Time-Encoding Machine

• Time-Encoding Machines are asynchronous systems that encode continuous-time signals into time-stamps at threshold-crossing events.
• They use mechanisms like crossing and integrate-and-fire to ensure robust signal reconstruction under specific density and Nyquist constraints.
• TEMs provide energy-efficient, hardware-friendly alternatives for ADCs, driving innovations in neuromorphic sensing and sub-Nyquist sampling.

A Time-Encoding Machine (TEM) is an asynchronous, event-driven sampler that encodes continuous-time signals into sequences of time-stamps corresponding to threshold-crossing or event-generation instants, rather than discrete-amplitude values at prescribed clocked times. The core concept is to map amplitude or structural features of the input signal directly into timing information. TEMs do not require a global clock and are robust to various sources of system imperfection, offering an energy-efficient alternative to classical analog-to-digital conversion and clock-based sampling. They have a foundational role in event-based sensory hardware, neuromorphic systems, efficient analog-to-digital converters (ADCs), and modern signal acquisition and processing for bandwidth- and energy-constrained environments.

1. Mathematical Definition and Main Architectures

A TEM can be formally defined as a mapping from a function space $\mathcal{E}$ to the space of strictly increasing sequences of real numbers (the time-stamps), $T:f \rightarrow \{t_n\}$. For inputs $f$ in appropriate function spaces—most commonly, bandlimited (Paley–Wiener) spaces or shift-invariant subspaces—a TEM produces spike sequences such that, under density constraints on inter-event intervals, perfect and robust inversion (signal reconstruction) is possible (Gontier et al., 2011, Adam et al., 2022).

The two canonical types are:

• Crossing-TEM (C-TEM): Emits events when $f(t)$ crosses a moving or fixed reference function $\Phi_n(t)$, i.e., $f(t_n) = \Phi_n(t_n)$.
• Integrate-and-Fire TEM (IF-TEM): Emits an event whenever the integrated input (possibly biased and scaled), e.g.,

$\int_{t_{n-1}}^{t_n} f(u)\,du = \delta,$

crosses a preset threshold $\delta$. After firing, the integrator is reset.

For multidimensional signals (e.g., image or video fields), a TEM is instantiated at each spatial location to encode local temporal traces (Adam et al., 2022).

2. Sampling Theory and Reconstruction Guarantees

The fundamental recovery condition for a bandlimited input $f$ is that the maximum inter-event interval across the output time-sequence must be less than the Nyquist interval, i.e.,

$\max_n (t_{n+1} - t_n) < \frac{\pi}{\Omega}$

for $f$ bandlimited to $[-\Omega, \Omega]$ (Gontier et al., 2011, Adam et al., 2022, Naaman et al., 2021). For general shift-invariant spaces generated by $\lambda$, explicit frame bounds and a critical interval $\tau$ involving the generator's analysis functions become the relevant density constraint (Gontier et al., 2011).

In finite-rate-of-innovation (FRI) models, perfect recovery is governed by collecting a sufficient number of events over the observation interval, determined by the signal's innovation rate and the chosen prefilter (Naaman et al., 2021, Kamath et al., 2021).

Reconstruction is typically cast as a problem of inverting a set of linear functionals—the integrals of $f$ over unknown intervals—via one of:

• Iterative projections onto convex sets (POCS) (Thao et al., 2019).
• Solving Vandermonde or generalized linear systems in a Fourier or time-domain representation (Naaman et al., 2023, Naaman et al., 2021, Naaman et al., 2024).
• Explicit Neumann-series contraction mappings for bandlimited cases (Gontier et al., 2011, Naaman et al., 2021, Omar et al., 2024).

For video or multichannel TEMs, a spatial grid ensures invertibility via sampling at or above the critical spatial density, then combining temporal recovery per channel (pixel) with spatial bandlimited interpolation (Adam et al., 2022).

3. Signal Classes and TEM Extensions

Bandlimited and Shift-Invariant Inputs

For $f$ in $\mathcal{B}_\Omega$ (bandlimited), or $f\in V^2(\lambda)$ (shift-invariant subspace with generator $\lambda$), density criteria for spike sequences and explicit reconstruction formulas are available (Gontier et al., 2011, Thao et al., 2019). The underlying theory leverages reproducing kernels and frame operators associated with non-uniform samples.

Bandpass and I/Q Extraction

TEMs support bandpass signal sampling using analogous inter-event interval constraints, with unique recovery possible at rates dictated by the Landau lower bound $2B$ (for bandwidth $B$), independent of center frequency—a qualitative departure from classical uniform bandpass sampling (Liu et al., 2023, Shao et al., 2024). Alternating POCS is employed for in-phase and quadrature extraction.

Finite-Rate-of-Innovation Signals

Combining IF-TEMs with prefiltering kernels (such as sum-of-sincs) produces sufficient statistics for FRI models—signals described by linear combinations of pulses with unknown positions and amplitudes. Constructing measurements as specific integrals of the filtered signal, one recovers the innovation parameters via Fourier analysis and Prony-like annihilation methods, achieving sub-Nyquist sampling rates (Naaman et al., 2021, Naaman et al., 2023, Naaman et al., 2024).

4. Quantization, Compression, and Architectural Variants

Quantization in TEMs primarily targets the inter-event intervals, rather than amplitudes. For bandlimited/FRI signals, the dynamic range of intervals diminishes with increasing input bandwidth/energy, yielding finer effective quantization for a fixed bit budget than classical ADCs (Naaman et al., 2021).

Adaptive and compressed IF-TEM variants offer further bit-rate reduction with minimal loss:

• Adaptive IF-TEM (AIF-TEM): Dynamically tracks the local amplitude; reduces oversampling and quantization MSE beyond classical IF-TEM, providing up to 14 dB empirical performance gain (Omar et al., 2024, Karp et al., 4 Nov 2025).
• Compressed IF-TEM (CIF-TEM): Implements analog domain compression prior to quantization by windowing the interval distribution, yielding 5–20 dB MSE improvement or 10–20% bit savings (Tarnopolsky et al., 2022).
• Adaptive Compressed IF-TEM (ACIF-TEM): Integrates adaptive biasing and compression, optimizing both hardware and bit efficiency; achieves 3-bit savings over AIF-TEM and ~60% compression over IF-TEM for equal MSE (Karp et al., 4 Nov 2025).

Non-uniform quantization, matched to the empirical interval distribution of the TEM output, halves the transmission cost over naive uniform quantization, for the same distortion (Yashaswini et al., 4 Nov 2025).

For hardware implementation, TEM-based ADCs use minimal analog circuitry—integrators, comparators, capacitive reset, and low-rate time-to-digital conversion—eliminating high-frequency clock trees and thus drastically reducing power consumption. TEM ADCs have been demonstrated to operate robustly at rates far below Nyquist, with empirical error floors and energy-per-conversion lower than classical architectures (Naaman et al., 2023, Naaman et al., 2024).

5. Practical Benefits and Limitations

Advantages:

• True asynchronous, event-driven operation; no global clock.
• Graceful data-rate adaptation to signal activity: high in transients, low in static regions.
• Exact recovery and predictable worst-case error in noiseless and quantized settings for admissible signals (Adam et al., 2022, Naaman et al., 2021).
• Hardware efficiency (e.g., tens of μW in ADC applications) and high resilience to timing jitter/quantization (Naaman et al., 2023, Naaman et al., 2024, Gontier et al., 2011).
• Direct integration with distributed sensors and event-based computational systems (e.g., spiking neural networks) (Adam, 2021).

Limitations:

• Reconstruction may become ill-conditioned for signals incompatible with the density condition or in the presence of severe noise.
• FIR-truncation errors limit practical accuracy of real-time iterative decoders (Thao et al., 2019).
• Implementation requires precise integrator and comparator circuits; in practice, device mismatch and analog drift can occur, mitigated by self-calibrating architectures (Mekel et al., 13 Sep 2025).
• Design and optimization of prefiltering and non-uniform quantization require empirical modeling.

6. Recent Extensions and Research Directions

• Self-Calibrating TEMs: Addressing device parameter uncertainty and drift, S-IF-TEMs estimate system parameters inline, yielding near-ideal recovery in practical settings under mild contraction conditions (Mekel et al., 13 Sep 2025).
• Spiking Neural Network Training: TEM theory has been leveraged to develop gradient-free training procedures for SNNs, interpreting synaptic weights as solutions to linear systems imposed by the spike-timing rules, connecting neural network learning to time-encoding constraints (Adam, 2021).
• Symbol Detection and Communications: IF-TEMs have been utilized in communication receiver front-ends, extracting information directly from event timing without full waveform reconstruction, with analytical error probability formulations and design guidelines for spectral efficiency vs. error robustness (Bernardo, 25 Aug 2025).
• Event-Based Video Sensing: Multichannel TEMs underlie fast, high-dynamic-range event cameras, enabling new spatial-temporal resolution trade-offs not possible in frame-based architectures (Adam et al., 2022).

7. Summary Table: Key TEM Variants and Characteristic Features

Variant	Principle	Adaptive/Compressed	Quantization	Application Domain
IF-TEM	Integrate-to-threshold	No	Uniform/NUQ	BL/FRI signals, ADC
AIF-TEM	Adaptive threshold/bias	Yes	Uniform/Dynamic	Energy-efficient ADC, sub-Nyquist
CIF-TEM	Stationary interval windows	No	Pre-quant. comp.	Low-bit-rate/energy ADC
ACIF-TEM	Adaptive + compressed	Yes	Joint	Minimal-bit, robust ADC
S-IF-TEM	Self-calibrating parameters	Yes (in calibration)	Uniform	Imperfect hardware, drift
BP-TEM	Bandpass, IQ extraction	No	Differential	RF/communications, event video

In summary, the Time-Encoding Machine offers a mathematically rigorous, hardware-efficient, and algorithmically tractable framework for asynchronous signal acquisition and neuromorphic computation, with provable guarantees and broad applicability across modern signal processing, communications, and event-driven sensing (Adam et al., 2022, Shao et al., 2024, Omar et al., 2024, Karp et al., 4 Nov 2025, Tarnopolsky et al., 2022, Mekel et al., 13 Sep 2025, Naaman et al., 2021, Naaman et al., 2023, Gontier et al., 2011).