FRI-TEM: Time Encoding Sampling of Finite-Rate-of-Innovation Signals

Abstract

Classical sampling is based on acquiring signal amplitudes at specific points in time, with the minimal sampling rate dictated by the degrees of freedom in the signal. The samplers in this framework are controlled by a global clock that operates at a rate greater than or equal to the minimal sampling rate. At high sampling rates, clocks are power-consuming and prone to electromagnetic interference. An integrate-and-fire time encoding machine (IF-TEM) is an alternative power-efficient sampling mechanism which does not require a global clock. Here, the samples are irregularly spaced threshold-based samples. In this paper, we investigate the problem of sampling finite-rate-ofinnovation (FRI) signals using an IF-TEM. We provide theoretical recovery guarantees for an FRI signal with arbitrary pulse shape and without any constraint on the minimum separation between the pulses. In particular, we show how to design a sampling kernel, IF-TEM, and recovery method such that the FRI signals are perfectly reconstructed. We then propose a modification to the sampling kernel to improve noise robustness. Our results enable designing low-cost and energy-efficient analogto-digital converters for FRI signals.

Time-Based Quantization for FRI and Bandlimited signals

“…T . The parameters {a , τ } L =1 can be uniquely computed from a minimum of 2L and 2L + 2 Fourier series coefficients (FSCs), for the classical and IF-TEM setup respectively, by using spectral analysis methods, such as the annihilating filter (AF) [16,17,22,23]. Hence, with 2L FSCs X[k] can uniquely determine the FRI signal x(t) [23].…”

“…Thus, the FRI signal reconstruction problem is reduced to uniquely determining the desired number of FSCs from the signal measurements. Reconstruction from IF-TEM outputs have been considered for cases where the FRI input is c-bounded and is guaranteed if the IF-TEM parameters satisfy [16,17].…”

“…This condition is similar to the classical FRI method where perfect recovery is achieved when sampling at the rate of innovation. For details on sampling and perfect reconstruction of FRI signals from classical and IF-TEM outputs, we refer to [17,19,22].…”

“…Time encoding has been studied for bandlimited signals [6,8], signals in shift-invariant spaces [15], and FRI signals [9,16,17]. FRI signals are of particular interest due to their prevalence in a variety of applications, such as radar [18], ultrasound [19], code-division multiple-access, and ultra-wideband [20].…”

Bandlimited signal reconstruction from leaky integrate-and-fire encoding using POCS

“…1). Time encoding is attracting more and more interest in data acquisition [1], [2], [3], [4], [5], [6], [7], [8], [7], [9], [10] as the downscaling of semiconductor integration is increasing time precision while resulting in less amplitude accuracy [11], [12], [13]. This is also part of the trend on event-based sampling [14] with more general objectives such as having acquisition activity dependent on input activity for power efficiency.…”