Temporal Resampling in Signal Processing

• Temporal resampling is the process of mapping data from one temporal grid to another while preserving or altering spectral and statistical properties.
• Key methods include classical sinc interpolation, periodic nonuniform sampling, and FFT-based schemes, ensuring accurate reconstruction for both uniform and irregular grids.
• This technique underpins advanced applications like video frame interpolation, time super-resolution, and adaptive model synchronization in diverse fields such as remote sensing and physics instrumentation.

Temporal resampling encompasses the transformation of discrete sequences or continuous processes, from one temporal grid or sampling structure to another, with the intent of preserving — or purposefully altering — underlying information, spectral content, or statistical structure. This concept appears across digital signal processing, time-series analysis, machine learning, remote sensing, video processing, event-based vision, and physics instrumentation. Techniques span classical uniform resampling, nonuniform density compensation, self-supervised learning-based upsampling, and task-adaptive model synchronization. Temporal resampling is central to aliasing mitigation, frame-rate transformation, time super-resolution, and data augmentation.

1. Mathematical Foundations and Resampling Schemes

Resampling is fundamentally a mapping between two time grids, typically from a source set of samples $\{x_n\}$ at times $\{t_n\}$ to a target grid $\{\tau_m\}$ with desired properties (rate, regularity, information density). Methods range from Shannon-Nyquist interpolation (sinc-based), to more general frameworks handling irregular grids and multiband spectra.

Classical uniform resampling uses the sinc kernel, but the periodic nonuniform sampling (PNS) formalism generalizes reconstruction by treating irregular sequences as local concatenations of periodic subgrids, and solves for $g(t)$ via matrix inversion, under the Landau density criterion. For irregular grids, the sampling density must meet or exceed the spectral support's Lebesgue measure. Truncation and asymptotic analysis show error decaying polynomially with block size $N$ (Lacaze, 2019).

Table: Resampling Modes and Requirements

Method	Sample Regularity	Spectral Support Requirement
Shannon/Nyquist	Uniform	Baseband, width $1/T$
PNS (order $N$)	Locally periodic	$N$ disjoint bands, width $N/T$
Irregular (PNS)	Arbitrary	Landau density: $|A|$

Signal processing frameworks also exploit optimized frequency-domain techniques, such as FFT-based LMN resampling, which achieves arbitrary rational rate conversion directly in the spectral domain, with strict amplitude and energy invariants (Gerlach et al., 2024).

2. Temporal Upsampling, Downsampling, and Super-Resolution

Temporal upsampling seeks to infer missing samples between observed data, increasing resolution beyond the native acquisition rate. High-fidelity methods exploit regularization, physical models, and learned dynamics:

• Video temporal super-resolution uses deep neural networks for frame interpolation based on optical flow, visibility masks, and multi-scale convolution blocks, outperforming naive linear interpolation in both RMSE and SSIM for severe weather satellite imagery (Vandal et al., 2019).
• Dynamic sensing and 3D frequency-selective reconstruction (3D-FSR): CMOS sensors acquire subsets of pixels in temporally-complementary patterns, with reconstruction across spatio-temporal blocks using adaptive Fourier methods. Effective frame rate increases by up to $N$ (Jonscher et al., 2022).
• Active super-resolution via multi-channel illumination: Light sources are modulated in $M$ channels with binary flicker codes. The detected channel vector in each exposure enables recovery of $N > M$ subframe samples via regularized linear inversion, extending spectral bandwidth up to $N f_s/2$, significantly beyond standard Nyquist (Cohen et al., 2022).
• Energy data and time-series: Self-supervised Generative Adversarial Transformers (GATs) bridge granularity gaps without ground-truth HR data, outperforming interpolation and GP-based upsampling for load, PV, and wind time series (Mu et al., 14 Aug 2025).
• Model-based turbulent flow super-sampling: Proper Orthogonal Decomposition (POD), followed by Galerkin projection, yields an empirical ODE system whose integration reconstructs time-continuous velocity fields from sparsely sampled snapshots, with continuity preserved via forward–backward blending (Li-Hu et al., 20 Feb 2025).

3. Task-Adaptive and Data-Adaptive Temporal Resampling

Temporal resampling is increasingly addressed via models that synchronize their internal update logic to the data's observed temporal structure:

• Task-Synchronized Recurrent Neural Networks (TSRNNs): Instead of interpolating or imputing missing timestamps, model updates are scaled by observed time intervals. This is implemented in Echo State Networks (TSESN) and GRUs (TSGRU) via Euler discretization with variable $\Delta t_n$, yielding robust handling of irregularly sampled data and competitive accuracy versus resampled or input-augmented baselines (Lukoševičius et al., 2022).
• Efficient spatio-temporal scan-resampling encoders: Alternating cross-attention with discounted cumulative sum enables efficient accumulation of historical information under variable observation set sizes and nonuniform temporal sampling, supporting O(1) inference and competitive metrics for multi-agent event streams (Ferenczi et al., 2024).
• Reservoir computing emulators: In geophysical turbulence, temporal subsampling diminishes small-scale spectral fidelity by biasing models towards averaged dynamics. Systematic and SSP resampling kernels minimize Monte Carlo variance in weak-potential particle filters, as shown via Markov generator analysis (Smith et al., 2023, Chopin et al., 2022).

4. Temporal Resampling in Event-Based Vision and Data Augmentation

In asynchronous event cameras and sequences, temporal resampling not only fills information gaps but augments complex downstream tasks:

• Event upsampling via trajectory-conditional point processes: Contrast maximization estimates motion trajectories, followed by Hawkes or self-correcting processes generating new event timestamps. This yields higher reconstructed image quality and detection accuracy, validated in diverse scenes (Xiang et al., 2022).
• SeqAug for modality-agnostic augmentation: Feature resampling by intra-sequence dimension-wise temporal permutation breaks spurious correlations and promotes robustness in RNNs and Transformers, with empirically verified gains in sentiment analysis benchmarks (Georgiou et al., 2023).

5. Learned Spatio-Temporal Resampling for Video and Multimedia

End-to-end frameworks jointly learn the downsampling kernel and restoration/inversion logic to explicitly mitigate aliasing and information loss:

• Joint spatio-temporal downsampling and upsampling: 3D convolutional low-pass kernels with enforced softmax constraints, differentiable quantization, and upsampler modules (ConvLSTM, deformable convolution, space-time pixel shuffle) enable alias-free reconstruction and arbitrary temporal/spatial resampling ratios, robust across storage formats (Xiang et al., 2022).
• Optimization for video re-timing with speediness prediction: Neural networks localize temporally-varying speed, scoring slowness, and optimizing non-uniform frame skips under smoothness and plausibility constraints, yielding controllable re-timed videos with high naturalness and accuracy (Jenni et al., 2022).

6. Applications, Empirical Performance, and Limitations

Temporal resampling is deployed in:

• Satellite meteorology (super-resolution, event interpolation).
• High-throughput particle physics signal chains.
• Event-based vision and autonomous robotics.
• Energy data analytics and smart grid predictive control.
• Reinforcement learning, agent-based modeling, video enhancement, and industrial sensing.

Reported gains include up to $8.55$ dB PSNR in dynamic sensing (Jonscher et al., 2022), $9\%$ RMSE reduction in self-supervised energy upsampling (Mu et al., 14 Aug 2025), and 87.4% speediness prediction accuracy for time-remapping (Jenni et al., 2022). Limitations arise from underdetermined inverse problems at high upsampling ratios, spectral density requirements not met in sparse or highly bandlimited domains, and nontrivial stability concerns for aggressive resampling in nonlinear dynamical emulation (Smith et al., 2023).

Temporal resampling continues to evolve, with research trending toward self-supervised mechanisms, more physically-grounded interpolation, adaptive or learned density compensation, and multi-modal generalization. Challenges remain around handling arbitrary nonuniform sampling, provable spectral recovery in nonstationary fields, and integrating resampling as native logic within time-dependent neural models and encoders.