  This letter considers the problem of sparse signal reconstruction from the
timing of its Level Crossings (LC)s. We formulate the sparse Zero Crossing (ZC)
reconstruction problem in terms of a single 1-bit Compressive Sensing (CS)
model. We also extend the Smoothed L0 (SL0) sparse reconstruction algorithm to
the 1-bit CS framework and propose the Binary SL0 (BSL0) algorithm for
iterative reconstruction of the sparse signal from ZCs in cases where the
number of sparse coefficients is not known to the reconstruction algorithm a
priori. Similar to the ZC case, we propose a system of simultaneously
constrained signed-CS problems to reconstruct a sparse signal from its Level
Crossings (LC)s and modify both the Binary Iterative Hard Thresholding (BIHT)
and BSL0 algorithms to solve this problem. Simulation results demonstrate
superior performance of the proposed LC reconstruction techniques in comparison
with the literature.
