Sampling and Reconstruction of Bandlimited Signals With Multi-Channel Time Encoding

Abstract

Sampling is classically performed by recording the amplitude of the input at given time instants; however, sampling and reconstructing a signal using multiple devices in parallel becomes a more difficult problem to solve when the devices have an unknown shift in their clocks.Alternatively, one can record the times at which a signal (or its integral) crosses given thresholds. This can model integrateand-fire neurons, for example, and has been studied by Lazar and Tóth under the name of "Time Encoding Machines". This sampling method is closer to what is found in nature.In this paper, we show that, when using time encoding machines, reconstruction from multiple channels has a more intuitive solution, and does not require the knowledge of the shifts between machines. We show that, if single-channel time encoding can sample and perfectly reconstruct a 2Ω-bandlimited signal, then M -channel time encoding can sample and perfectly reconstruct a signal with M times the bandwidth. Furthermore, we present an algorithm to perform this reconstruction and prove that it converges to the correct unique solution, in the noiseless case, without knowledge of the relative shifts between the machines. This is quite unlike classical multi-channel sampling, where unknown shifts between sampling devices pose a problem for perfect reconstruction.

Encoding and Decoding Mixed Bandlimited Signals Using Spiking Integrate-and-Fire Neurons

“…We extended the work to understand single-signal Mchannel time encoding by building on the approach that Lazar and Tóth developed [3]. We showed that if such a 2Ωbandlimited, c-bounded signal x(t) is sampled noiselessly using M IF-TEMs with the same parameters κ, δ and b but with nonzero shifts between their integrators α (j) , j = 1 · · · M , then x(t) can be reconstructed from its samples t [9]. Essentially, if a 2Ω-bandlimited signal can be reconstructed using one TEM, then a 2M Ω-bandlimited signal can be reconstructed using M TEMs with the same parameters.…”

“…In previous work, the constraints for reconstructability were written in terms of the bandwidth Ω, the signal bound c, and the parameters of the TEMs κ, δ and b [3,9,10,11,13]. These constraints arise because of a relationship between the spiking rate of each machine r (j) and the parameters of the machine.…”

“…As we did previously [9], we use a projection onto convex sets algorithm to reconstruct a signal from its spike times.…”

“…More specific definitions of time encoding can adopt a more biological approach. Time encoding machines can resemble integrate-and-fire neurons with perfect integrators [9], or leaky integrate-and-fire neurons with refractory periods [10] or even Hodgkin-Huxley neurons for more biological resemblence [11]. Moreover, time encoding machines can be used in different configurations, such as single-signal single-channel encoding, or single-signal multi-channel encoding which improves signal reconstruction [6,9,10].…”

Guaranteed Reconstruction from Integrate-and-Fire Neurons with Alpha Synaptic Activation

“…[8,9]. Time encoding systems have been previously used for reconstructing signals in shift-invariant spaces [10][11][12][13][14][15], as well as FRI signals in [16][17][18][19] using exponential spline sampling kernels and in [20] using hyperbolic secant kernels.…”