Gap Triple Filtering in Modal & Signal Analysis

• Gap Triple Filtering is a set of advanced methods that exploit triple-wise spectral gaps, modal structures, or energy intervals for precise component separation.
• It is applied in CFD, quantum information, and nanoelectronics to enhance signal extraction, reduce uncertainty, and control multi-modal degeneracies.
• Techniques like least-squares minimization and Bayesian clustering optimize its performance by robustly resolving overlaps and improving inference in complex systems.

Gap Triple Filtering encompasses a set of methodologies in physics, applied mathematics, and engineering for analyzing signals, states, or transmission properties by exploiting triple-wise relationships related to spectral gaps, modal structures, or energy intervals. The concept finds specific realization in areas such as modal decomposition of convective/acoustic pressure fields in turbomachinery, multi-modal electronic transport filtering in graphene, and higher-order statistical correlation analysis of arithmetic sequences. Gap triple filtering typically aims to resolve, separate, or selectively analyze components corresponding to distinct gaps or intervals, or to exploit triple-wise combinatorial structures for advanced inference and system control.

1. Triple-Plane Pressure Mode Filtering and Convective-Acoustic Separation

A prominent implementation of gap triple filtering occurs in the context of time-periodic CFD simulations of turbomachinery components via the extended triple-plane pressure mode matching (XTPP) technique (Wohlbrandt et al., 2015). The standard TPP method extracts the acoustic field by decomposing measured pressure signals across three adjacent axial planes in a duct into acoustic eigenmodes—where the underlying gap is a spectral separation between acoustic and non-acoustic (convective) modes.

The XTPP method analytically adds a "convective pseudo-mode" basis to the decomposition, defined by axial wavenumber $k_x^c$ associated with the mean flow and using the same orthogonal radial eigenfunctions as the acoustics:

$$p'(x, r, \theta, t) = \mathrm{Re}\left\{ \sum_n \left[ A_n^+ e^{i k_x^+ x} + A_n^- e^{i k_x^- x} + A_n^c e^{i k_x^c x} \right] f_n(r) \right\} e^{i(m\theta - \omega t)}$$

By exploiting three spatially separated "planes"—analogous to a triple-gap structure—the method fits both the acoustic and convective pressure fields, using a least-squares minimization over the modes. This enables robust filtering: the pseudo-sound (convective pressure fluctuations) are assigned to the $A^c$ amplitudes, and true acoustic modes ($A^\pm$) are stabilized and denoised. The result is highly reliable separation near sources, drastic reduction in uncertainty (sound power variance drops from $10\,$dB to $2\,$dB), and a significant reduction in required simulation domain size.

2. Graph-Based Structured Filtering and Triple Gap Degeneracy Resolution

Gap triple filtering also arises in advanced parameter estimation scenarios in quantum information and signal processing—where the likelihood or posterior depends on gap (difference) structures, typically between three spectral values. In the structured filtering framework (Granade et al., 2016), sequential Monte Carlo (SMC) particle filters are generalized by AI-guided weighted clustering in the parameter space, allowing dynamic inference of arbitrary multi-modal posterior structures.

For problems such as randomized gap estimation (RGE), the likelihood depends only on differences (gaps) between eigenvalues, potentially leading to triple-wise degeneracies: several sets of parameters yield identical probability. Standard SMC collapses to a single mean, losing modes. Structured filtering recursively clusters particles in the posterior by weighted k-means, with the number of clusters (representing gaps and their permutations) selected by Bayesian model selection via Bayes factors. The graphical model is a rooted tree encoding all plausible clusterings, where splitting and pruning rules preserve or collapse clusters based on evidence:

Aspect	Structured Filtering	Standard SMC/Liu–West
Posterior	Dynamically multi-modal	Unimodal approximation
Clustering	Weighted/AI-driven	None
Degeneracy	Robust	Fails for gap degeneracy

This approach explicitly supports gap triple filtering—preserving three-way gap relationships and allowing the estimator to represent all symmetry-equivalent solutions induced by triple-wise gap degeneracies.

3. Triple Barrier Electronic Filtering in Bilayer Graphene

In AB bilayer graphene, triple electrostatic barriers realize gap triple filtering of charge carrier transport (Saley et al., 2022). The system consists of three adjacent gate-defined barrier regions, possibly each with independent interlayer potential differences (biases) $\delta_j$. These generate one, two, or three discrete energy gaps in the electronic spectrum, tunable via the barrier configuration:

• For energies $E < \gamma_1$ (interlayer coupling), only one mode propagates; at normal incidence, transmission is suppressed in the gap region for sufficiently large barriers—anti-Klein tunneling.
• For $E > \gamma_1$, four transmission channels exist; asymmetric barriers and selective biasing enable triple-wise control over gap formation.
• Transmission probability through channels is given by:

$$T_{\pm}^{\tau} = \frac{k_0^{\pm}}{k_0^{\tau}} |t_{\pm}^{\tau}|^2$$

• By applying bias in $n$ ($n=1,2,3$) regions, $n$ gaps are opened ($U_j-\delta_j < E < U_j+\delta_j$). Large-width triple barriers yield strong transmission suppression inside gaps, beyond the double barrier case.
• Enhanced transmission resonances, especially in $T_{-}^{+}$, and increased conductance peak count emerge, providing precise multi-gap spectral filtering for electronic transistors and energy filter designs.

4. Triple Correlation Functions and Arithmetic Gap Filtering

Gap triple filtering is conceptually embodied in higher-order correlation statistics of sequences, such as the triple correlation function of fractional parts $\{n^2\alpha\}$ modulo $1$ (Technau et al., 2020). For $$S_{\alpha, N} = \{\alpha, 4\alpha, \ldots, N^2\alpha\} \bmod 1$$, the triple correlation function

$$R_3(\alpha, L, N, g) = \frac{1}{N}\sum_{\substack{1 \leq x_1, x_2, x_3 \leq N \ \ \text{distinct}}} g\left( \frac{N}{L} \{\alpha(x_1^2 - x_2^2)\}_{\text{sgn}}, \frac{N}{L} \{\alpha(x_2^2 - x_3^2)\}_{\text{sgn}} \right)$$

models gap filtering via triple intervals. The main result establishes that, for almost all $\alpha$ and $L > N^{1/4+\varepsilon}$,

$\mathbb{E} W_{\alpha, L, N}^3 = L^3 (1 + o(1))$

corresponding to Poissonian triple-gap statistics. Standard discrepancy methods only yield this for $L > N^{1/2+\varepsilon}$; the new techniques connect triple correlations to modular counting and exploit cancellations over polynomial congruences to extend the Poisson regime—a genuine improvement in gap-triple filtering for arithmetic dynamics.

5. Methodological Implications and Generalizations

Gap triple filtering can be interpreted as a methodological template for problems where components are distinguished or manipulated based on triple-wise relations in spectral, spatial, modal, or statistical domains. All approaches above share key principles:

• Structural enrichment: Additional modes, clusters, or channels are explicitly modeled to capture triple-wise relations.
• Orthogonality: Eigenfunction choices reinforce separation and stability, visible in XTPP and graphene transport models.
• Optimization and inference: Least-squares or Bayesian strategies are used to estimate modal or cluster amplitudes, separating overlaps—e.g., solving Hermitian systems in XTPP, weighted clustering in structured filtering.
• Performance gain: Robustness to degeneracy, consistency across positions, and reduction in uncertainty are empirically validated.

This suggests that the triple gap structure is optimal for certain separation/filtering problems, especially where binary methods are insufficient due to overlying degeneracy or multi-modal phenomena.

6. Applications, Limitations, and Outlook

Applications traverse CFD-based aeroacoustic analysis, quantum parameter estimation, nanoelectronic engineering, and analytic number theory. Noted limitations include:

• Extension to more complex mean flow or multi-modal profiles may result in loss of orthogonality and system ill-conditioning, as seen for generalizations of XTPP.
• Structured filtering requires regulation of tree depth and tuning of cluster selection parameters; statistical performance depends on particle count and resampler design.
• For arithmetic triple correlations, control extends to generic polynomial sequences only for large enough intervals; more refined cancellation methods may be needed for smaller $L$ or larger $k$.

A plausible implication is that future work will build on triple filtering concepts to address higher-order degeneracy, extend to spatio-temporal and multi-level filtering regimes, and develop scalable algorithms for real-time control and analysis leveraging these structures. Gap triple filtering continues to be foundational for advanced separation, detection, and inference in fields exploiting multi-gap or multi-channel structures.