A Generalized Approach for Recovering Time Encoded Signals With Finite Rate of Innovation

In this paper, we consider the problem of reconstructing a function <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</i>(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i>) from its direct time encoding machine (TEM) measurements in a general scenario in which the signal is represented as an infinite sum of weighted generic functions ϕ(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i>) shifted in real time points. These functions belong to the class of signals with finite rate of innovation (FRI), which is more general than shift-invariant or bandlimited spaces, for which recovery guarantees were already introduced. For an FRI signal <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</i>(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i>), recovery guarantees from their direct TEM samples were introduced for particular functions ϕ(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i>) or functions ϕ(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i>) with alias cancellation properties leading to <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</i>(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i>) being periodic and bandlimited. On the theoretical front, this work significantly increases the class of functions for which reconstruction is guaranteed, and provide a condition for perfect input recovery depending on the first two local derivatives of ϕ(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i>). We extend this result with reconstruction guarantees in the case of noise corrupted FRI signals. On the practical front, we validate the proposed method via numerical simulations with filters previously used in the literature, as well as filters that are not compatible with the existing results. In cases where the filter has an unknown mathematical function and is only measured, the proposed method streamlines the recovery process by bypassing the filter modelling stage. Additionally, we validate the proposed method using a TEM hardware implementation.