Time Domain Sampling Methods

• Time domain sampling methods are techniques that operate directly on time-resolved signals to achieve high temporal resolution and robust recovery even under nonuniform conditions.
• They incorporate strategies like cubic spline re-gridding, modified Shannon interpolation, and Fisher information-based design to mitigate noise and handle sparse events.
• Applications span ultrafast spectroscopy, inverse scattering, graph signal processing, and model reduction, enabling efficient, accurate signal analysis across diverse domains.

Time domain sampling methods comprise a broad arsenal of theoretical frameworks and computational strategies for acquiring, processing, and reconstructing signals from time-resolved measurements, operating directly in the time domain rather than relying on spectral or frequency-domain transformations. These methods are critical across experimental physics, engineering, statistical signal processing, and inverse problems, especially where high temporal resolution, sparse event recovery, or minimal data acquisition are paramount. Time domain sampling approaches encompass techniques for nonuniform sampling correction, information-optimal experiment design, analysis under noise or sampling jitter, problem-specific direct/inverse imaging methods, graph signal processing, and extensions to generalized transform domains.

1. Theoretical Foundations and Motivation

Time domain sampling exploits the structure of signals measured as functions of time, with the principal objective of perfect (or stable) reconstruction under specified signal constraints. Classical results include the Nyquist–Shannon sampling theorem—dictating that a bandlimited signal can be uniquely reconstructed from uniform samples at twice its maximal frequency. However, practical and generalized scenarios often preclude ideal sampling. Time-domain methodologies directly address challenges such as sample nonuniformity (Potts et al., 2017), sparsity (Pavlíček et al., 29 Apr 2024), structured signals on graphs (Ji et al., 2020, Sheng et al., 29 Aug 2025), and robustness to noise or system imperfections (Takeuchi et al., 2023, Kircheis et al., 2023).

A major theme throughout is the tension between theoretical critical sampling density (the minimum possible for stable recovery) and practical limits imposed by measurement noise, nonuniformity, and computational constraints. Modern developments integrate signal processing, operator theory, and statistical inference to formulate both necessary and sufficient sampling criteria—driven either by intrinsic signal degrees of freedom (e.g., spectral support) or by specific recovery objectives such as parameter estimation or imaging contrast.

2. Correction and Reconstruction from Nonuniform Sampling

In systems where ideal uniform sampling cannot be achieved, corrective re-gridding techniques are fundamental. In time-domain terahertz spectroscopy (TDTS), mechanical delay stages cause inherent nonuniformity in sample positions, which directly translates into errors and degraded signal-to-noise ratio (SNR) when performing FFT-based spectral analysis. Two methodologies have been established and experimentally validated (Potts et al., 2017):

• Cubic Spline Re-gridding: Piecewise cubic polynomial interpolants are constructed between adjacent (irregular) time samples. The coefficients are imposed by continuity in value and (typically) first and second derivatives, with boundary conditions enforcing zero curvature (natural splines). The smoothness of the spline ensures robust recovery of the underlying temporal waveform, which is crucial for ultrafast optics applications.
• Modified Shannon Interpolation: Recognizing that the THz pulses are effectively bandlimited, the uniform-sample reconstruction can be rephrased via a matrix equation: $y = R Y$, where $R_{ij} = \mathrm{sinc}((x_i-jX)/X)$, $y(x_i)$ are measured samples, and $Y(jX)$ the values on the ideal grid. Provided that $R$ is invertible, the uniform samples are determined as $Y = R^{-1} y$. In this approach, each reconstructed point is a nonlocal function of all measured data due to the infinite support of the sinc kernel.

These methods improve SNR and reduce processing error, particularly as SNR and operational frequencies increase; imaging modalities benefit from the correction by enabling faster acquisition at relaxed precision (Potts et al., 2017).

3. Optimal Sampling Design via Statistical Information

When experimental constraints, measurement costs, or desired precision motivate non-uniform or reduced sampling, information-theoretic sampling design is employed. The Fisher information matrix (FIM) quantifies the information content each sample contributes toward parameter estimation (Bolzonello et al., 2023). Given a model $f(t, θ)$ (e.g., exponential decay, oscillatory signal), and measurement noise model (usually Gaussian), the FIM is calculated as: $$I = \sum_n \frac{1}{σ_o^2(t_n)} \begin{bmatrix} e^{-2γ t_n} & a t_n e^{-2γ t_n}\ a t_n e^{-2γ t_n} & a^2 t_n^2 e^{-2γ t_n} \end{bmatrix}$$ for parameters $(a, γ)$ of a decaying exponential. The Cramér–Rao lower bound provides a theoretical minimum for the variance of unbiased estimators: $\mathrm{Cov}(\hat{θ}) \geq I^{-1}$ Sampling schemes are optimized by selecting a subset of time-points that minimizes the expected uncertainty (e.g., as a weighted trace of $I^{-1}$). This approach can dramatically reduce the number of measurements—by orders of magnitude—without substantial loss of statistical power in parameter estimation, classification, or multidimensional spectroscopy (Bolzonello et al., 2023).

4. Direct Sampling for Inverse Problems

A prominent development in time-domain inverse scattering and source reconstruction is the turn to direct sampling methods, particularly for imaging the support of scatterers or sources without iterative or optimization-heavy inversion. Central methods include:

• Indicator Functionals Based on Green's Function and Convolution: For wave propagation in acoustics or electromagnetics, the indicator at a sampling point $z$ is constructed by convolving the Green's function (or its time-delayed version) with observed data, e.g.,

$$I(z) = \int_{-\infty}^\infty \left| \int_\Gamma u(x, t + c_0^{-1}|x - z|) \varphi_\sigma(x, t, z) ds(x) \right|^2 dt$$

where $\varphi_\sigma$ is a test function (often an exponentially damped fundamental solution), $u$ is the observed field, and $\Gamma$ the measurement surface (Yu et al., 2023, Guo et al., 2023, Geng et al., 10 Oct 2024).

• Time Domain Linear Sampling Method (TD-LSM): An ill-posed operator equation is set for each sampling point; the solution norm forms the indicator, large for points outside the scatterer and smaller for internal points. The approach is justified via Laplace transform tools, connecting to frequency-domain methods and guaranteeing stability and rigorous support recovery under mild assumptions (Lähivaara et al., 2021, Song et al., 8 Dec 2024, Liu et al., 30 Dec 2024).
• Sparse Source Recovery: For signals modeled as sparse impulse trains, time-domain recovery via annihilating filters or difference operators provides exact spike localization and amplitude estimation, with explicit sampling theorems dictating the minimal number of samples required (dependent on kernel support and sparsity level) (Pavlíček et al., 29 Apr 2024).

These indicator-based methods exhibit robustness to noise, do not require detailed knowledge of scatterer geometry or material parameters, and provide computationally efficient alternatives to optimization-based reconstruction.

5. Time-Domain Sampling on Graph and Manifold Structures

Generalization to structured data—where measurements are functions over irregular domains (e.g., graphs, networks) varying in time—necessitates joint time-domain and graph-domain sampling theory. The principal notions are:

• Continuous-Time Graph Signals: For signals $f: V \times \mathbb{R} \to \mathbb{R}$ (with $V$ the vertex set), a joint bandlimit is imposed in both time and graph frequency (via the GFT). The minimal sampling set is determined by both the time-bandwidths per vertex and the graph’s spectral constraints, leading to a formula for minimal rate: $r^* = \min_{V_0} 2 \sum_{v \in V_0} B[v]$ where $B[v]$ is the time-bandwidth at $v$ and $V_0$ a uniqueness set of vertices (Ji et al., 2020).
• Joint Time-Vertex Graph Signal (TVGS) Sampling: The joint time-vertex Fourier transform enables characterization of signals with support on subsets of the joint spectral plane $(i, f)$, where $i$ is the graph spectral index. For a jointly bandlimited signal with spectrum restricted to sets $\mathcal{F}_i$ in frequency for each spectral index, the critical density is

$D(\mathcal{S}) \geq \frac{B}{N}$

where $B = \sum_{i \in \mathcal{I}} \mu(\mathcal{F}_i)$ (Sheng et al., 29 Aug 2025). Multi-band sampling strategies adapt sampling rates per subband and select vertex subsets, achieving minimum redundancy and stable recovery in both synthetic and real data (e.g., EEG, traffic networks).

6. Numerical and Regularization Strategies for Practical Sampling

Operationalizing time-domain sampling theorems requires robust numerical schemes in the presence of finite data, noise, and computational constraints.

• Oversampling and Window Regularization: In practical implementations of Shannon’s theorem, poor decay of the sinc kernel leads to slow convergence and noise amplification. Oversampling—sampling above the critical rate—together with compactly supported regularization windows (e.g., sinh-type or Kaiser–Bessel), localizes computation and achieves exponential error decay in the truncation parameter. This controlled regularization regime outperforms frequency-domain windows, which provide only algebraic error decay (Kircheis et al., 2023).
• Jitter and Uncertainty Analysis: High-precision applications (e.g., audio, clocks) require direct characterization and separation of sampling jitter from amplitude or phase-independent (PI) noise. Time-domain analysis of zero-crossings (using interpolated continuous signals from windowed FFT) quantifies jitter with picosecond sensitivity, separating player and recorder contributions through multi-channel and cross-correlation approaches (Takeuchi et al., 2023).
• Data Informativity and Conditioning: In model reduction and system identification, time-domain data-informativity frameworks allow estimation of transfer function values and derivatives from a single time-resolved experiment, provided that the input trajectories are persistently exciting. Analysis of conditioning (e.g., via eigenvalue characterization of rank-1 projector perturbations) enables optimal scaling and robust numerical solution of the associated linear systems (Ackermann et al., 17 Jul 2024).

7. Applications and Impact

Time-domain sampling methods enable high-fidelity reconstruction, imaging, and parameter estimation in domains including:

• Ultrafast and Multidimensional Spectroscopy: Correction and optimal design of sampling optimize SNR, classification, and acquisition speed in THz, Raman, and 2D electronic spectra (Potts et al., 2017, Bolzonello et al., 2023).
• Inverse Scattering and Source Localization: Direct time-domain indicator methods and linear sampling extend efficiently to acoustic/electromagnetic imaging, laser ultrasonics, and passive source reconstruction, showing robustness even under limited aperture or high noise (Lähivaara et al., 2021, Guo et al., 2023, Geng et al., 10 Oct 2024, Song et al., 8 Dec 2024, Liu et al., 30 Dec 2024).
• Graph-Based Sensor Networks: Joint time-vertex sampling strategies inform optimal sensor placement, dynamic monitoring, and reduced-data acquisition in spatial networks, with demonstrated efficacy in EEG and traffic sensor analysis (Ji et al., 2020, Sheng et al., 29 Aug 2025).
• Reduced Order Modeling: Time-domain Krylov/IRKA methods enable H₂-optimal, data-driven model reduction from single time-series datasets, circumventing repeated frequency-domain system solves—critical in large-scale or black-box systems (Ackermann et al., 17 Jul 2024).
• Sparse Signal and Super-Resolution Recovery: Time-domain sparse sampling in fractional Fourier domains extends recovery guarantees, provides explicit CRBs for spike localization, and avoids transform-domain artifacts (Pavlíček et al., 29 Apr 2024).

In aggregate, these frameworks constitute a mathematically rigorous, computationally efficient, and empirically validated foundation for time-domain signal acquisition and recovery in contemporary scientific and engineering practice.