WD-AMF: Adaptive Waveform-Domain Filtering

• WD-AMF is a framework that extends classical matched filtering by adaptively regulating local waveform segments based on statistical and physical insights to isolate true signal contributions.
• Adaptive thresholding, gating, and compensation techniques refine the convolution integral, reducing interference while preserving signal energy in diverse applications like radar and geophysical inversion.
• Analytic and neural extensions enable joint estimation of amplitude, delay, and deformation, with simulations demonstrating high jamming suppression and improved performance metrics.

Waveform-Domain Adaptive Matched Filtering (WD-AMF) is a framework for extending classical matched filtering to operate at a fine-grained level within the time-domain convolution (the "waveform domain"). By adaptively censoring or deforming portions of the matched-filter integral based on statistical, structural, or physical knowledge of the data, WD-AMF enables robust signal estimation and interference suppression in a variety of contexts, including radar anti-jamming, experimental waveform analysis, free-space optics, and geophysical inversion. The central innovation lies in introducing adaptivity within the convolution integral (or its discrete analog), typically by gating, weighting, or deforming the local integrand as informed by the instantaneous waveform behavior.

1. Mathematical Foundations and Generalization of Matched Filtering

Standard matched filtering computes the convolution (or cross-correlation) of a received signal with a known template, maximizing signal-to-noise ratio (SNR) under additive noise. In WD-AMF, the matched-filter operation is "opened up" in its running variable (often denoted $\mu$ in continuous time), allowing the filter to act locally and adaptively on different segments of the integrand. For a received signal $x(t)$ and a template $s(t)$, the classical matched filter output is: $$y_o(t) = \int_{-\infty}^{\infty} x(\tau) s^*(\tau - t) d\tau.$$ WD-AMF defines the pointwise integrand as

$v^{(t)}(\mu) = x(t - \mu) h(\mu)$

with $h(\mu) = s^*(-\mu)$. Instead of integrating over the full $\mu$ domain, WD-AMF partitions $\mu$ at every $t$ into subsets reflecting putative signal and interference regions, integrating only over trusted supports and appropriately compensating for censored regions (Su et al., 2024, Su et al., 2023). In the discrete-time setting, similar operations apply: $$z_o[k] = \sum_{n \in U_s[k]} v_k[n] + \sum_{n \in \Psi[k]} \widehat{v}_k[n],$$ where $U_s[k]$ are adaptively selected indices and $\Psi[k]$ are compensation sets.

Beyond gating, WD-AMF can also involve learning residual deformations (filter shape perturbations) from the received waveform, as in neural WD-AMF (Haigh, 6 Feb 2026), or jointly estimating amplitude, delay, and local dilation parameters via analytic extensions (Cappelli et al., 2024).

2. Adaptive Filtering Algorithms and the Waveform Domain

The core mechanism of WD-AMF is adaptive processing within the waveform domain integral. Each value $v^{(t)}(\mu)$ can be seen as the local contribution of the received signal to the matched-filter output at time $t$, mediated by the template. The strategies include the following (Su et al., 2023):

• Adaptive thresholding: Local increments $\left|v^{(t)}(\mu)\right|$ are tested against a dynamically computed threshold $E^{(t)}$; values deemed to reflect interference are censored by setting a binary weight $w_i^{(t)} = 0$.
• Compensation term: Since censoring portions of the sum loses total signal energy, a compensation is introduced by randomly sampling from trusted segments to maintain proper normalization and noise statistics.
• Interpretation as filter bank: The set of local contributions $\{v^{(t)}(\mu)\}$ can be seen as outputs of a bank of narrow bandpass filters, each corresponding to a temporal slice of the received data.

An explicit pseudocode for the algorithm is provided in (Su et al., 2023), involving computation of integrand values, reference slope estimation, thresholding, compensation, and final reconstruction.

3. Waveform-Domain Complementarity and Anti-Jamming in Radar

A major application of WD-AMF is in electronic counter-countermeasures (ECCM), specifically for oppression of interrupted-sampling repeater jamming (ISRJ) in radar. ISRJ is characterized by sliced retransmission of the radar waveform, producing coherent false targets in traditional matched filtering. WD-AMF counters this via two coupled strategies (Su et al., 2024):

• Partitioned matched filter integral: By adaptively censoring integrand regions likely associated with jamming slices, WD-AMF robustly isolates the true echo.
• Transmit-side waveform-domain complementarity: Specially designed sets of phase-coded waveforms (typically derived via column-orthogonal submatrices of the Walsh–Hadamard matrix) ensure that, when summed over $D$ waveforms, the combined response $w_s^{(t)}(\mu)$ is strictly supported on the true echo’s region with perfect nulls elsewhere:

$$w_s^{(t)}(\mu) = \sum_{i=0}^{D-1} s_i(t + \mu) s_i^*(\mu)$$

with $$b^{(m)}(n) = \sum_{i=0}^{D-1} a_i(n+m) a_i^*(n) = D$$ for $m=0$, $0$ otherwise. This "waveform-domain complementarity" completely decouples the true echo from all jamming-induced slices. When $D \gg N$ (number of waveforms $\gg$ code length), such sets can be constructed for arbitrary $N$ by column selection from a Walsh–Hadamard matrix.

In simulation, this combination eliminates all jamming false peaks and achieves full matched-filter gain on true targets, independent of the jammer’s parameters (Su et al., 2024). Performance metrics such as main-lobe loss (MLL), side-lobe level (SLL), and peak side-lobe ratio (PSLR) demonstrate >50 dB jamming suppression with negligible loss of target gain for complementary waveform sets, and resilience across SNR/JNR variations.

4. Neural and Analytic Extensions: Deformable and Feature-Driven WD-AMF

Extensions of WD-AMF have been proposed to generalize the adaptive kernel from hard gating to continuous deformation, leveraging analytic and data-driven techniques.

• Analytic deformation estimation (Cappelli et al., 2024): In pulse detection for experimental physics, WD-AMF jointly estimates amplitude, time delay, and a small global time-scale deformation parameter. The pulse shape is generalized to

$$w(t) = A\, s\bigl((1-\varepsilon)(t - t_0)\bigr) + n(t)$$

and the optimal filter is derived via first-order Taylor expansion and $\chi^2$ minimization. The result is a closed-form estimator for amplitude and deformation, with improved discrimination of pile-up and non-standard backgrounds in low and high SNR regimes.

• Neural residual filtering (Deformable WD-AMF) (Haigh, 6 Feb 2026): In optical communications, the matched filter is augmented with a learned complex residual $\delta h[n]$, mapped from a 16-dimensional feature vector extracted from the received waveform. The mapping is a compact two-layer neural network, trained end-to-end to minimize error vector magnitude (EVM), with regularization to enforce smoothness. This approach achieves substantial EVM and bit-error rate improvements under bandwidth-limited channels, with the network naturally reverting to the classical matched filter when conditions are favorable.

5. Geophysical Model Inversion: Adaptive Waveform Inversion

In the context of full-waveform inversion (FWI) for subsurface model estimation, WD-AMF manifests as adaptive waveform inversion (AWI) (Symes, 2024). For each source–receiver pair, a "matching filter" $g_{sr}(t)$ is computed to minimize the $L^2$ misfit (with Tikhonov regularization),

$$g_{sr} = \arg\min_g \; \|g * d^{sr}_{\text{pred}} - d^{sr}_{\text{obs}}\|_2^2 + \varepsilon(\lambda)\, \|g\|_2^2,$$

normalized to unit energy, and the inversion objective is to minimize the mean-squared time dispersion of the matching filters: $$J_{\text{AWI}}(m) = \sum_{s,r} \int t^2\, |\bar{g}_{sr}(t)|^2\,dt.$$ In the high-frequency limit with single arrivals, this reduces to mean squared traveltime error, linking AWI to classical traveltime tomography. This approach suppresses amplitude and phase-cycle misfits, increasing robustness against local minima. The lack of unit-norm scaling (MSWI) leads to loss of this direct correspondence.

In multiple-arrival scenarios, AWI loses its robustness as matching filters are no longer localized in time, potentially reintroducing local minima and cycle-skipping.

6. Implementation, Algorithmic Structure, and Performance Benchmarks

Implementation details for WD-AMF depend on the domain and application. The central loop involves, for each evaluation time:

• Computing the waveform-domain integrand (or residual),
• Adaptive gating or deformation (via thresholding rules, learned feature maps, or filter estimation),
• Summation/integration of trusted supports and compensation,
• (If necessary) joint estimation of amplitude, time, and deformation parameters or filter coefficients,
• Performance evaluation via SNR/SINR, EVM, BER, PSLR, and discrimination metrics.

Simulation case studies include:

• Radar: L-band radar with 256×160 Walsh–Hadamard complementary waveforms under ISRJ achieved PSLR ≈52 dB and target main-lobe loss of 0 dB (full energy recovery), independent of SNR/JNR (Su et al., 2024).
• Particle physics: WD-AMF yielded at least a factor 2 improvement in discrimination over standard decay/rise time or $\chi^2$ metrics, for both low and high SNR (Cappelli et al., 2024).
• Optical communications: WD-AMF reduced EVM from >50% (fixed filter) to <30% under severe bandlimitation (relative gain >40%) (Haigh, 6 Feb 2026).
• Geophysics: AWI linked mean-square filter spread to traveltime error, robustly connecting FWI and tomography in single-arrival regimes (Symes, 2024).

7. Assumptions, Applicability, and Limitations

WD-AMF provides a flexible and generalizable extension to matched filtering for digitized waveforms with additive, stationary noise and a-priori known or estimated template shapes. Key assumptions, caveats, and deployment considerations include (Cappelli et al., 2024, Su et al., 2024, Symes, 2024):

• Stationarity and calibration accuracy of the noise power spectral density and template waveform are critical.
• In analytic deformation estimation, the Taylor-expansion based approach is valid only for small ($\lesssim10\%$) global deformations.
• In waveform-domain complementarity, code construction (e.g., Walsh–Hadamard, Golay, Tseng–Liu) must strictly satisfy the orthogonality/zero-correlation constraints.
• In multichannel settings or with multiple arrivals (as in AWI/MSWI), the correspondence to desired physical quantities can degrade.
• For neural approaches, care must be taken with feature extraction and regularization to avoid unintended distortions or instabilities.

Applicability spans radar ECCM, experimental waveform discrimination, advanced receivers in impaired channels, and geophysical inverse problems, with the common thread of exploiting adaptive local structure within the convolution integral for robust signal recovery and interference suppression.

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Key sources: (Su et al., 2024, Su et al., 2023, Cappelli et al., 2024, Haigh, 6 Feb 2026, Symes, 2024).