ACIF-TEM: Adaptive Compressed IF-TEM

• Adaptive Compressed IF-TEM (ACIF-TEM) is a clockless, adaptive architecture that combines dynamic bias tuning and segmented analog compression to digitize amplitude-bounded signals efficiently.
• It synergistically merges features of Adaptive IF-TEM and Compressed IF-TEM, reducing oversampling and quantization error while achieving up to 60% lower bit-rates compared to traditional methods.
• ACIF-TEM employs a two-step pulse-shrinking TDC and iterative reconstruction to maintain high precision, making it ideal for energy-constrained edge sensing and distributed acquisition systems.

Adaptive Compressed Integrate-and-Fire Time Encoding Machine (ACIF-TEM) is a signal acquisition architecture that synergistically combines adaptive sampling and analog compression within a fully integrated, clockless time-to-digital conversion framework. Building upon the power-efficient Integrate-and-Fire Time Encoding Machine (@@@@1@@@@), ACIF-TEM integrates adaptive bias control from Adaptive IF-TEM (AIF-TEM) and pre-quantization analog compression from Compressed IF-TEM (CIF-TEM), with each component reinforcing the performance and efficiency of the other. ACIF-TEM is shown to realize substantial bit- and energy-reductions compared to all prior IF-TEM variants, particularly in practical audio-band and quasi-stationary signal regimes (Karp et al., 4 Nov 2025).

1. Signal Model and Motivations

ACIF-TEM targets amplitude-bounded and bandlimited analog signals $x(t)$, satisfying $|x(t)|\leq c_{\max}$ and $X(\omega)=0$ for $|\omega|>\Omega$, with total finite energy $E$. In standard analog-to-digital conversion, Nyquist sampling at $2\Omega$ Hz leads to uniform bit allocation and power-intensive, high-rate digital circuitry. IF-TEM offers a clockless, asynchronous alternative by integrating $x(t)$ plus a static bias until a threshold is reached and emitting nonuniform firing times, with subsequent intervals $T_n = t_n - t_{n-1}$ encoding local signal amplitude (Karp et al., 4 Nov 2025).

However, classical IF-TEM reveals two core inefficiencies. First, a constant bias $b_{\mathrm{IF}} > c_{\max}$ results in oversampling during low-amplitude periods. Second, the time-to-digital converter (TDC) must quantize over the full dynamic range $\Delta t_{\max} - \Delta t_{\min}$, leading to non-optimal bit usage. These limitations motivate adaptive and compressed variants.

2. Component Architectures: AIF-TEM and CIF-TEM

Adaptive IF-TEM (AIF-TEM)

AIF-TEM addresses oversampling inefficiency by dynamically tuning the bias $b_n$ to local amplitude statistics. At each firing $n$, the bias is set with a margin: $b_n = \hat{c}_n + \beta$, using a Max-Amplitude Predictor (MAP) that estimates the maximum $c_n$ over a preceding window. This adaptation ensures the integrator output remains strictly increasing and the inter-firing interval $T_n$ is tightly bounded: $$\Delta t_{n,\min} = \frac{\kappa\delta}{b_n + c_n} \leq T_n \leq \frac{\kappa\delta}{b_n - c_n} = \Delta t_{n,\max}$$ where $\kappa$ is an integrator gain and $\delta$ the comparator threshold. Amplitude adaptation reduces average oversampling and enables local control over both rate and quantization (Omar et al., 2024).

Compressed IF-TEM (CIF-TEM)

CIF-TEM applies a compression transformation prior to quantization by partitioning the interval range $D = \Delta t_{\max} - \Delta t_{\min}$ into $L$ segments. Each $T_n$ is mapped to a segment index $w_{i,n}$ and a residual $r_n$ via

$$w_{i,n} = 1 + \left\lfloor \frac{T_n - \Delta t_{\min}}{D/L} \right\rfloor, \qquad r_n = T_n - [\Delta t_{\min} + (w_{i,n} - 1)\tfrac{D}{L}]$$

with only infrequent index updates (i.e., $w_{i,n} \neq w_{i,n-1}$) incurring additional bits. The residual is then uniformly quantized within its segment, substantially reducing quantization error and total bit-rate over conventional IF-TEM for stationary or slowly varying signals (Tarnopolsky et al., 2022).

3. ACIF-TEM: Integrated Synergistic Design

ACIF-TEM fuses AIF-TEM's bias adaptation and CIF-TEM's segmental compression into a unified workflow. In Stage 1, the MAP-driven adaptive bias produces a dynamically range-restricted set of intervals $T_n$. In Stage 2, a bias-driven segmentator—tied to local amplitude statistics—parcels the remaining interval range into $L_{AC}$ segments. Both adaptation and compression logic are physically integrated in a clockless, two-step pulse-shrinking (2PS) TDC, eliminating the need for high-frequency clocks and extra latches.

Notably, segment granularity in ACIF-TEM is computed as

$$L_b = \left\lceil \frac{c_{\max}}{\sigma(c_n)} \right\rceil, \qquad L_{AC} = \min\!\left[\, \phi(c_n)\, L_b,\, \tfrac{K}{2} \right], \quad \phi(c_n) = \frac{(2\,\mathbb{E}[c_n]+\beta)^2}{2\,\beta\,(2\,c_{\max}+\beta)}$$

where $\sigma(c_n)$ is the run-time estimate of local amplitude variance and $K$ the total quantization levels. This self-tuning ensures locally optimal trade-offs between segment width and quantization precision. Updates occur every $M$ events, amortizing computational overhead and ensuring adaptation tracks abrupt signal changes (Karp et al., 4 Nov 2025).

2PS TDC Operation

Each firing interval $T_n$ passes through $F=L_{AC}$ coarse and $G=\log_2(K/L_{AC})$ fine shrinking stages. The output is reconstructed as

$T_n = f\,\Delta T_1 + g\,\Delta T_2 + Q_e$

where $\Delta T_1 = D_n/L_{AC}$, $\Delta T_2 = \Delta T_1/\log_2(K/L_{AC})$, and $|Q_e| \leq \Delta T_2/2$. Bit usage is minimized since only the segment index (coarse) and fine position need to be stored per event.

4. Sampling–Quantization Analysis and Reconstruction

The ACIF-TEM reconstruction pipeline employs standard IF-TEM inversion algorithms: the (possibly nonlinear) mapping from digitized $\{T_n\}$ to the analog signal $x(t)$ is solved using iterative kernel expansions over sinc-basis centered at appropriately chosen time points, with convergence conditions enforced by construction via bias and segmentation strategies.

Distortion decomposes as $$\mathrm{MSE}_{\mathrm{total}} = \mathrm{MSE}_{\mathrm{sampling}} + \mathrm{MSE}_{\mathrm{quant}}$$, with quantization error for the two-step TDC bounded by $(\Delta T_2)^2/12$. The overall bit-rate is analytically characterized as

$$N_{AC} = \mathrm{OS}_{AC} \cdot S \left( \log_2 K + \log_2(L_{AC}) [P_M - \log_2\phi(c_n) - 1] \right)$$

where $\mathrm{OS}_{AC}$ denotes ACIF-TEM's oversampling ratio, $S$ the number of Nyquist-rate samples, and $P_M$ the segment-change probability. In all cases, $N_{AC} < N_{IF}$ and $N_{AC} < N_A$ for the same quantization resolution (Karp et al., 4 Nov 2025).

5. Empirical Performance and Comparative Evaluation

Empirical evaluation on audio signals (44.1 kHz bandwidth) demonstrates that ACIF-TEM achieves fixed MSE targets with substantially reduced bit usage compared to all baselines. For MSE $-30$ dB, the key metrics are as follows:

Sampler	Oversampling	Bits/event	Total bits	Compression
IF-TEM	$OS$	9	100 M	baseline
AIF-TEM	$0.8\,OS$	7	78 M	22% vs IF-TEM
CIF-TEM	$1.0\,OS$	7	70 M	30% vs IF-TEM
ACIF-TEM	$0.8\,OS$	4	40 M	60% vs IF-TEM

“Bits per event” includes segment, residual, and amortized bias update bits. ACIF-TEM yields a 3-bit/event reduction compared to AIF-TEM and 60% total bit-rate reduction compared to IF-TEM (Karp et al., 4 Nov 2025). Direct-comparison experiments confirm that similar MSE levels require 1–2 fewer bits in ACIF-TEM than CIF-TEM, and that ACIF-TEM matches or exceeds “oracle” MAP performance (Omar et al., 2024, Tarnopolsky et al., 2022).

6. Implementation Considerations and Limitations

The ACIF-TEM architecture is fully asynchronous due to the 2PS TDC design, leading to low power dissipation and high immunity to process-voltage-temperature variations; no high-frequency clock is required. Correct operation requires the MAP block to conservatively estimate local amplitude maxima, as underestimation can violate the fundamental sampling criterion $\Delta t_{n,\max}<\pi/\Omega$ and compromise reconstruction.

Segment update interval $M$ trades off between control-loop stability and adaptation speed. In rapidly varying signals, run-length savings in segment index transmissions diminish, moderately attenuating compression efficiency. For extremely low-amplitude signals, the segment number $L_{AC}$ drops to its minimum and ACIF-TEM reduces to a simple TDC (Karp et al., 4 Nov 2025).

7. Context and Relation to Broader Research

ACIF-TEM extends the paradigm of asynchronous, nonuniform analog-to-digital sampling by integrating adaptive, compressive, and fully digital-efficient techniques within a unified converter. The synergistic design resolves both oversampling and quantization inefficiency. ACIF-TEM stands distinct from other variable-bias, variable-threshold, and event-based analog encoders by guaranteeing geometric convergence in iterative reconstruction while achieving bit-minimal, MSE-constrained operation on real signals (Yashaswini et al., 22 Jan 2026, Omar et al., 2024).

A plausible implication is that ACIF-TEM architectures are particularly attractive in edge sensing, neural recording, and bandwidth-constrained distributed acquisition systems where nonstationary signal variation and energy budget are critical constraints. However, adaptation speed and estimator reliability remain practical limitations, especially in hostile or fast-fluctuating environments.

In summary, ACIF-TEM provides a rigorously analyzed, empirically validated, and hardware-realistic solution for compressed, adaptive, and asynchronous analog-to-digital conversion in modern information processing pipelines (Karp et al., 4 Nov 2025, Tarnopolsky et al., 2022, Omar et al., 2024).