Filter Feedback Matrices: Theory & Applications

• Filter Feedback Matrices (FFMs) are matrix-valued constructs where each entry is a stable FIR/IIR filter, enabling precise spatio-temporal mixing in feedback delay networks.
• FFMs maintain losslessness through paraunitarity constraints, ensuring energy preservation and controlled modal decay while emulating rough acoustic reflections.
• FFMs enhance artificial reverberation by accelerating echo density and managing modal decay via innovative designs like velvet feedback matrices that balance delay and computational efficiency.

Filter Feedback Matrices (FFMs) are matrix-valued structures in which each entry represents a filter—typically finite impulse response (FIR) or infinite impulse response (IIR)—generalizing the conventional scalar feedback matrix utilized in Feedback Delay Networks (FDNs). FFMs enable precise spatio-temporal mixing and dispersal of energy in recursive filter structures, facilitating applications such as artificial reverberation and decorrelation while maintaining losslessness through paraunitarity constraints (Schlecht et al., 2019).

1. Mathematical Definition and Formulation

Let $M$ denote the number of channels in the network. A filter feedback matrix is an $M\times M$ matrix $A(z) = [H_{ij}(z)]_{1\le i, j \le M}$, where $H_{ij}(z)$ is a stable FIR or IIR filter. The FFM operates in conjunction with a diagonal delay matrix $$D(z) = \operatorname{diag}[z^{-d_1}, ..., z^{-d_M}]$$, resulting in the transfer function:

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

where $b, c \in \mathbb{R}^M$ are input/output gain vectors, and $d$ is a direct-path gain. When all $H_{ij}(z)$ are scalars, $A(z)$ reduces to a traditional feedback matrix. By making $H_{ij}(z)$ general filters—often with short support—each reflection in the network can be temporally smeared, supporting physical and perceptual modeling of scattering and decorrelation (Schlecht et al., 2019).

2. Losslessness and Paraunitarity

An FFM is lossless if and only if it is paraunitary, i.e.,

$$A(e^{j\omega})A(e^{-j\omega})^H = I_M, \quad \forall \omega.$$

This ensures all system poles are on the unit circle, equivalent to energy preservation in the delay network. An equivalent allpass condition for each $i,j$:

$$\sum_{k=1}^M H_{ik}(z) H_{jk}^*(z^{-1}) = \delta_{ij}.$$

Construction of paraunitary matrices can proceed via several equivalent methods:

• Vaidyanathan’s elementary-block factorization: $A(z) = \prod_{\ell=1}^K G_\ell(z)$ where $$G_\ell(z) = I - v_\ell v_\ell^H + z^{-1}v_\ell v_\ell^H$$ and $v_\ell$ are unit vectors. Each $G_\ell(z)$ yields a minimal McMillan degree-1 paraunitary matrix.
• Cascaded delay–unitary mixing: $A(z) = D_K(z)U_K \cdots D_1(z)U_1D_0(z)$ with $D_k(z)$ diagonal delay matrices and $U_k$ constant $M \times M$ unitary matrices. The interleaving of delays and unitary transforms yields an FIR paraunitary matrix with efficient implementation (Schlecht et al., 2019).

3. Applications in Feedback Delay Networks and Reverberation

In an FDN with $M$ delay lines, the use of FFMs rather than scalar matrices allows each feedback coefficient to convolve its input with a short FIR, spreading each impulse over time and emulating the time-dispersive, scattering character of rough acoustic surfaces. This has several direct effects:

• Echo density acceleration: Even with few delay lines ($M=4$), an FFM produces reverberation tails with echo densities comparable to scalar FDNs with $M=16$, but at reduced computational and memory cost.
• Modal decay control: FFMs introduce minimal variation in modal decay distribution, typically a few percent RMS spread for practical filter orders (up to 2000), which remains imperceptible for audio applications.
• Scattering emulation: Algebraically, traversal through an FFM is equivalent to repeated convolution by short FIR filters, yielding a burst of low-coherence, temporally distributed echoes that accumulate into a dense, noise-like tail (Schlecht et al., 2019).

4. Velvet Feedback Matrix (VFM) Construction

A "velvet feedback matrix" (Editor's term) is a sparse, structured FFM designed to maximize echo density growth while minimizing arithmetic complexity. Its entry construction follows:

$V_{ij}(z) = \sum_{p=1}^P s_{ij,p} z^{-d_{ij,p}},$

where $s_{ij,p}$ are pulses (typically $\pm 1$), $d_{ij,p}$ are delays at irregular intervals, and $P$ is the number of pulses per filter. VFMs employ Hadamard-type mixing and staged delay choices, enabling O($M\log M$) complexity per stage by exploiting fast Walsh–Hadamard transforms, and use pulse densities $\rho = P/L$ calibrated to ensure echo-density growth within target perceptual thresholds (Schlecht et al., 2019).

5. Practical Implementation and Trade-Offs

Implementation strategies for FFMs in FDNs include:

• Cascaded form with ring buffers: Delays are implemented as circular buffers; unitary mixing is realized by additions only—no explicit FIR convolution is required.
• FFT-based mixing: For longer FIRs, each FFM block is computed via parallel FFTs and per-bin unitary mixing, at O($MF\log F + M^2$) cost per block (where $F$ is FIR length).
• Density/delay trade-offs: Increasing FFM order (number of mixing stages) or pulse density accelerates echo density at mild computational cost, but excessive group-delay or FIR length may degrade modal decay control.

A typical design targets total group-delay to "smear" each reflection by a few tens of samples, striking a balance between density and decay accuracy (Schlecht et al., 2019).

6. Theoretical and Physical Interpretation

From a geometric acoustics perspective, FFMs model rough surface interactions, translating a specular reflection into a cluster of temporally distributed, lower-coherence echoes. Each echo path through the FFM accumulates temporal spread proportional to both delay and FIR length, so repeated traversals result in highly overlapping, noise-like responses. This mechanism drives rapid approach to the statistical echo density of white noise, as quantified by the power-law exponent of the log echo density curve (Schlecht et al., 2019).

7. Performance Metrics and Comparative Results

Monte Carlo simulations demonstrate that four-line scalar FDNs require roughly 0.4 s to reach 90% Gaussian echo density, whereas a four-line VFM FDN with $\rho = 1/30$ and two mixing stages achieves the same in 0.04 s—a tenfold improvement. The mean reverberation time can be controlled within 5% of target by appropriate placement of diagonal loss after the VFM, ensuring perceptual transparency. Modal RMS spread remains minimal ($\lesssim$ a few percent) for FIR orders up to 2000, confirming FFMs’ practical viability in high-fidelity artificial reverberation (Schlecht et al., 2019).