Self-Calibrating IF-TEM: Mismatch-Aware Sampling

• The paper introduces S-IF-TEM as a calibration-aware extension of the classical IF-TEM, jointly estimating varying integration gains and discharge times to correct hardware mismatches.
• It employs an online calibration mechanism by intermittently injecting a known reference signal to solve for unknown parameters, ensuring robust signal reconstruction.
• Empirical results show S-IF-TEM achieves near-ideal reconstruction performance with significantly lower NMSE compared to blind approaches, highlighting its practical advantage.

Searching arXiv for the specified papers and closely related IF-TEM/self-calibration work. The self-calibrating integrate-and-fire time encoding machine (S-IF-TEM) is a threshold-based, event-driven sampling framework in which signal acquisition and sampler-parameter identification are performed jointly from spike times. In the formulation introduced in "Self-Calibrating Integrate-and-Fire Time Encoding Machine" (Mekel et al., 13 Sep 2025), S-IF-TEM is built on a practical IF-TEM (P-IF-TEM) model that extends the classical integrate-and-fire sampler to include device mismatches and imperfections that can otherwise lead to significant reconstruction errors. Unlike ideal IF-TEM settings, the practical model accounts for inaccurately known or time-varying system parameters, nonzero integrator discharge time after firings, and nonlinear operation under large input dynamic ranges. Calibration is performed online by intermittently injecting a known reference signal and using the additional spike timings to estimate the unknown parameters needed for reconstruction (Mekel et al., 13 Sep 2025).

1. Position within time encoding theory

Time encoding machines replace amplitude samples at clocked time instants by event times determined by the signal and the sampler dynamics. In integrate-and-fire operation, the input is biased, integrated, compared to a threshold, and reset on each firing, so the output is a sequence of time instants carrying the analog information. This event-based paradigm is asynchronous and does not require a global clock, which distinguishes it from classical uniform sampling (Naaman et al., 2021).

Within that general framework, S-IF-TEM is specifically a calibration-aware extension of IF-TEM. The classical model assumes fixed and known analog parameters, whereas S-IF-TEM assumes that the effective sampler law may drift and must be inferred from the same hardware during operation (Mekel et al., 13 Sep 2025).

Model	Defining relation	Distinguishing feature
Classical IF-TEM	$$\frac{1}{\kappa}\int_{t_n}^{t_{n+1}} (x(t)+b)\,dt=\delta$$	Known fixed $\kappa$, immediate reset
P-IF-TEM	$$\frac{1}{\sigma_n}\int_{t_n+\Delta_n^{\mathrm{dis}}}^{t_{n+1}} (x(t)+b)\,dt=\delta$$	Unknown $\sigma_n$, nonzero $\Delta_n^{\mathrm{dis}}$
S-IF-TEM	P-IF-TEM with online calibration	Simultaneous parameter estimation and reconstruction

The practical importance of this distinction is that mismatch alters the spike-generation law itself. In an ideal IF-TEM, a bandlimited signal $x(t)\in \mathcal{B}_{\omega_M,E}$ with $|x(t)|\le c$ is biased by $b>c$, and exact recovery is possible if all inter-spike intervals satisfy $T_n=t_{n+1}-t_n<\pi/\omega_M$. S-IF-TEM retains the same reconstruction objective but no longer assumes that the effective integration constant and reset behavior are known a priori (Mekel et al., 13 Sep 2025).

2. Practical IF-TEM dynamics and mismatch model

The P-IF-TEM model formalizes three concrete nonidealities: an unknown or time-varying integration constant, nonzero discharge or reset time, and nonlinear integrator operation for larger input ranges. These effects are treated as first-order departures from the classical IF-TEM law and are represented directly in the firing relation (Mekel et al., 13 Sep 2025).

The ideal implicit timing law is

$$\frac{1}{\kappa}\int_{t_n}^{t_{n+1}} \bigl(x(t)+b\bigr)\,dt = \delta.$$

The practical extension is written as

$\kappa$0

where $\kappa$1 is the effective integration time constant at the $\kappa$2-th interval, $\kappa$3 is the discharge time after spike $\kappa$4, and $\kappa$5 captures nonlinear operating-region effects. The paper then compresses the nonlinearity into an averaging factor $\kappa$6, defines

$\kappa$7

and rewrites the firing rule as

$\kappa$8

This reformulation is central because it makes the practical sampler look like a classical IF-TEM with interval-dependent effective gain and dead time. The unknown sequences are $\kappa$9 and $$\frac{1}{\sigma_n}\int_{t_n+\Delta_n^{\mathrm{dis}}}^{t_{n+1}} (x(t)+b)\,dt=\delta$$0, and the reconstruction problem becomes one of recovering the signal despite those hidden practical parameters (Mekel et al., 13 Sep 2025).

The paper assumes bounded parameter ranges

$$\frac{1}{\sigma_n}\int_{t_n+\Delta_n^{\mathrm{dis}}}^{t_{n+1}} (x(t)+b)\,dt=\delta$$1

which imply

$$\frac{1}{\sigma_n}\int_{t_n+\Delta_n^{\mathrm{dis}}}^{t_{n+1}} (x(t)+b)\,dt=\delta$$2

For signals in $$\frac{1}{\sigma_n}\int_{t_n+\Delta_n^{\mathrm{dis}}}^{t_{n+1}} (x(t)+b)\,dt=\delta$$3 with $$\frac{1}{\sigma_n}\int_{t_n+\Delta_n^{\mathrm{dis}}}^{t_{n+1}} (x(t)+b)\,dt=\delta$$4, the inter-spike intervals satisfy

$$\frac{1}{\sigma_n}\int_{t_n+\Delta_n^{\mathrm{dis}}}^{t_{n+1}} (x(t)+b)\,dt=\delta$$5

with

$$\frac{1}{\sigma_n}\int_{t_n+\Delta_n^{\mathrm{dis}}}^{t_{n+1}} (x(t)+b)\,dt=\delta$$6

These bounds quantify how signal amplitude and hardware mismatch jointly control event density (Mekel et al., 13 Sep 2025).

3. Online calibration mechanism

S-IF-TEM augments P-IF-TEM with an online calibration procedure. The basic idea is to intermittently replace the true input $$\frac{1}{\sigma_n}\int_{t_n+\Delta_n^{\mathrm{dis}}}^{t_{n+1}} (x(t)+b)\,dt=\delta$$7 with a known calibration or reference signal $$\frac{1}{\sigma_n}\int_{t_n+\Delta_n^{\mathrm{dis}}}^{t_{n+1}} (x(t)+b)\,dt=\delta$$8, so that the unknown practical parameters can be inferred from the resulting spike intervals. The paper assumes that the unknown parameters are approximately constant over a short window of $$\frac{1}{\sigma_n}\int_{t_n+\Delta_n^{\mathrm{dis}}}^{t_{n+1}} (x(t)+b)\,dt=\delta$$9 consecutive spikes, which allows local estimation by solving a pair of linear equations (Mekel et al., 13 Sep 2025).

Within such a segment, the system injects a reference voltage twice, at firing instants $\sigma_n$0 and $\sigma_n$1, with amplitudes

$\sigma_n$2

During calibration, the comparator threshold is switched from $\sigma_n$3 to segment-specific calibration thresholds $\sigma_n$4 and $\sigma_n$5. The next firings caused by those injected reference signals are $\sigma_n$6 and $\sigma_n$7, and the measured calibration intervals are

$\sigma_n$8

Under the P-IF-TEM law and the piecewise-constant assumption, the calibration intervals satisfy

$\sigma_n$9

and

$\Delta_n^{\mathrm{dis}}$0

These two equations are solved for the two unknowns $\Delta_n^{\mathrm{dis}}$1 and $\Delta_n^{\mathrm{dis}}$2, yielding estimates $\Delta_n^{\mathrm{dis}}$3 and $\Delta_n^{\mathrm{dis}}$4. The paper describes this stage as a simple linear-equation solver (Mekel et al., 13 Sep 2025).

A key quantity is the non-sampling interval $\Delta_n^{\mathrm{dis}}$5, defined as the time after a firing during which the true signal is not sampled. In the non-calibration phase,

$\Delta_n^{\mathrm{dis}}$6

In calibration phases, the extra reference-induced firing adds dead time, so

$\Delta_n^{\mathrm{dis}}$7

This creates a direct design tradeoff: calibration improves parameter knowledge but consumes admissible sampling time (Mekel et al., 13 Sep 2025).

4. Reconstruction equations and exact recovery conditions

The practical t-transform relation in P-IF-TEM is

$\Delta_n^{\mathrm{dis}}$8

Under calibration, the corresponding quantity is

$\Delta_n^{\mathrm{dis}}$9

These relations are the measurements fed into the standard iterative IF-TEM reconstruction algorithm from the earlier literature; the paper does not introduce a brand-new inversion method, but adapts the known iterative reconstruction to practical and calibrated measurements (Mekel et al., 13 Sep 2025).

For P-IF-TEM, the sufficient condition for perfect recovery is

$x(t)\in \mathcal{B}_{\omega_M,E}$0

with

$x(t)\in \mathcal{B}_{\omega_M,E}$1

This is the practical counterpart of the classical spacing constraint; the additional $x(t)\in \mathcal{B}_{\omega_M,E}$2 term accounts for the non-sampling dead time (Mekel et al., 13 Sep 2025).

For S-IF-TEM, if

$x(t)\in \mathcal{B}_{\omega_M,E}$3

then perfect reconstruction is possible from

$x(t)\in \mathcal{B}_{\omega_M,E}$4

provided

$x(t)\in \mathcal{B}_{\omega_M,E}$5

where

$x(t)\in \mathcal{B}_{\omega_M,E}$6

and

$x(t)\in \mathcal{B}_{\omega_M,E}$7

The calibration itself must also satisfy an admissibility constraint. The paper requires

$x(t)\in \mathcal{B}_{\omega_M,E}$8

with

$x(t)\in \mathcal{B}_{\omega_M,E}$9

Equivalently,

$|x(t)|\le c$0

This expresses the central S-IF-TEM design condition: the reference signal and calibration thresholds must be chosen so that calibration does not violate the reconstruction regime (Mekel et al., 13 Sep 2025).

In the appendix, the reconstruction proof is formulated through an operator $|x(t)|\le c$1 and its adjoint $|x(t)|\le c$2, with midpoint

$|x(t)|\le c$3

and sinc kernel

$|x(t)|\le c$4

A Neumann-series argument shows that if

$|x(t)|\le c$5

then the signal can be reconstructed iteratively; the derived bound is exactly the condition $|x(t)|\le c$6 for the practical model and its calibrated analogue for S-IF-TEM (Mekel et al., 13 Sep 2025).

5. Relation to earlier IF-TEM, TEM, and threshold-sampling results

S-IF-TEM is best understood as a practical extension of several earlier research threads rather than an isolated construction. In "Sampling and Reconstruction of Bandlimited Signals with Multi-Channel Time Encoding" (Adam et al., 2019), the self-calibrating aspect concerns a different nuisance parameter: unknown relative time offsets between multiple integrate-and-fire channels. That work shows that reconstruction from multiple channels does not require prior knowledge of the shifts between machines, and that if single-channel time encoding can sample and perfectly reconstruct a $|x(t)|\le c$7-bandlimited signal, then $|x(t)|\le c$8-channel time encoding with shifted integrators can sample and perfectly reconstruct a signal with $|x(t)|\le c$9 times the bandwidth. In that setting, the unknown shifts are absorbed into the observed event times. In S-IF-TEM, by contrast, the unknowns are interval-dependent effective gains, discharge times, and nonlinear operating-region effects that alter the spike law itself (Adam et al., 2019).

"FRI-TEM: Time Encoding Sampling of Finite-Rate-of-Innovation Signals" (Naaman et al., 2021) provides a complementary algebraic framework for asynchronous IF-TEM reconstruction. It studies periodic FRI signals of the form

$b>c$0

with known pulse shape $b>c$1, and uses a sampling kernel $b>c$2 chosen to preserve a finite index set of Fourier series coefficients. For the IF-TEM, the threshold law is

$b>c$3

and the derived interval measurement is

$b>c$4

Stacking the resulting equations yields a linear system whose measurement matrix is left-invertible if

$b>c$5

The paper also proposes a zero-excluded kernel design for robustness and reports about $b>c$6–$b>c$7 dB lower MSE in noisy simulations. It does not solve self-calibration, but it supplies a clean IF-TEM measurement model, explicit invertibility conditions, and a robustness perspective that are directly relevant to calibration-aware extensions (Naaman et al., 2021).

"Integrate-and-Fire from a Mathematical and Signal Processing Perspective" (Moser et al., 20 Jan 2025) develops a broader mathematical foundation. It defines the IF operator

$b>c$8

represents spike trains as

$b>c$9

and studies several reset rules, especially reset-by-subtraction and reset-to-mod. A key identity is

$T_n=t_{n+1}-t_n<\pi/\omega_M$0

which makes precise the statement that SOD can be understood as a differential version of IF. The paper’s central geometry is based on the Alexiewicz semi-norm

$T_n=t_{n+1}-t_n<\pi/\omega_M$1

and it proves quasi-isometry, explicit error bounds, and a maximal sparsity property:

$T_n=t_{n+1}-t_n<\pi/\omega_M$2

That work does not present a named S-IF-TEM algorithm, but it provides foundational theory for threshold sampling with discontinuities, impulses, and sparse regularization (Moser et al., 20 Jan 2025).

A common source of confusion is therefore terminological. Earlier TEM work uses a self-calibrating viewpoint for unknown channel shifts, whereas the 2025 S-IF-TEM paper uses self-calibration for simultaneous estimation of practical hardware mismatches and signal reconstruction. The two uses are related at the level of nuisance-parameter absorption, but they target different unknowns and different failure modes (Adam et al., 2019).

6. Empirical behavior, limitations, and scope

The simulations in the S-IF-TEM paper use synthetic bandlimited signals built as sums of shifted sinc pulses. The main setting is $T_n=t_{n+1}-t_n<\pi/\omega_M$3 rad/s, i.e. $T_n=t_{n+1}-t_n<\pi/\omega_M$4 Hz, with $T_n=t_{n+1}-t_n<\pi/\omega_M$5, threshold $T_n=t_{n+1}-t_n<\pi/\omega_M$6, bias $T_n=t_{n+1}-t_n<\pi/\omega_M$7 normally, and, in one representative example, $T_n=t_{n+1}-t_n<\pi/\omega_M$8. The discharge time varies in $T_n=t_{n+1}-t_n<\pi/\omega_M$9, the nonlinearity values are $$\frac{1}{\kappa}\int_{t_n}^{t_{n+1}} \bigl(x(t)+b\bigr)\,dt = \delta.$$0, calibration uses $$\frac{1}{\kappa}\int_{t_n}^{t_{n+1}} \bigl(x(t)+b\bigr)\,dt = \delta.$$1, and a representative maximum admissible non-sampling interval is $$\frac{1}{\kappa}\int_{t_n}^{t_{n+1}} \bigl(x(t)+b\bigr)\,dt = \delta.$$2 (Mekel et al., 13 Sep 2025).

The paper compares four reconstruction modes.

Case	Average NMSE	Additional note
Ideal IF-TEM	about $$\frac{1}{\kappa}\int_{t_n}^{t_{n+1}} \bigl(x(t)+b\bigr)\,dt = \delta.$$3 dB	worst-case about $$\frac{1}{\kappa}\int_{t_n}^{t_{n+1}} \bigl(x(t)+b\bigr)\,dt = \delta.$$4 dB
Blind IF-TEM	about $$\frac{1}{\kappa}\int_{t_n}^{t_{n+1}} \bigl(x(t)+b\bigr)\,dt = \delta.$$5 dB	best-case around $$\frac{1}{\kappa}\int_{t_n}^{t_{n+1}} \bigl(x(t)+b\bigr)\,dt = \delta.$$6 dB
S-IF-TEM	about $$\frac{1}{\kappa}\int_{t_n}^{t_{n+1}} \bigl(x(t)+b\bigr)\,dt = \delta.$$7 dB	worst-case about $$\frac{1}{\kappa}\int_{t_n}^{t_{n+1}} \bigl(x(t)+b\bigr)\,dt = \delta.$$8 dB
Genie S-IF-TEM	about $$\frac{1}{\kappa}\int_{t_n}^{t_{n+1}} \bigl(x(t)+b\bigr)\,dt = \delta.$$9 dB	worst-case about $\kappa$00 dB

Over $\kappa$01 synthetic signals, the proposed self-calibrating method nearly matches the genie and ideal cases, while blind reconstruction is much worse. The abstract and evaluation section highlight improvements exceeding $\kappa$02 dB relative to blind reconstruction. The paper also states that the proposed method has only about a $\kappa$03 dB average gap and a $\kappa$04 dB worst-case gap relative to the ideal case, and that in the example figure for a $\kappa$05 Hz signal the estimated $\kappa$06 and $\kappa$07 closely track the true values (Mekel et al., 13 Sep 2025).

The principal limitations are structural rather than incidental. The unknown sequences $\kappa$08, $\kappa$09, $\kappa$10, and $\kappa$11 are assumed bounded and slowly varying enough to be treated as constant over a segment of $\kappa$12 consecutive firings. The nonlinear effect is absorbed into $\kappa$13 and then into $\kappa$14, rather than modeled in finer circuit detail. The calibration signal must be injected in the same operating region as the true signal so that the estimated mismatch reflects the actual nonlinear behavior. Most importantly, calibration consumes time, so the calibration thresholds and reference amplitudes cannot be arbitrary; they must satisfy the admissibility bound ensuring that the extra non-sampling time remains within the perfect-reconstruction regime (Mekel et al., 13 Sep 2025).

The resulting picture is precise. S-IF-TEM is not merely an IF-TEM made robust by heuristic correction, nor is it identical to earlier self-calibrating multichannel TEM formulations. It is a mismatch-aware IF-TEM architecture in which online estimation of effective integration gain and discharge time is embedded into the sampling process itself, and exact reconstruction is guaranteed when the calibrated timing law satisfies the stated interval and contractivity conditions (Mekel et al., 13 Sep 2025).