[Paper Review] Time Encoding of Finite-Rate-of-Innovation Signals

Abijith Jagannath Kamath, Sunil Rudresh+1 more|arXiv (Cornell University)|Jul 7, 2021

Sparse and Compressive Sensing Techniques37 references4 citations

TL;DR

This paper proposes a time-encoding framework for finite-rate-of-innovation (FRI) signals using crossing-time-encoding machines (C-TEM) and integrate-and-fire time-encoding machines (IF-TEM), transforming nonuniform time-encoded measurements into a sum-of-sinusoids structure via Fourier domain analysis. It establishes sufficient conditions for perfect reconstruction using standard FRI techniques and introduces a robust optimization method to handle measurement noise, achieving reconstruction at the theoretical minimum sampling rate matching the signal's rate of innovation.

ABSTRACT

Time-encoding of continuous-time signals is an alternative sampling paradigm to conventional methods such as Shannon's sampling. In time-encoding, the signal is encoded using a sequence of time instants where an event occurs, and hence fall under event-driven sampling methods. Time-encoding can be designed agnostic to the global clock of the sampling hardware, which makes sampling asynchronous. Moreover, the encoding is sparse. This makes time-encoding energy efficient. However, the signal representation is nonstandard and in general, nonuniform. In this paper, we consider time-encoding of finite-rate-of-innovation signals, and in particular, periodic signals composed of weighted and time-shifted versions of a known pulse. We consider encoding using both crossing-time-encoding machine (C-TEM) and integrate-and-fire time-encoding machine (IF-TEM). We analyze how time-encoding manifests in the Fourier domain and arrive at the familiar sum-of-sinusoids structure of the Fourier coefficients that can be obtained starting from the time-encoded measurements via a suitable linear transformation. Thereafter, standard FRI techniques become applicable. Further, we extend the theory to multichannel time-encoding such that each channel operates with a lower sampling requirement. We also study the effect of measurement noise, where the temporal measurements are perturbed by additive noise. To combat the effect of noise, we propose a robust optimization framework to simultaneously denoise the Fourier coefficients and estimate the annihilating filter accurately. We provide sufficient conditions for time-encoding and perfect reconstruction using C-TEM and IF-TEM, and furnish extensive simulations to substantiate our findings.

Motivation & Objective

• To develop a time-encoding framework for FRI signals that enables perfect reconstruction using nonuniform, event-driven sampling.
• To analyze how time-encoding via C-TEM and IF-TEM maps to a sum-of-sinusoids structure in the Fourier domain, enabling application of standard FRI techniques.
• To derive sufficient conditions for perfect reconstruction using both C-TEM and IF-TEM, ensuring sampling rates match the signal's rate of innovation.
• To extend the framework to multichannel time-encoding, reducing per-channel sampling requirements.
• To design a robust optimization method that jointly denoises Fourier coefficients and estimates the annihilating filter under additive measurement noise.

Proposed method

• Transforms time-encoded measurements from C-TEM and IF-TEM into a Fourier-domain representation that exhibits a sum-of-sinusoids structure via a linear transformation of the time instants.
• Applies Prony’s method to estimate the signal parameters (amplitudes and time shifts) from the reconstructed Fourier coefficients.
• Uses a left-inverse of the measurement matrix derived from the sampling kernel and time instants to ensure unique recovery of Fourier coefficients.
• Introduces a robust optimization framework that jointly estimates the annihilating filter and denoises Fourier coefficients under additive noise, improving reconstruction accuracy.
• Derives sufficient conditions on sampling rate and reference signal amplitude to guarantee at least one measurement per oscillation period of the reference signal.
• Extends the theory to multichannel time-encoding, where each channel operates at a reduced sampling rate, maintaining overall reconstruction performance.

Experimental results

Research questions

• RQ1 Can time-encoding via C-TEM and IF-TEM be mapped to a sum-of-sinusoids structure in the Fourier domain, enabling use of standard FRI reconstruction techniques?
• RQ2 What are the sufficient conditions on the sampling rate and system parameters to ensure perfect reconstruction of periodic FRI signals using C-TEM and IF-TEM?
• RQ3 How can measurement noise in time-encoded signals be effectively mitigated to preserve reconstruction accuracy?
• RQ4 Can multichannel time-encoding reduce per-channel sampling requirements while maintaining perfect reconstruction of FRI signals?
• RQ5 What is the theoretical minimum sampling rate required for perfect reconstruction using time-encoding machines, and does it match the signal’s rate of innovation?

Key findings

• Perfect reconstruction of periodic FRI signals is achievable using C-TEM and IF-TEM when the number of time-encoded measurements $ L $ satisfies $ L \geq 2K+1 $, where $ K $ is the number of signal components.
• The sampling rate requirement matches the finite rate of innovation $ \frac{2K}{T} $, confirming sub-Nyquist sampling is possible with time-encoding.
• For C-TEM, a sufficient condition for unique recovery is $ A_r \geq \|x*g\|_\infty $ and $ f_r > \frac{2K+1}{T} $, ensuring at least one measurement per half-period of the reference signal.
• For IF-TEM, the condition $ t_1 + (L-1)\frac{\kappa\gamma}{b - \|y\|_\infty} < T $ must hold, with $ y = x*g $, to ensure sufficient measurements in the interval $[0,T)$.
• The proposed robust optimization framework successfully denoises Fourier coefficients and improves annihilating filter estimation under additive noise, enhancing reconstruction fidelity.
• Multichannel time-encoding reduces per-channel sampling rate while preserving perfect reconstruction, enabling scalable and energy-efficient signal acquisition.