S4D-DFouT: Discrete Fourier Init for SSMs

• S4D-DFouT is defined as a discrete-domain Fourier initialization for diagonal state space models that decouples decay and frequency, ensuring uniform spectral coverage.
• It addresses spectral bias and aliasing in traditional continuous-time SSM initializations by designing poles directly in the z-domain with a separate learnable damping parameter.
• Empirical results on long-range tasks like LRA and PathX-256 demonstrate S4D-DFouT’s enhanced robustness, scalability, and efficiency relative to earlier diagonal methods.

S4D-DFouT is a discrete-domain, frequency-aware initialization for diagonal state space models (SSMs) that places poles uniformly at discrete Fourier bins while controlling decay with a separate damping parameter $\xi$. It was introduced to address a specific spectral bias in diagonal SSMs: under conventional continuous-time initialization followed by discretization, decay and oscillation frequency are both scaled by the learnable sampling interval $\Delta$, which makes spectral sensitivity non-uniform, can induce aliasing, and can collapse representations into narrow low-frequency bands. By designing poles directly in the $z$-domain, S4D-DFouT decouples frequency from decay at initialization, yields uniform spectral coverage, and improves robustness and scalability on long-range sequence tasks (Solozabal et al., 28 Aug 2025). The term is not used uniformly across the later literature: some works deploy S4D without defining “DFouT,” while others treat it either as a Fourier-grid diagonal variant or as shorthand for Fourier-domain kernel computation (Zhao et al., 14 Jun 2025, Khacef et al., 27 Nov 2025, Tscherniak et al., 28 Nov 2025, Shikder et al., 22 Apr 2026).

1. Terminology and place within the S4D family

S4D-DFouT belongs to the broader family of diagonal SSMs that emerged as simpler and more efficient alternatives to structured SSMs such as S4. Earlier foundational work on diagonal SSMs did not use the name “S4D-DFouT.” Instead, it introduced variants such as S4D-LegS, S4D-Inv, and the damped Fourier-style S4D-Lin with $\lambda_n = -\tfrac{1}{2} + i \pi n$; the related non-diagonal DPLR variant was named S4-FouT (Gu et al., 2022). In that genealogy, S4D-DFouT is a later, more explicit discrete-Fourier reformulation: rather than approximating Fourier-oriented structure through continuous-time diagonal spectra and subsequent discretization, it initializes the model directly on the discrete Fourier grid (Solozabal et al., 28 Aug 2025).

This terminological distinction is material. Standard S4D formulations typically train continuous-time parameters and a time step $\Delta$, then discretize during the forward pass, whereas S4D-DFouT is defined by direct pole placement in the discrete domain (Zhao et al., 14 Jun 2025, Solozabal et al., 28 Aug 2025). Later operator analysis also treats S4D-DFouT as a named diagonal SSM variant with a uniform discrete Fourier frequency grid, indicating that the term has acquired a specific technical meaning beyond generic “Fourier-like S4D” language (Shikder et al., 22 Apr 2026).

At the same time, the literature remains heterogeneous.

Work	Name used	Relation to S4D-DFouT
(Gu et al., 2022)	S4D-Lin / S4-FouT	Earlier Fourier-style diagonal or DPLR variants
(Solozabal et al., 28 Aug 2025)	S4D-DFouT	Discrete Fourier-domain initialization
(Shikder et al., 22 Apr 2026)	S4D-DFouT	Fourier-grid diagonal variant in operator analysis
(Khacef et al., 27 Nov 2025)	S4D	No DFouT module introduced
(Tscherniak et al., 28 Nov 2025)	S4D, with “DFouT” as DFT/FFT shorthand	Execution-path interpretation rather than separate architecture

A recurring misconception is therefore to treat S4D-DFouT as a universally standardized module name. The later application literature does not support that view. In edge AI, neuromorphic fall detection, mobile sensing, and EEG-based BCI, several systems use S4D or Fourier-oriented diagonal initializations without introducing a distinct DFouT block or acronym expansion (Zhao et al., 14 Jun 2025, Khacef et al., 27 Nov 2025, Mei et al., 2024, Tscherniak et al., 28 Nov 2025).

2. State-space formulation and frequency-domain view

The formal setting is a diagonal discrete-time SSM. For a single-input single-output system,

$$x_{t+1} = A x_t + B u_t, \qquad y_t = C x_t + D u_t.$$

The induced convolution kernel is

$k_t = C A^t B + D\,\mathbf{1}_{\{t=0\}}.$

When $A = \operatorname{diag}(\lambda_1,\dots,\lambda_N)$, the kernel decomposes as

$k_t = \sum_{i=1}^N (C_i B_i)\,\lambda_i^t.$

This is the basic diagonal SSM mechanism: the model becomes a superposition of damped complex exponentials, and the kernel can be evaluated through Vandermonde multiplication rather than the more elaborate machinery used by earlier structured SSMs (Solozabal et al., 28 Aug 2025).

The transfer function is

$H(z) = D + C\,(I - z^{-1}A)^{-1} B,$

and on the unit circle,

$\Delta$0

In the continuous-to-discrete view that underlies standard S4D parameterizations, one starts from a continuous diagonal system,

$\Delta$1

and discretizes, typically with zero-order hold, so that

$\Delta$2

This mapping is exactly where the spectral bias arises: if a continuous pole is written as $\Delta$3, then the discrete pole is $\Delta$4, so both decay and oscillation are tied to the same $\Delta$5 (Solozabal et al., 28 Aug 2025).

The resulting discrete frequency response can be written as

$\Delta$6

Each mode contributes a resonant peak centered at $\Delta$7, with bandwidth controlled by the decay term $\Delta$8. This makes the relation between pole placement and spectral selectivity explicit: angle sets the center frequency, radius sets memory and sharpness (Solozabal et al., 28 Aug 2025).

3. The discrete Fourier-domain initialization

S4D-DFouT replaces continuous-time initialization followed by discretization with direct initialization in the discrete Fourier domain. For a single SSM of order $\Delta$9, the poles are

$z$0

with learnable damping $z$1 (Solozabal et al., 28 Aug 2025). The angles form a uniform grid on the discrete unit circle, while the radii are determined independently by the damping. In this construction, oscillation frequency is fixed by the discrete Fourier bins and memory is set separately by $z$2.

The special case $z$3 places poles exactly on the unit circle. In that limit, the modes form an exact DFT basis and provide perfect frequency coverage together with circular-convolution capacity. Positive damping widens resonant peaks and stabilizes training. Stability follows from $z$4, since then

$z$5

A standard parameterization is to enforce positivity through a softplus transformation (Solozabal et al., 28 Aug 2025).

For a layer with embedding dimension $z$6, S4D-DFouT also introduces layer-wise synchronization through phase offsets

$z$7

so that

$z$8

This yields a uniform grid of $z$9 distinct angles across the layer and is designed to prevent multiple parallel SSMs from competing over the same frequency bands (Solozabal et al., 28 Aug 2025).

The initialization of the remaining parameters is correspondingly simple. The reported practical scheme sets $\lambda_n = -\tfrac{1}{2} + i \pi n$0, initializes $\lambda_n = -\tfrac{1}{2} + i \pi n$1 with $\lambda_n = -\tfrac{1}{2} + i \pi n$2 so that $\lambda_n = -\tfrac{1}{2} + i \pi n$3, and often takes $\lambda_n = -\tfrac{1}{2} + i \pi n$4. Real-valued outputs are ensured by taking

$\lambda_n = -\tfrac{1}{2} + i \pi n$5

so conjugate-paired poles are not required. For real signals, a “half-plane” DFouT variant uses only

$\lambda_n = -\tfrac{1}{2} + i \pi n$6

modes on $\lambda_n = -\tfrac{1}{2} + i \pi n$7, exploiting Hermitian symmetry to halve computation without loss of information (Solozabal et al., 28 Aug 2025).

4. Spectral bias, aliasing, and computational properties

The primary motivation for S4D-DFouT is the spectral bias induced by continuous-to-discrete mapping in diagonal SSMs. Under zero-order hold, a continuous pole $\lambda_n = -\tfrac{1}{2} + i \pi n$8 becomes $\lambda_n = -\tfrac{1}{2} + i \pi n$9 with

$\Delta$0

Distinct analog frequencies remain distinct only if

$\Delta$1

If $\Delta$2 is too large, aliasing occurs; if $\Delta$3 is too small, modes collapse toward low frequencies. Because conventional diagonal initializations must also learn or tune $\Delta$4, they inherit this entanglement between timescale and frequency (Solozabal et al., 28 Aug 2025).

The empirical evidence reported for this effect is specific. On serialized CIFAR, learned kernels exhibit local-attention profiles with peaks at the row stride, such as 32, and little response beyond early lags, indicating concentrated spectral sensitivity. An $\Delta$5-norm analysis further shows that only SSMs whose $\Delta$6 lands in a narrow range contribute substantially, while many modes remain effectively inactive, especially in deeper layers. These observations motivate direct discrete-domain placement of poles rather than indirect control through $\Delta$7 (Solozabal et al., 28 Aug 2025).

S4D-DFouT mitigates this by fixing initialization on an alias-free discrete frequency grid and using $\Delta$8 alone to control memory. This eliminates the initial $\Delta$9-dependence of frequency coverage, reduces the need to sample or learn wide ranges of $$x_{t+1} = A x_t + B u_t, \qquad y_t = C x_t + D u_t.$$0, and makes the model less sensitive to sampling changes. A separate robustness result in raw-audio Speech Commands under zero-shot 8 kHz subsampling is consistent with this design goal: DFouT surpasses earlier diagonal schemes by approximately two points without retraining (Solozabal et al., 28 Aug 2025).

The computational profile remains that of a diagonal SSM. With diagonal $$x_{t+1} = A x_t + B u_t, \qquad y_t = C x_t + D u_t.$$1, the kernel

$$x_{t+1} = A x_t + B u_t, \qquad y_t = C x_t + D u_t.$$2

can be evaluated by direct Vandermonde multiplication in $$x_{t+1} = A x_t + B u_t, \qquad y_t = C x_t + D u_t.$$3 or by FFT-based convolution once the kernel has been formed in $$x_{t+1} = A x_t + B u_t, \qquad y_t = C x_t + D u_t.$$4. Because the angular grid aligns with DFT harmonics, the initialization is reported to often improve numerical conditioning as well (Solozabal et al., 28 Aug 2025). A related robustness literature on diagonalizing HiPPO-derived SSMs also emphasizes that transfer-function behavior in the Fourier domain is a central diagnostic, although it does not use the S4D-DFouT name (Yu et al., 2023).

Typical reported hyperparameter ranges for S4D-DFouT include $$x_{t+1} = A x_t + B u_t, \qquad y_t = C x_t + D u_t.$$5 for many tasks, Adam with learning rates in $$x_{t+1} = A x_t + B u_t, \qquad y_t = C x_t + D u_t.$$6, weight decay $$x_{t+1} = A x_t + B u_t, \qquad y_t = C x_t + D u_t.$$7–$$x_{t+1} = A x_t + B u_t, \qquad y_t = C x_t + D u_t.$$8, batch sizes $$x_{t+1} = A x_t + B u_t, \qquad y_t = C x_t + D u_t.$$9–$k_t = C A^t B + D\,\mathbf{1}_{\{t=0\}}.$0, six layers, embedding sizes $k_t = C A^t B + D\,\mathbf{1}_{\{t=0\}}.$1–$k_t = C A^t B + D\,\mathbf{1}_{\{t=0\}}.$2, and hidden state dimension $k_t = C A^t B + D\,\mathbf{1}_{\{t=0\}}.$3–$k_t = C A^t B + D\,\mathbf{1}_{\{t=0\}}.$4. For PathX-256, one reported configuration uses $k_t = C A^t B + D\,\mathbf{1}_{\{t=0\}}.$5, $k_t = C A^t B + D\,\mathbf{1}_{\{t=0\}}.$6, $k_t = C A^t B + D\,\mathbf{1}_{\{t=0\}}.$7, batch size $k_t = C A^t B + D\,\mathbf{1}_{\{t=0\}}.$8, learning rate $k_t = C A^t B + D\,\mathbf{1}_{\{t=0\}}.$9, and $A = \operatorname{diag}(\lambda_1,\dots,\lambda_N)$0 (Solozabal et al., 28 Aug 2025).

5. Empirical results and comparative behavior

The principal empirical claim for S4D-DFouT is that it improves over earlier diagonal initializations on the Long Range Arena (LRA) while enabling from-scratch training on PathX-256. The reported benchmark scores are as follows (Solozabal et al., 28 Aug 2025).

Task	S4D-DFouT
ListOps (2048)	61.89
Text (4096)	87.41
Retrieval (4000)	90.95
Image (1024)	88.48
Pathfinder (1024)	94.30
PathX-128 (16384)	94.17
PathX-256 (65536)	87.89

Relative to diagonal baselines, the comparison is uneven but informative. S4D-Lin reports 60.52 on ListOps, 86.97 on Text, 90.96 on Retrieval, 87.93 on Image, 93.96 on Pathfinder, and 55 on PathX-128, with a reproduced 96.02 noted in the source summary. S4D-Inv reports 60.18, 87.34, 91.09, 87.83, 93.78, and 92.80 on the same sequence of tasks; S4D-LegS reports 60.47, 86.18, 89.46, 88.19, 93.06, and 91.95. The distinguishing empirical point is PathX-256: S4D-DFouT is reported to allow training from scratch on this benchmark, whereas earlier accounts associated success there with self-pretraining (Solozabal et al., 28 Aug 2025).

Several ablations sharpen the interpretation. On delay or continuous copying tasks, S4D-DFouT reconstructs ideal delays independently of $A = \operatorname{diag}(\lambda_1,\dots,\lambda_N)$1, whereas S4D-Lin succeeds only at a specific setting, $A = \operatorname{diag}(\lambda_1,\dots,\lambda_N)$2, and degrades sharply when the period is misaligned. In this sense, DFouT’s discrete-bin initialization removes a tuning dependence that remains visible in continuous-time diagonal initializations (Solozabal et al., 28 Aug 2025).

The discrete-domain alternatives also matter. S4D-RndImag, which uses random angles, and S4D-Token, with $A = \operatorname{diag}(\lambda_1,\dots,\lambda_N)$3, underperform or produce biased spectral coverage. A batched-DFouT layout, in which each SSM receives contiguous angle blocks, reduces uniformity; synchronized DFouT performs best. The gains are also task-dependent. On permuted sCIFAR, which is reported as lacking pronounced frequency structure, DFouT reaches 65.72 and is described as comparable but not superior to some alternatives (Solozabal et al., 28 Aug 2025).

These findings support a narrow but important interpretation. S4D-DFouT is not a universal replacement for all diagonal or non-diagonal SSM initializations. It is most directly advantageous when task structure benefits from broad, uniform initial frequency coverage, robustness to discretization or sampling changes, and efficient scaling to very long contexts (Solozabal et al., 28 Aug 2025).

6. Later interpretations, deployment context, and limitations

Later theoretical work has broadened the interpretive frame around S4D-DFouT. An exact operator analysis of diagonal S4 variants embeds S4D in a one-dimensional ring of nonlinear oscillators and states explicitly that S4D-DFouT corresponds to a diagonal SSM variant with spectrum

$A = \operatorname{diag}(\lambda_1,\dots,\lambda_N)$4

In that account, the latent dynamics decompose into traveling-wave Fourier modes on the ring, and the nonlinear decoder induces wave–wave interactions through a Carleman expansion. The forward pass remains an exactly analyzable diagonal SSM, and the DFouT choice is interpreted as a uniform DFT frequency allocation across modes rather than a change in the surrounding operator formalism (Shikder et al., 22 Apr 2026).

Application literature, however, has mostly deployed S4D without introducing S4D-DFouT as a separate named component. A neuromorphic fall-detection system on Intel Loihi 2 uses S4D as a temporal head over CNN or MCUNet spatial features, with recurrent on-chip updates

$A = \operatorname{diag}(\lambda_1,\dots,\lambda_N)$5

and reports that MCU13B+S4D achieves 83.6% F1, rounded to 84% in the abstract, with approximately $A = \operatorname{diag}(\lambda_1,\dots,\lambda_N)$6 SynOps sparsity and about 90 mW total power on a single Loihi 2 chip. That work states explicitly that “DFouT” does not appear in the system description (Khacef et al., 27 Nov 2025). Likewise, an EEG-based mobile BCI describes “DFouT” only as a common shorthand for computing the state-space convolution kernel in the Fourier domain via DFT or FFT, not as a distinct model class (Tscherniak et al., 28 Nov 2025).

Deployment implications have also been drawn indirectly. A quantization study on small-scale S4D states that it does not define “DFouT,” but adds that if S4D-DFouT denotes a Fourier- or DFT-based output or parameterization layered on S4D, then the same quantization rules apply: quantize complex-valued quantities with paired real/imaginary quantizers using shared scales and zero-points, keep the state transition matrix $A = \operatorname{diag}(\lambda_1,\dots,\lambda_N)$7 and recurrent state $A = \operatorname{diag}(\lambda_1,\dots,\lambda_N)$8 at higher precision, and rely on quantization-aware training rather than post-training quantization for recurrent edge deployment. In that study, heterogeneous quantization reduces memory by approximately $A = \operatorname{diag}(\lambda_1,\dots,\lambda_N)$9, or about $k_t = \sum_{i=1}^N (C_i B_i)\,\lambda_i^t.$0 compression versus floating point, without sacrificing sMNIST performance in the reported S4D-64h configuration (Zhao et al., 14 Jun 2025).

The limitations of S4D-DFouT are correspondingly specific. Tasks without pronounced frequency structure do not necessarily benefit; textual and large-language-model settings were not explored in the originating study; and non-diagonal models such as S5 can outperform diagonal SSMs on some LRA tasks because low-rank structure yields richer dynamics and better conditioning (Solozabal et al., 28 Aug 2025). The broader interpretability literature on S4D also shows that trained diagonal kernels may realize low-pass, band-pass, or high-pass behavior depending on architecture, which indicates that DFouT’s principal contribution is an initialization and coverage strategy rather than a fixed terminal spectrum (Ravikumar et al., 19 Jan 2026). In that sense, S4D-DFouT is best understood as a discrete spectral design principle for diagonal SSMs: it regularizes the starting pole geometry, not the full learned dynamics.