Sampling based on timing: Time encoding machines on shift-invariant subspaces

Abstract

Sampling information using timing is an approach that has received renewed attention in sampling theory. The question is how to map amplitude information into the timing domain. One such encoder, called time encoding machine, was introduced by Lazar and Tóth (2004 [23]) for the special case of band-limited functions. In this paper, we extend their result to a general framework including shift-invariant subspaces. We prove that time encoding machines may be considered as non-uniform sampling devices, where time locations are unknown a priori. Using this fact, we show that perfect representation and reconstruction of a signal with a time encoding machine is possible whenever this device satisfies some density property. We prove that this method is robust under timing quantization, and therefore can lead to the design of simple and energy efficient sampling devices.

Reconstruction, Identification and Implementation Methods for Spiking Neural Circuits

“…These results have been extended for inputs belonging to the more general shift-invariant spaces (SIS) (Gontier and Vetterli, 2014).…”

“…Under certain conditions, functions belonging to the more general shift-invariant spaces (SIS) can be perfectly reconstructed from uniform (Aldroubi et al, 1994) as well as nonuniform samples Feichtinger, 1998, Aldroubi andGröchenig, 2001). Furthermore, Gontier and Vetterli (2014) have provided sufficient conditions for reconstructing the input of an IF neuron belonging to a SIS from the associated output sequence.…”

“…Gontier and Vetterli (2014) extended the results of Lazar and Tóth (2003) to SIS using the non-uniform sampling framework developed by Aldroubi and Feichtinger (1998), Gröchenig (2001), Feichtinger et al (1995), Gröchenig (1992Gröchenig ( , 1993, and Gröchenig and Schwab (2003). The TEMs considered by Gontier and Vetterli (2014) The shift invariant space of order 2 generated by function λ is defined by (Unser, 2000) V 2…”

“…A dual method, sampling based on timing, records the time location when the amplitude of a function, or an operator applied to the function, exceeds a threshold value (Gontier and Vetterli, 2014). The time encoding machine (TEM), which performs sampling based on timing, has first been defined by Lazar and Tóth (2003) as a real-time asynchronous mechanism that encodes the amplitude information of a function into a time sequence.…”

“…(2.22) Gontier and Vetterli (2014) consider shift invariant spaces with integer shifts V 2 (λ) = V 2 1 (λ), for which the inner product is defined as…”