VBT-IF-TEM: Adaptive Time Encoding Machine

• VBT-IF-TEM is an adaptive time encoding architecture that leverages local signal and derivative energies to dynamically adjust bias and threshold for precise, energy-aware sampling.
• It integrates encoder parameter design with iterative reconstruction algorithms to ensure convergence even under non-uniform, sub-Nyquist sampling conditions.
• Parameter tuning in VBT-IF-TEM allows a trade-off between sampling density and convergence speed, demonstrating efficiency in simulation and real-data experimental scenarios.

A Variable-Bias, Variable-Threshold Integrate-and-Fire Time Encoding Machine (VBT-IF-TEM) is an adaptive time encoding architecture explicitly designed for non-uniform sampling of real bandlimited signals. It leverages local signal and derivative energies to set both the bias and threshold dynamically, enabling energy-aware firing density and guaranteeing iterative reconstruction even under sub-Nyquist sampling regimes. The methodology is characterized by a co-design approach between encoder parameters and reconstruction algorithms, systematically enforcing a local convergence condition that ensures accurate analog signal recovery while minimizing sampling density (Yashaswini et al., 22 Jan 2026).

1. Local Energy-Based Sufficient Condition for Perfect Reconstruction

VBT-IF-TEM is fundamentally grounded in a local, energy-based sufficient condition for the convergence of iterative reconstruction algorithms. Consider a real bandlimited signal $f(t) \in B_{\Omega_0, c}$ ($|F(\omega)|=0$ for $|\omega|>\Omega_0$, $|f(t)|\leq c$), and denote firing times by $t_n$ and inter-firing intervals by $T_n = t_{n+1} - t_n$. Over each interval, the signal energy and derivative energy are given by:

• $E_n = \int_{t_n}^{t_{n+1}} |f(t)|^2 dt$
• $D_n = \int_{t_n}^{t_{n+1}} |f'(t)|^2 dt$

A sufficient condition for guaranteed convergence is:

$T_n < \pi\sqrt{\frac{E_n}{D_n}}$

This condition dynamically permits longer sampling gaps when the signal is nearly flat ($D_n$ is small), while mandating shorter gaps for regions of rapid signal variation. This local control contrasts with global, Nyquist-type bounds, directly shaping both the rate and density of time encodings.

2. VBT-IF-TEM Encoder Architecture

The VBT-IF-TEM employs a reset-on-fire integrator that accumulates a biased version of the analog input signal. Specifically, beginning at $t_n$ (post-firing reset), the integrator computes:

• Integrated signal: $x_i(t) = f(t) + b_n(t)$
• Next firing time $t_{n+1}$ is the smallest $t > t_n$ such that:

$\int_{t_n}^t [f(u) + b_n(u)] du = \Delta_n(t)$

Both bias $b_n(t)$ and threshold $\Delta_n(t)$ are functions specified over $[t_n, t_{n+1}]$ and adaptively depend on running local energies. In the unshifted baseline design:

• $e_n(t) = \int_{t_n}^t |f(u)|^2 du$
• $d_n(t) = \int_{t_n}^t |f'(u)|^2 du$
• Bias: $b_n(t) = c + \frac{1}{\pi\sqrt{\alpha e_n(t)}}$, for $\alpha \in (0,1)$
• Threshold: $\Delta_n(t) = 1/\sqrt{d_n(t) + \beta e_n(t)}$, for $\beta > 0$

The algorithm ensures that the minimum bias $b_{n,\text{min}}$ and the computed $\Delta_n(t_{n+1})$ together enforce $T_n < \pi\sqrt{E_n/D_n}$ by construction.

3. Robust Shifted-Signal Formulation

To ensure robustness in regions where $f(t)$ approaches zero (and prevent unbounded bias/threshold), a shifted-signal methodology is introduced. With a constant $s > c$:

• $\tilde{f}(t) = f(t) + s > 0$
• Running energies: $\tilde{e}_n(t) = \int_{t_n}^t |\tilde{f}(u)|^2 du$, $\tilde{d}_n(t) = \int_{t_n}^t |\tilde{f}'(u)|^2 du$
• Shifted bias: $$b_n(t) = c-s + \frac{1}{\pi \sqrt{\alpha \tilde{e}_n(t)}}$$
• Shifted threshold: $$\Delta_n(t) = 1/\sqrt{\tilde{d}_n(t) + \beta \tilde{e}_n(t)}$$

Replacing $E_n$, $D_n$ by their shifted variants maintains bounded $b_n,\Delta_n$ and strictly enforces the energy-based local condition necessary for algorithmic convergence.

4. Algorithmic Workflow and Iterative Reconstruction

The VBT-IF-TEM operational sequence is captured by the following pseudocode:

Initialization: t₀=0 Loop for n=0,1,2,...: 1. Compute running energies eₙ(t), dₙ(t) (or ˜eₙ, ˜dₙ). 2. Set bₙ(t), Δₙ(t) per shifted design. 3. Integrate y=0; t←tₙ while y < Δₙ(t): y ← y + (f(t) + bₙ(t))·dt; t ← t + dt end tₙ₊₁ ← t 4. Record firing time tₙ₊₁ and signal average yₙ = ∫ₜₙᵗₙ₊₁ f(u) du = Δₙ(tₙ₊₁) – ∫ₜₙᵗₙ₊₁ bₙ(u) du 5. Reset integrator; n ← n+1

Reconstruction from time encodings $\{t_n, y_n\}$ proceeds by iterative application of a standard algorithm. With $g(t) = \sin(\Omega_0 t)/(\pi t)$ and $s_n = (t_{n+1} + t_n)/2$:

• $A[f](t) = \sum_n y_n \cdot g(t - s_n)$
• $f_0 = A[f]$
• Iterative update: $f_{l+1}(t) = f_l(t) + A[f - f_l](t)$

Convergence is governed by $\|I - A\| \leq \alpha < 1$, leading to exponential error decay: $\|f - f_l\| \leq \alpha^{l+1}\|f\| \to 0$.

5. Trade-Offs and Parameter Selection Guidelines

The architecture admits explicit tuning of sampling density and convergence speed via parameters $(\alpha, \beta, s)$. Extreme bounds on $T_n$ under the shifted design are as follows:

Signal Variation Regime	Bound on $T_n$	Mechanism
Low Variation	$T_n \leq \pi\sqrt{\alpha/\beta}$	Set by flat regions, minimum sampling
High Variation	$$T_n \leq \pi\sqrt{\frac{\alpha (s+c)^2}{\beta (s+c)^2 + (c\Omega_0)^2}}$$	Driven by signal transitions

Smaller $\alpha$ or larger $\beta$ induces denser sampling and faster convergence, while the converse yields greater sparsity and slower convergence. Practical parameter choice is guided by desired maximum and minimum gap characteristics in respective signal regimes.

6. Simulation and Experimental Performance

Empirical assessment demonstrates the substantial sampling density reduction and precise reconstruction properties of VBT-IF-TEM across diverse signal classes:

• For a chirp-like signal ($\Omega_0 = 100$ Hz, $c=1$): Uniform Nyquist (180 samples, NMSE ≈ $-62$ dB), conventional IF-TEM (796 samples, NMSE ≈ $-60$ dB), VBT-IF-TEM (169 timestamps + 169 averages, NMSE ≈ $-54$ dB), with sub-Nyquist gaps in flat regions.
• For sum-of-sincs: Uniform (90 samples, NMSE ≈ $-52$ dB), conventional IF-TEM (729 samples, NMSE ≈ $-56$ dB), VBT-IF-TEM (96 samples, NMSE ≈ $-61$ dB).
• Adaptive two-level selection (state-dependent $(\alpha, \beta)$): sampling further reduced to 136 (from 169 with fixed), NMSE ≈ $-42$ dB.
• Real-data experiments: Ultrasonic guided waves (120 kHz) — uniform 1966 vs adaptive 1856 samples (NMSE ≈ $-42.7$ dB); ECG (100 Hz) — uniform 2000 vs adaptive 1837 samples (NMSE ≈ $-32.5$ dB).

7. Broader Significance and Methodological Impact

The VBT-IF-TEM exemplifies the benefits of algorithm-encoder co-design by enforcing local conditions that directly govern the convergence properties of reconstruction. The integration of signal- and derivative-energy-driven bias and threshold adapts the time encoding density to the local characteristics of $f(t)$, ensuring that the standard iterative algorithm reconstructs the input signal perfectly even with sub-Nyquist local rates. This approach substantially outperforms both uniform sampling and conventional fixed-bias IF-TEMs regarding sample efficiency without compromising reconstruction fidelity. A plausible implication is the facilitation of resource-aware sensing systems for temporally sparse or bandwidth-variable signals, where adaptive compression and accurate digital representation are essential (Yashaswini et al., 22 Jan 2026).