Time Encoding Quantization of Bandlimited and Finite-Rate-of-Innovation Signals

TL;DR

The paper addresses the impact of quantization on IF-TEM samplers for BL and FRI signals, deriving an MSE bound that highlights when time-difference quantization can outperform uniform amplitude quantization. By linking signal energy $E$, bandwidth $\Omega$, and maximum amplitude $c$ (with $c=\sqrt{E\Omega/\pi}$) and introducing a bias-to-amplitude relation $b=\alpha c$, it shows the IF-TEM quantization step $\Delta_{IF-TEM}$ shrinks as $E$ or $\Omega$ grows, leading to improved MSE performance. The work provides a rigorous upper bound on the IF-TEM reconstruction error, presents a sufficient condition under which IF-TEM beats Nyquist ADC at fixed bits, and extends the analysis to FRI signals where increasing the number of pulses $L$ further reduces quantization error. Extensive simulations and experiments validate the theory, reporting up to ~8 dB MSE gains for BL, and at least 5–8 dB gains for FRI under comparable bit budgets. Overall, the results suggest that energy- and bandwidth-aware parameter design in IF-TEM offers a practical route to lower bit-rate requirements without sacrificing reconstruction accuracy in sub-Nyquist, asynchronous sampling systems.

Abstract

This paper studies the impact of quantization in integrate-and-fire time encoding machine (IF-TEM) sampler used for bandlimited (BL) and finite-rate-of-innovation (FRI) signals. An upper bound is derived for the mean squared error (MSE) of IF-TEM sampler and is compared against that of classical analog-to-digital converters (ADCs) with uniform sampling and quantization. The interplay between a signal's energy, bandwidth, and peak amplitude is used to identify how the MSE of IF-TEM sampler with quantization is influenced by these parameters. More precisely, the quantization step size of the IF-TEM sampler can be reduced when the maximum frequency of a bandlimited signal or the number of pulses of an FRI signal is increased. Leveraging this insight, specific parameter settings are identified for which the quantized IF-TEM sampler achieves an MSE bound that is roughly 8 dB lower than that of a classical ADC with the same number of bits. Experimental results validate the theoretical conclusions.

Figures (11)

• Figure 1: Schematic diagram of (a) classical sampler and (b) IF-TEM sampler.
• Figure 2: Sampling mechanism of a signal $x(t)$ using an IF-TEM sampler. (a) the signal $x(t)$. (b) the signal with an addition of a bias $b$ such that $x(t)+b$ is a non negative signal. (c) the signal $x(t)+b$ is integrated and scaled, each time the threshold $\delta$ is reached the integrator resets and the time differences between consecutive time instances $T_n = t_n - t_{n-1}$ are recorded. (d) The IF-TEM series of time instances is calculated by summing up the time differences $T_n$ starting from an initial time instant $t_0=0$.
• Figure 3: A kernel-based FRI sampling framework: An FRI signal $x(t)$ is first filtered by a sampling kernel $g(t)$ and then instantaneous uniform samples are measured at a sub-Nyquist rate. Parameters of the FRI signal are estimated from the sub-Nyquist samples.
• Figure 4: Generalized sampling with quantization system mode: Filtered continuous-time signal $y(t)=(x*g)(t)$ is sampled by a sampler $S$ that results in a discrete representation $\{\theta_n\}_{n\in I}$, where $I$ is a countable set. The representation is quantized by a quantizer $Q$, resulting in $\{f_Q(\theta_n)\}_{n\in I}$.
• Figure 5: Time instances differences in IF-TEM with BL signals as a function of the frequency band. In red: the difference $\Delta t_{\max} - \Delta t_{\min}$, as defined in \ref{['eq:consecutive_time']}. In blue: the solid line shows the average values of $\Delta t_{\max}^{*} - \Delta t_{\min}^{*}$, which are the real difference. See the range in Table \ref{['Tab:Tcr2']}.

Theorems & Definitions (13)

• Definition 1
• Definition 2
• Theorem 1
• proof
• Theorem 2
• proof
• Lemma 1
• Theorem 3
• proof
• Theorem 4