C}^{N \times N}$. Losslessness (all poles on the unit circle) is preserved if $A(z)$ is paraunitary:

$A(z^{-1})^H A(z) = I.$

A particular FFM instance, the velvet feedback matrix (VFM), arranges Hadamard mixing stages interleaved with short delays, yielding a matrix whose entries are sparse, pulse-dense “velvet-noise” FIRs.

4. Scattering Emulation, Echo Density, and Mixing

The fundamental motivation for SDNs is the emulation of non-specular, physically diffuse scattering. In a physical environment, rough boundaries cause a micro-diffusion of reflections—clusters of echoes rather than single delayed replicas. SDNs replicate this effect mathematically by replacing pointwise mixing with paraunitary filter matrices. The spread of energy through the system can be quantified by the group-delay matrix:

$$\Gamma(\omega) = \frac{\partial}{\partial \omega} \arg A(e^{j\omega}).$$

Large off-diagonal values in $\Gamma(\omega)$ indicate effective temporal spreading analogous to broadband scattering (Schlecht et al., 2019).

Metrics for echo density and modal distribution are critical: the generalized characteristic polynomial

$g(z) = z^{\text{deg} A} \det [ D(z^{-1}) - A(z) ]$

encodes the system’s poles, each with decay rate

$\tau_k = -\frac{1}{F_s} \ln |p_k|.$

The normalized echo density (NED) traces how rapidly the impulse response approaches a fully diffuse, white-noise-like regime. For the VFM, mixing times are reduced by factors of 5–10 over scalar FDNs, with four-line velvet FDNs reaching perceptual mixing in 50–100 ms (Schlecht et al., 2019).

5. Room-Acoustic Simulation, Accuracy, and Energy Decay

SDNs provide physically interpretable models closely aligned with theoretical energy decay predictions. The total energy decay rate follows the classical Sabine and Eyring formulas:

$$T_{60, \text{Sab}} = \frac{0.161 V}{\sum_i A_i \alpha_i}, \qquad T_{60, \text{Eyr}} = -\frac{0.161 V}{\sum_i A_i \ln(1-\alpha_i)},$$

where $V$ is volume, $A_i$ is surface area, and $\alpha_i$ is absorption. The SDN’s ensemble decay curves and echo density evolution closely match those produced by the Image Method (IM), and remain physically consistent over a wide range of absorption parameters (Sena et al., 2015).

First-order reflections are rendered exactly: a two-segment path from source to wall to receiver, with physically correct time-of-flight and amplitude, closely matches geometric acoustics. Higher-order reflections become increasingly approximate but maintain the correct global energy decay and statistical echo density.

6. Implementation, Complexity, and Practical Considerations

SDNs support both FIR and IIR implementations. For velvet FFMs, the system state at each time step comprises $N$ parallel ring buffers updated by cascaded Hadamard transforms and pointer-shifted delays. Computational complexity with a Householder mixing matrix per node is $O(K)$ per node per sample; overall SDN cost for $K$ wall nodes is comparable to an FDN with $Q \approx 12$ for $K=5$ (rectangular room) (Sena et al., 2015).

Memory usage consists primarily of delay-line buffers, with an upper bound

$Q \leq [K(K-1)+2K+1] \cdot (F_s R / c),$

where $R$ is the room diameter. For a 5 m cubic room and $F_s=44.1$ kHz, this remains below 170 kB. Compared to time-domain IM, which requires storage for the entire room impulse response (often millions of samples), SDN memory requirements are minor.

A summary of the main SDN computational and physical mapping features:

System	Delay Buffers Used	Mixing Matrix	Computational Cost
SDN	Inter-node plus source/mic	Householder/A	$O(K)$ per node/sample
FDN	$N$ ring buffers	Scalar/FFM	$O(N^2)$ or $O(N \log N)$
Image Method (IM)	Full room impulse response	N/A	$O(\text{images})$

7. Significance and Limitations

SDNs bridge the gap between statistically motivated artificial reverberators and physically principled geometric acoustics methods. They enable efficient, parameterizable, real-time simulation of room acoustics, supporting features such as:

• Exact rendering of first-order reflections
• Energy and echo density evolution matching physical predictions
• Direct parameter mapping from room geometry, wall properties, and source/receiver directivity
• Substantially reduced computational and memory demands compared to direct physical modeling

A plausible implication is that SDNs are well-suited for embedded or low-latency environments requiring realistic reverberation under strong computational constraints. One limitation is the approximation of higher-order reflections: while energy and statistical properties are preserved, fine spatial correlation for late reverberation may diverge from full geometric methods at long times or extremely diverse room topologies. Additionally, SDNs' reliance on idealized boundary scattering and delay quantization can introduce perceptual artifacts if not carefully parameterized.

Recent research demonstrates that the use of paraunitary, sparse FIR feedback matrices such as the velvet feedback matrix further improves early echo density and reduces mixing time, enabling more pronounced micro-diffusion effects with negligible additional overhead (Schlecht et al., 2019). This family of techniques offers extensibility to simulate anisotropic scattering, frequency-dependent absorption, and multi-band reverberant effects.