  We investigate time encoding as an alternative method to classical sampling,
and address the problem of reconstructing classes of non-bandlimited signals
from time-based samples. We consider a sampling mechanism based on first
filtering the input, before obtaining the timing information using a time
encoding machine. Within this framework, we show that sampling by timing is
equivalent to a non-uniform sampling problem, where the reconstruction of the
input depends on the characteristics of the filter and on its non-uniform
shifts. The classes of filters we focus on are exponential and polynomial
splines, and we show that their fundamental properties are locally preserved in
the context of non-uniform sampling. Leveraging these properties, we then
derive sufficient conditions and propose novel algorithms for perfect
reconstruction of classes of non-bandlimited signals such as: streams of
Diracs, sequences of pulses and piecewise constant signals. Next, we extend
these methods to operate with arbitrary filters, and also present simulation
results on synthetic noisy data.
