Spectral Pre-Filtering for Context-Adaptive Sensor Fusion: A Four-Role FFT-GDCB Integration for High-Stakes Decision Systems

Oleg Miroshnichenko

cs.LGAug 22, 2026

Spectral Pre-Filtering for Context-Adaptive Sensor Fusion: A Four-Role FFT-GDCB Integration for High-Stakes Decision Systems

Authors: Oleg Miroshnichenko

Abstract

Context-adaptive Kalman filters calibrate their noise covariance matrices Q and R from innovation residuals via online regression. When the underlying sensor or signal carries periodic structure -- mechanical LiDAR rotation harmonics, engine vibration, ground multipath, weekly and annual demand cycles, dosing-interval rhythms, weekly media-buying cadence -- the regression input is contaminated and the fitted covariance models structural modes rather than genuine state uncertainty. We introduce a four-role FFT pre-filter that solves this problem at $O(N\log N)$ cost and serves three additional roles "for free": (i) it whitens coloured noise before the Kalman update, restoring the optimality assumption; (ii) it cleans innovations before covariance regression, preventing periodic contamination of $\hat{R}$ and $\hat{Q}$ ; (iii) it generates spectral context features that enrich the downstream bandit's regime-selection state; (iv) it deseasonalises the input feature vector before any supervised regression that produces a sensitivity coefficient (beta, dose offset, bid modifier). We position the algorithm inside the Gated Decoupled Compositional Bandits (GDCB) family, where it acts as a preprocessing layer for the supervised scaler. The single $O(N\log N)$ FFT call thereby serves four downstream consumers, fits in <0.1% of the sensor-fusion or pricing-pipeline compute budget, and is a drop-in addition with no changes to the Kalman filter, bandit, or runtime composition operator. We summarise empirical validation across six independent domains (rocket descent, autonomous-vehicle tracking, short-term rental pricing, clinical drug dosing, airline fare distribution, and ad-operations bid calibration), all returning a PROVES verdict under a pre-registered evaluation protocol.

Figures & tables

Figure 1: The four-role FFT pre-filter. A single

O(N\log N)

FFT call on a calibration window of

N

samples produces four products that feed four distinct downstream consumers: measurement whitening (Role 1, blue), covariance regression cleaning (Role 2, orange), spectral bandit context features (Role 3, purple), and feature deseasonalisation for the supervised scaler (Role 4, green). The downstream consumers are off-the-shelf: the Kalman filter, the GDCB scaler

\delta_{\theta}(\mathbf{x})

( Miroshnichenko 2026a ) , the bandit posterior, and the revenue

\beta

-vector. No modifications to any of these components are required.

Figure 2: The notch algorithm on a synthetic contaminated innovation. (a) Time domain of the 100 Hz innovation

\nu_{t}

. (b) PSD with two clear peaks at 15 and 20 Hz, well above the

\mu+3\sigma

detection threshold (orange dotted). (c) PSD after notching: floor restored to white-noise level. The empirical variance of

\nu^{\mathrm{aper}}

matches the true measurement noise variance to within

11\%

Signal class	Symptom	Recommended role(s)
Narrow-band periodic harmonics in $z_{t}$	spike at known $f_{*}$ in spectrogram	Role 1 (whitening) or Role 2 (notch)
Broadband coloured (1/f, pink) noise in $z_{t}$	sloped PSD over full band	Role 1 (Wiener-filter whitening)
Periodic structure in innovation $\nu_{t}$ after KF	NIS distribution biased upward	Role 2 (notch on $\nu_{t}$ before regression)
Multiple operating regimes (rain, MaxQ, propellant low)	context $\mathbf{x}$ does not predict noise level	Role 3 (spectral features into $\mathbf{x}$ )
Daily revenue regressors with calendar cycles	$\hat{\beta}$ shrinks to $\sim 0$	Role 4 (per-feature notch before regression)
Forward-looking demand prior	no signal of upcoming peak	Role 4 by-product (Fourier forecast)

Table 1: Which role to use, by signal class.

Application	$W$	$\Delta t$	Update rate	Cost relative to downstream pipeline
Rocket GNC (IMU)	100	10 ms	100 Hz	$<\!0.01\%$ of KF compute
AV sensor fusion (LiDAR)	500	10 ms	100 Hz	$<\!0.10\%$ of fusion budget
Clinical PK/PD	144	1 h	hourly	negligible (microseconds)
STR / Airline / Ad Ops daily	365–730	1 d	monthly	negligible (microseconds)

Table 2: Computational cost of the four-role FFT call across deployment regimes.

Domain	Role(s)	Headline metric	Achieved	Verdict
Rocket descent (50 MC runs)	2	$\widehat{R}$ improvement factor (raw $\to$ FFT)	$5.16\times$ (117% of theoretical ceiling)	PROVES
AV pedestrian tracking (KITTI-like)	2 + 3	$\widehat{R}$ improvement factor	Dry: $4.16\times$ ; 53% of 10 Hz-harmonic power. Rain: $3.10\times$ ; spectral entropy 4.04 (anomaly)	PROVES
Clinical PK/PD (20 patients $\times$ 60 d)	2 + 3	$\widehat{R}$ factor; false-alarm reduction	$13.5\times$ ; $13.8\%$ false alarms removed	PROVES
STR pricing (730 d, 30 properties)	4	mean $\beta$ recovery (4 params, raw $\to$ FFT)	$+111$ – $134\%$ ; $4/4$ closer to truth	PROVES
Airline fare (365 d, 20 routes)	4	mean $\beta$ recovery (4 params)	$+60$ – $115\%$ ; $4/4$ closer to truth	PROVES
Ad operations (365 d, 20 campaigns)	4	mean $\beta$ recovery (4 params)	$+57$ – $112\%$ ; $4/4$ closer to truth	PROVES

Table 3: Six-domain validation summary (April 2026 evaluation, all PROVES).

Figure 3: Six-domain validation summary, seven evaluation conditions. Expected (theory, grey) vs. achieved (empirical, coloured) for each condition; the AV domain is evaluated under two weather regimes (dry, rain), so six domains yield seven bars. Role 2 domains (rocket / AV / clinical, orange) report the

\widehat{R}

improvement factor; Role 4 domains (STR / airline / ad ops, green) report the mean

\beta

recovery improvement (proportion). All seven conditions return a PROVES verdict under the pre-registered evaluation protocol.

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FW-NKF: Frequency-Weighted Neural Kalman Filters

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Department of Computer Science, ETH Zürich, Switzerland

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Robust state estimation is central to robotic autonomy, yet classical Kalman filters struggle with frequency-dependent disturbances and model mismatch such as sensor vibrations, electromagnetic interference, and periodic noise. Although Deep Kalman Filter (DKF) variants extend the Extended Kalman Filtering (EKF) framework by learning latent transitions, they lack explicit mechanisms to suppress band-limited noise components that typically corrupt sensor measurements in real-world scenarios. We introduce the Frequency-Weighted Neural Kalman Filter (FW-NKF), a unified hybrid approach that embeds a causal spectral-shaping operator into the Kalman measurement residual and jointly learns observation, and transition networks. By adapting both the filter spectrum and the latent state representation, FW-NKF attenuates the noise-dominated frequency bands while capturing complex residual structures. We conduct extensive experiments on four heterogeneous benchmarks, including chaotic systems such as multi-dimensional Lorenz systems and full-body inertial pose estimation, and find a reduction in localization error of up to 10% as well as marked improvements in orientation accuracy. Our ablation studies confirm that frequency weighting and deep latent-state modeling contribute to overall performance.

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