Recursive Parametric Smoothing Strategy

• RPSS is a design pattern of recursive procedures that replace discrete estimates with smoothed surrogates to stabilize inference, prediction, and signal processing.
• It employs techniques like Gaussian KDE, recursive IIR filters, and incomplete-data score recursions to address challenges such as particle depletion and noise bias.
• Applications span recursive Bayesian smoothing, digital filter design, state-space models, and sequential Monte Carlo, enabling efficient real-time estimation.

Recursive Parametric Smoothing Strategy, or RPSS, is best understood here as an Editor’s term for a family of recursive procedures that replace a discrete, filtered, or otherwise local representation by a smoothed parametric or semi-parametric surrogate and then propagate that surrogate into subsequent estimation, inference, or prediction. In the supplied literature, the most explicit instance is Smoothed Prior‑Proposal Recursive Bayes (SPP‑RB), which uses Gaussian or regularized KDE approximations of transient posteriors to avoid particle depletion in recursive Bayesian inference (Scharf, 3 Aug 2025). Closely related constructions appear in recursive IIR smoothing and prediction for polynomial-phase signals (Kennedy, 2023), incomplete-information maximum-likelihood smoothing in state-space models (Surya, 2023), and forward smoothing with sequential Monte Carlo for additive functionals (Moral et al., 2010). The term also appears in the abstract of “An inverse random diffraction grating problem for the Helmholtz equation,” but the supplied details state that the provided source contains no definition, description, or occurrence of RPSS (Sun et al., 26 Jul 2025).

1. Terminological status and scope

The literature represented here does not use a single standardized meaning of RPSS. One source associates the phrase with inverse reconstruction of random-surface statistics in a Helmholtz diffraction setting, but the supplied source text for that paper is described as a placeholder template rather than the scientific manuscript itself, so no technical account of the method can be extracted from it (Sun et al., 26 Jul 2025). By contrast, the other papers provide explicit recursive smoothing mechanisms, even when they use different names.

Source	Explicit label	Smoothing object
(Sun et al., 26 Jul 2025)	RPSS in abstract; supplied source lacks definition	random-surface statistics
(Scharf, 3 Aug 2025)	SPP‑RB	transient posterior sample
(Kennedy, 2023)	recursive IIR/FIR smoothing and prediction	phase or frequency signal
(Surya, 2023)	ML smoother from incomplete information	latent state trajectory
(Moral et al., 2010)	forward smoothing SMC	additive functional of latent path

This terminological dispersion matters. In one line of work, smoothing means replacing an empirical posterior by a Gaussian or Gaussian-mixture proposal. In another, it means designing fixed-lag filters that exactly reproduce low-order polynomials while minimizing coloured-noise gain. In a third, it means backward or forward recursion for latent-state estimation using score functions, information matrices, or particle approximations. A plausible implication is that RPSS is not a single algorithmic object but a recurrent design pattern: recursive inference is stabilized by a smoothed surrogate that preserves essential structure while avoiding degeneracy, excessive variance, or bias.

2. Recursive Bayesian posterior smoothing

In “A strategy to avoid particle depletion in recursive Bayesian inference,” the recursive structure is

$$[\boldsymbol{\theta}\mid \boldsymbol{y}_{1:j}] \propto [\boldsymbol{y}_j\mid \boldsymbol{\theta},\boldsymbol{y}_{1:j-1}][\boldsymbol{\theta}\mid \boldsymbol{y}_{1:j-1}],$$

with $\boldsymbol{y}$ partitioned into batches $\boldsymbol{y}_1,\dots,\boldsymbol{y}_J$. The paper’s starting point is Prior‑Proposal Recursive Bayes (PP‑RB), where the posterior sample from stage $j-1$ is used directly as the proposal at stage $j$, causing particle depletion because distinct support points can only decrease over stages. SPP‑RB replaces the raw empirical distribution by a smoothed proposal $q_{j-1}(\boldsymbol{\theta})$, while retaining the simplified Metropolis–Hastings ratio

$$r_j^t = \frac{[\boldsymbol{y}_j\mid \boldsymbol{\theta}^*]}{[\boldsymbol{y}_j\mid \boldsymbol{\theta}^{t-1}]}$$

in the conditionally independent case (Scharf, 3 Aug 2025).

The smoothing mechanism begins from a sample $$\{\boldsymbol{\theta}^1,\ldots,\boldsymbol{\theta}^M\}$$. A particle is selected categorically and then perturbed. Three proposal families are discussed. The pure nonparametric option is a Gaussian KDE,

$$[\boldsymbol{\theta}^*\mid \{\boldsymbol{\theta}\}^M] = \frac{1}{M}\sum_{i=1}^M \mathrm{N}\!\left(\boldsymbol{\theta}^* \mid \boldsymbol{\theta}^i,\mathbf{H}_\theta\right).$$

The fully parametric option is a joint Gaussian approximation,

$$[\boldsymbol{\theta}^*\mid \{\boldsymbol{\theta}\}^M] = \mathrm{N}\!\left(\boldsymbol{\theta}^* \mid \boldsymbol{\bar{\theta}},\mathbf{S}_\theta\right).$$

The semi-parametric option is the regularized KDE,

$\boldsymbol{y}$0

typically with $\boldsymbol{y}$1, $\boldsymbol{y}$2. The limiting cases are explicit: $\boldsymbol{y}$3 yields the Gaussian approximation, $\boldsymbol{y}$4 recovers pure multinomial resampling, and intermediate values generate a Gaussian mixture whose centers are shrunk toward the sample mean.

Two properties are central. First, the proposal is continuous, so repeated draws are almost surely unique even if the stored particles are not. Second, the paper shows moment matching: $\boldsymbol{y}$5 under mild dependence conditions. This directly targets the depletion mechanism of PP‑RB, where samples collapse onto a shrinking discrete support and can ultimately degenerate to a single repeated value.

The same logic extends to blocked updates. If $\boldsymbol{y}$6 is partitioned into blocks, the regularized KDE induces a conditional Gaussian-mixture proposal for each block,

$\boldsymbol{y}$7

with weights proportional to Gaussian densities in the fixed complement block. This gives a conditional approximation to the transient full conditional without reverting to a discrete empirical prior.

The numerical studies are unusually concrete. In logistic regression with $\boldsymbol{y}$8 predictors plus intercept ($\boldsymbol{y}$9) and $\boldsymbol{y}_1,\dots,\boldsymbol{y}_J$0, stage $\boldsymbol{y}_1,\dots,\boldsymbol{y}_J$1 uses $\boldsymbol{y}_1,\dots,\boldsymbol{y}_J$2 observations and stages $\boldsymbol{y}_1,\dots,\boldsymbol{y}_J$3–$\boldsymbol{y}_1,\dots,\boldsymbol{y}_J$4 use batches of size $\boldsymbol{y}_1,\dots,\boldsymbol{y}_J$5. All-at-once NUTS in Stan/brms provides transient posterior baselines. SPP‑RB is run with $\boldsymbol{y}_1,\dots,\boldsymbol{y}_J$6, $\boldsymbol{y}_1,\dots,\boldsymbol{y}_J$7 per stage, and $\boldsymbol{y}_1,\dots,\boldsymbol{y}_J$8 independent stage‑1 runs. Performance is summarized through marginal Kolmogorov–Smirnov statistics against the all-at-once posteriors. The reported finding is that $\boldsymbol{y}_1,\dots,\boldsymbol{y}_J$9 yields small KS discrepancies, while diagonal KDE alternatives generally perform worse. In a hierarchical species distribution model on a $j-1$0 grid with $j-1$1, $j-1$2, about $j-1$3 parameters, and $j-1$4 stages, SPP‑RB with $j-1$5 again yields transient and ultimate posteriors that agree well with all-at-once MCMC, and partition-based diagnostics show consistency of ultimate posteriors across different data splits.

3. Recursive smoother design in digital signal processing

A distinct RPSS interpretation appears in “Recursive and non-recursive filters for sequential smoothing and prediction with instantaneous phase and frequency estimation applications,” where smoothing is a filter-design problem for approximately polynomial signals in coloured noise. The observed complex analytic signal is

$j-1$6

with local polynomial phase

$j-1$7

If $j-1$8 two-point differentiators are used before or after the low-pass stage, the residual polynomial seen by the low-pass filter has degree $j-1$9. Unbiased steady-state tracking requires

$j$0

where $j$1 is the number of dc-derivative flatness constraints imposed on the low-pass filter (Kennedy, 2023).

The FIR design is cast as weighted least squares with whitening. With Vandermonde matrix $j$2, synthesis row $j$3, coloured-noise covariance $j$4, and $j$5, the optimal FIR coefficients are

$j$6

For unbiased estimators, the steady-state error variance is expressed in terms of the white-noise gain of the equivalent bandpass operator,

$j$7

This FIR construction is the reference “gold standard” against which the recursive design is compared.

The IIR construction is the recursive counterpart. The low-pass transfer function is expanded in first-order basis functions,

$j$8

where the poles $j$9 are obtained from an analogue low-pass prototype, with Bessel poles recommended for passband phase linearity. The coloured-noise gain becomes a quadratic form,

$q_{j-1}(\boldsymbol{\theta})$0

and unbiased polynomial tracking with group delay $q_{j-1}(\boldsymbol{\theta})$1 is enforced by

$q_{j-1}(\boldsymbol{\theta})$2

The constrained minimizer has the closed form

$q_{j-1}(\boldsymbol{\theta})$3

This recasts recursive smoothing as parametric quadratic optimization in the basis coefficients, with the pole set fixing stability, approximate bandwidth, and phase behavior.

A notable feature is explicit optimization of the lag $q_{j-1}(\boldsymbol{\theta})$4. The coloured-noise gain is

$q_{j-1}(\boldsymbol{\theta})$5

and the paper studies its stationary points. For $q_{j-1}(\boldsymbol{\theta})$6, there is a unique optimal $q_{j-1}(\boldsymbol{\theta})$7. For $q_{j-1}(\boldsymbol{\theta})$8, there are three stationary points, and the middle one is used in simulation because it provides the best phase linearity. The same pole set supports both a lagged smoother and a one-step predictor: the predictor is designed by fixing $q_{j-1}(\boldsymbol{\theta})$9, and both filters can share the same recursive state-space realization, differing only in the output vector.

The paper’s application focus is instantaneous phase and frequency estimation under angle unwrapping constraints. Three configurations are compared: pre-differentiator, aft-differentiator, and nil-differentiator. The aft-differentiator plus predictor is emphasized because the one-sample lead reduces unwrapping errors and lowers the SNR threshold at which failure occurs; the reported thresholds are around $$r_j^t = \frac{[\boldsymbol{y}_j\mid \boldsymbol{\theta}^*]}{[\boldsymbol{y}_j\mid \boldsymbol{\theta}^{t-1}]}$$0 for typical pre-differentiator settings and $$r_j^t = \frac{[\boldsymbol{y}_j\mid \boldsymbol{\theta}^*]}{[\boldsymbol{y}_j\mid \boldsymbol{\theta}^{t-1}]}$$1–$$r_j^t = \frac{[\boldsymbol{y}_j\mid \boldsymbol{\theta}^*]}{[\boldsymbol{y}_j\mid \boldsymbol{\theta}^{t-1}]}$$2 in many aft-differentiator scenarios. In the absence of unwrapping errors, IIR estimators with optimized $$r_j^t = \frac{[\boldsymbol{y}_j\mid \boldsymbol{\theta}^*]}{[\boldsymbol{y}_j\mid \boldsymbol{\theta}^{t-1}]}$$3 attain the FIR lower bound with significantly lower computational cost. The illustrative complexity comparison is $$r_j^t = \frac{[\boldsymbol{y}_j\mid \boldsymbol{\theta}^*]}{[\boldsymbol{y}_j\mid \boldsymbol{\theta}^{t-1}]}$$4 multiply-accumulate operations per sample for an FIR with $$r_j^t = \frac{[\boldsymbol{y}_j\mid \boldsymbol{\theta}^*]}{[\boldsymbol{y}_j\mid \boldsymbol{\theta}^{t-1}]}$$5 taps versus $$r_j^t = \frac{[\boldsymbol{y}_j\mid \boldsymbol{\theta}^*]}{[\boldsymbol{y}_j\mid \boldsymbol{\theta}^{t-1}]}$$6 for an IIR with $$r_j^t = \frac{[\boldsymbol{y}_j\mid \boldsymbol{\theta}^*]}{[\boldsymbol{y}_j\mid \boldsymbol{\theta}^{t-1}]}$$7, approximately a sixfold reduction.

4. Incomplete-information maximum-likelihood smoothing

In state-space models, RPSS can also denote recursive smoothing based on incomplete-data likelihoods. The model class in “Maximum likelihood smoothing estimation in state-space models: An incomplete-information based approach” is

$$r_j^t = \frac{[\boldsymbol{y}_j\mid \boldsymbol{\theta}^*]}{[\boldsymbol{y}_j\mid \boldsymbol{\theta}^{t-1}]}$$8

with both linear Gaussian and nonlinear examples considered. The smoother is defined as

$$r_j^t = \frac{[\boldsymbol{y}_j\mid \boldsymbol{\theta}^*]}{[\boldsymbol{y}_j\mid \boldsymbol{\theta}^{t-1}]}$$9

Its score equation is

$$\{\boldsymbol{\theta}^1,\ldots,\boldsymbol{\theta}^M\}$$0

where $$\{\boldsymbol{\theta}^1,\ldots,\boldsymbol{\theta}^M\}$$1 is the derivative of the incomplete-data log-density with respect to $$\{\boldsymbol{\theta}^1,\ldots,\boldsymbol{\theta}^M\}$$2 (Surya, 2023).

The paper’s main structural device is the incomplete-information identity

$$\{\boldsymbol{\theta}^1,\ldots,\boldsymbol{\theta}^M\}$$3

with complete data $$\{\boldsymbol{\theta}^1,\ldots,\boldsymbol{\theta}^M\}$$4 and incomplete data $$\{\boldsymbol{\theta}^1,\ldots,\boldsymbol{\theta}^M\}$$5. This turns the incomplete-data score into the conditional expectation, or orthogonal projection, of the complete-data score. The corresponding observed-information matrices satisfy

$$\{\boldsymbol{\theta}^1,\ldots,\boldsymbol{\theta}^M\}$$6

and $$\{\boldsymbol{\theta}^1,\ldots,\boldsymbol{\theta}^M\}$$7 is positive definite, so incomplete data entail a strict information loss.

The resulting smoothing recursion has a Newton-like form,

$$\{\boldsymbol{\theta}^1,\ldots,\boldsymbol{\theta}^M\}$$8

and the paper gives explicit iterative schemes. The Newton–Raphson variant uses the incomplete-data observed information $$\{\boldsymbol{\theta}^1,\ldots,\boldsymbol{\theta}^M\}$$9. The EM-gradient variant uses the complete-data information $$[\boldsymbol{\theta}^*\mid \{\boldsymbol{\theta}\}^M] = \frac{1}{M}\sum_{i=1}^M \mathrm{N}\!\left(\boldsymbol{\theta}^* \mid \boldsymbol{\theta}^i,\mathbf{H}_\theta\right).$$0 and is described as locally equivalent to an EM algorithm. A BHHH-type version replaces the information matrix by the conditional outer product of scores. This makes smoothing a recursive optimization problem driven by score and information rather than by direct conditional-mean propagation.

For general nonlinear or non-Gaussian models, the paper embeds these recursions in a particle approximation. Filtering densities are represented by weighted particles, and conditional expectations over previous states are approximated using weights

$$[\boldsymbol{\theta}^*\mid \{\boldsymbol{\theta}\}^M] = \frac{1}{M}\sum_{i=1}^M \mathrm{N}\!\left(\boldsymbol{\theta}^* \mid \boldsymbol{\theta}^i,\mathbf{H}_\theta\right).$$1

These weights feed directly into particle approximations of the incomplete-data score and observed information, producing the EM-gradient-particle algorithm.

A second major contribution is recursive covariance estimation. The covariance of the smoothing error satisfies a backward recursion in blocks of the expected incomplete-data information matrix. In the linear Gaussian case, the entire framework collapses to the Rauch–Tung–Striebel smoother: $$[\boldsymbol{\theta}^*\mid \{\boldsymbol{\theta}\}^M] = \frac{1}{M}\sum_{i=1}^M \mathrm{N}\!\left(\boldsymbol{\theta}^* \mid \boldsymbol{\theta}^i,\mathbf{H}_\theta\right).$$2 with covariance recursion identical to standard RTS. The paper states that the RTS smoother is fully efficient, its covariance coincides with the Cramér–Rao lower bound, and it has strictly smaller covariance than the Kalman filter.

5. Forward-only sequential Monte Carlo smoothing

A forward-only RPSS is developed in “Forward Smoothing using Sequential Monte Carlo.” The target is the smoothed expectation of an additive functional

$$[\boldsymbol{\theta}^*\mid \{\boldsymbol{\theta}\}^M] = \frac{1}{M}\sum_{i=1}^M \mathrm{N}\!\left(\boldsymbol{\theta}^* \mid \boldsymbol{\theta}^i,\mathbf{H}_\theta\right).$$3

in a general hidden Markov model with transition density $$[\boldsymbol{\theta}^*\mid \{\boldsymbol{\theta}\}^M] = \frac{1}{M}\sum_{i=1}^M \mathrm{N}\!\left(\boldsymbol{\theta}^* \mid \boldsymbol{\theta}^i,\mathbf{H}_\theta\right).$$4 and observation density $$[\boldsymbol{\theta}^*\mid \{\boldsymbol{\theta}\}^M] = \frac{1}{M}\sum_{i=1}^M \mathrm{N}\!\left(\boldsymbol{\theta}^* \mid \boldsymbol{\theta}^i,\mathbf{H}_\theta\right).$$5. Instead of a backward pass over the full trajectory, the paper defines

$$[\boldsymbol{\theta}^*\mid \{\boldsymbol{\theta}\}^M] = \frac{1}{M}\sum_{i=1}^M \mathrm{N}\!\left(\boldsymbol{\theta}^* \mid \boldsymbol{\theta}^i,\mathbf{H}_\theta\right).$$6

so that

$$[\boldsymbol{\theta}^*\mid \{\boldsymbol{\theta}\}^M] = \frac{1}{M}\sum_{i=1}^M \mathrm{N}\!\left(\boldsymbol{\theta}^* \mid \boldsymbol{\theta}^i,\mathbf{H}_\theta\right).$$7

The key identity is the forward recursion

$$[\boldsymbol{\theta}^*\mid \{\boldsymbol{\theta}\}^M] = \frac{1}{M}\sum_{i=1}^M \mathrm{N}\!\left(\boldsymbol{\theta}^* \mid \boldsymbol{\theta}^i,\mathbf{H}_\theta\right).$$8

with $$[\boldsymbol{\theta}^*\mid \{\boldsymbol{\theta}\}^M] = \frac{1}{M}\sum_{i=1}^M \mathrm{N}\!\left(\boldsymbol{\theta}^* \mid \boldsymbol{\theta}^i,\mathbf{H}_\theta\right).$$9 (Moral et al., 2010).

Under particle filtering, the backward kernel is approximated by

$$[\boldsymbol{\theta}^*\mid \{\boldsymbol{\theta}\}^M] = \mathrm{N}\!\left(\boldsymbol{\theta}^* \mid \boldsymbol{\bar{\theta}},\mathbf{S}_\theta\right).$$0

which yields the particle recursion

$$[\boldsymbol{\theta}^*\mid \{\boldsymbol{\theta}\}^M] = \mathrm{N}\!\left(\boldsymbol{\theta}^* \mid \boldsymbol{\bar{\theta}},\mathbf{S}_\theta\right).$$1

The estimator of the smoothed additive functional is then

$$[\boldsymbol{\theta}^*\mid \{\boldsymbol{\theta}\}^M] = \mathrm{N}\!\left(\boldsymbol{\theta}^* \mid \boldsymbol{\bar{\theta}},\mathbf{S}_\theta\right).$$2

The paper’s theoretical comparison with the path-space particle estimator is explicit. Under strong boundedness assumptions on the transition and observation densities, the proposed forward smoother has mean-square error bounded by a constant times $$[\boldsymbol{\theta}^*\mid \{\boldsymbol{\theta}\}^M] = \mathrm{N}\!\left(\boldsymbol{\theta}^* \mid \boldsymbol{\bar{\theta}},\mathbf{S}_\theta\right).$$3, up to a factor $$[\boldsymbol{\theta}^*\mid \{\boldsymbol{\theta}\}^M] = \mathrm{N}\!\left(\boldsymbol{\theta}^* \mid \boldsymbol{\bar{\theta}},\mathbf{S}_\theta\right).$$4. This implies asymptotic variance that grows linearly with time. By contrast, the standard path-space estimator suffers from particle path degeneracy and has asymptotic variance that grows at least quadratically with time under favorable mixing assumptions.

These forward smoothing recursions are then coupled to recursive parameter estimation. For recursive maximum likelihood, the required score can be expressed by Fisher’s identity as an additive functional of the latent path. For online EM, vector-valued sufficient statistics are updated recursively, and the particle implementation becomes

$$[\boldsymbol{\theta}^*\mid \{\boldsymbol{\theta}\}^M] = \mathrm{N}\!\left(\boldsymbol{\theta}^* \mid \boldsymbol{\bar{\theta}},\mathbf{S}_\theta\right).$$5

followed by $$[\boldsymbol{\theta}^*\mid \{\boldsymbol{\theta}\}^M] = \mathrm{N}\!\left(\boldsymbol{\theta}^* \mid \boldsymbol{\bar{\theta}},\mathbf{S}_\theta\right).$$6. In the stochastic volatility example with true parameter $$[\boldsymbol{\theta}^*\mid \{\boldsymbol{\theta}\}^M] = \mathrm{N}\!\left(\boldsymbol{\theta}^* \mid \boldsymbol{\bar{\theta}},\mathbf{S}_\theta\right).$$7, $$[\boldsymbol{\theta}^*\mid \{\boldsymbol{\theta}\}^M] = \mathrm{N}\!\left(\boldsymbol{\theta}^* \mid \boldsymbol{\bar{\theta}},\mathbf{S}_\theta\right).$$8 particles, and initial guess $$[\boldsymbol{\theta}^*\mid \{\boldsymbol{\theta}\}^M] = \mathrm{N}\!\left(\boldsymbol{\theta}^* \mid \boldsymbol{\bar{\theta}},\mathbf{S}_\theta\right).$$9, the reported parameter trajectories converge close to the true values over long data sequences.

6. Unifying interpretation, diagnostics, and limitations

Taken together, these papers suggest a common RPSS architecture. A recursive algorithm maintains a stagewise representation of uncertainty or signal state; that representation is then smoothed into a surrogate with better numerical or statistical behavior; the surrogate is propagated into the next update; and diagnostic or theoretical criteria are used to assess whether the surrogate remains faithful to the underlying target. In SPP‑RB, the surrogate is a Gaussian or Gaussian-mixture approximation of the transient posterior. In recursive IIR design, it is a constrained pole-basis model with coefficients chosen to minimize coloured-noise gain. In incomplete-information ML smoothing, it is a local score-information representation of the latent-state objective. In forward smoothing SMC, it is the auxiliary function $\boldsymbol{y}$00 propagated by a backward kernel [(Scharf, 3 Aug 2025); (Kennedy, 2023); (Surya, 2023); (Moral et al., 2010)].

Several misconceptions are addressed by this comparison. RPSS is not intrinsically Bayesian, because one instance is maximum-likelihood smoothing and another is deterministic filter design. It is not intrinsically nonparametric, because the strongest examples rely on multivariate normal approximations, fixed-pole IIR bases, or information-matrix recursions. Nor does recursive smoothing necessarily imply a backward pass through the full history: the SMC forward smoother is explicitly online and forward-only. A further misconception is that smoothing must sacrifice exactness. In the linear Gaussian state-space setting, the incomplete-information ML smoother reduces exactly to RTS, and in the polynomial-phase setting the imposed dc-derivative constraints guarantee unbiased steady-state reproduction of the specified polynomial class.

The limitations are equally structural. SPP‑RB depends on the quality of the approximation $\boldsymbol{y}$01, and the paper emphasizes sensitivity to $\boldsymbol{y}$02, to stage sample size $\boldsymbol{y}$03, to high-dimensional covariance estimation, and to partition strategy. The digital-filter formulation requires correct choice of $\boldsymbol{y}$04, $\boldsymbol{y}$05, $\boldsymbol{y}$06, bandwidth, and lag; low-SNR performance is ultimately limited by angle unwrapping. The incomplete-information ML smoother requires differentiability, integrability, and particle approximations of conditional expectations, and ML targets may differ materially from MMSE targets in multimodal settings. The forward SMC smoother has $\boldsymbol{y}$07 per-step complexity in its basic form and relies on strong boundedness assumptions for its cleanest variance bounds. The supplied details also indicate a terminological limitation: although (Sun et al., 26 Jul 2025) names RPSS in its abstract, the available source does not define it, so the article’s unified characterization necessarily rests on the explicit constructions in the other papers rather than on a canonical formulation from the diffraction-grating context [(Sun et al., 26 Jul 2025); (Scharf, 3 Aug 2025); (Kennedy, 2023); (Surya, 2023); (Moral et al., 2010)].