Perfect Recovery and Sensitivity Analysis of Time Encoded Bandlimited Signals

TLDR

A time encoding machine asynchronously encodes amplitude information into a time sequence. The study investigates the operating characteristics of a feedback-loop time encoding machine, designs a nonlinear inverse decoder, and analyzes its sensitivity to trigger parameters. The authors model the machine with an adder, linear filter, and Schmitt trigger, derive bounds on recovery error from time quantization, and compare these to amplitude quantization errors in irregular sampling. They prove that bandlimited signals can be perfectly recovered when consecutive time intervals are bounded by the inverse Nyquist rate, link the recovery to irregular sampling and nonlinear modulation schemes, show the algorithm is trigger-parameter insensitive, and find that time-domain and amplitude-domain quantization are largely equivalent under Nyquist-type.

Abstract

A time encoding machine is a real-time asynchronous mechanism for encoding amplitude information into a time sequence. We investigate the operating characteristics of a machine consisting of a feedback loop containing an adder, a linear filter, and a noninverting Schmitt trigger. We show that the amplitude information of a bandlimited signal can be perfectly recovered if the difference between any two consecutive values of the time sequence is bounded by the inverse of the Nyquist rate. We also show how to build a nonlinear inverse time decoding machine (TDM) that perfectly recovers the amplitude information from the time sequence. We demonstrate the close relationship between the recovery algorithms for time encoding and irregular sampling. We also show the close relationship between time encoding and a number of nonlinear modulation schemes including FM and asynchronous sigma-delta modulation. We analyze the sensitivity of the time encoding recovery algorithm and demonstrate how to construct a TDM that perfectly recovers the amplitude information from the time sequence and is trigger parameter insensitive. We derive bounds on the error in signal recovery introduced by the quantization of the time sequence. We compare these with the recovery error introduced by the quantization of the amplitude of the bandlimited signal when irregular sampling is employed. Under Nyquist-type rate conditions, quantization of a bandlimited signal in the time and amplitude domains are shown to be largely equivalent methods of information representation.