Abstract
Sampling is classically performed by recording the amplitude of an input signal 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 integrate-and-fire neurons, for example, and has been studied by Lazar and T\'oth 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 $\mathbf{2\Omega}$-bandlimited signal, then $\mathbf{M}$-channel time encoding with shifted integrators can sample and perfectly reconstruct a signal with $\mathbf{M}$ 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 integrators of the machines. This is quite unlike classical multi-channel sampling, where unknown shifts between sampling devices pose a problem for perfect reconstruction.
Highlights
A LMOST all current sampling theories represent a signal using pairs
We study multi-channel time encoding, where a bandlimited signal is input to M > 1 time encoding machines that generate different outputs because of a shift in their
We show that, if a bandlimited signal with bandwidth Ω can be reconstructed using one time encoding machine (TEM), using a Projection onto Convex Sets (POCS) algorithm [23], [24], a bandlimited signal with bandwith M Ω can be reconstructed from M TEMs with the same parameters, as long as the machines are shifted with nonzero shifts
Summary
A LMOST all current sampling theories represent a signal using (time, amplitude) pairs. We show that, in time encoding, reconstruction from multi-channel sampling with unknown initial conditions is no harder than reconstruction using single-channel sampling This is not the case in classical sampling. We study multi-channel time encoding, where a bandlimited signal is input to M > 1 time encoding machines that generate different outputs because of a shift in their. We show that the reconstruction algorithm and conditions do not require the knowledge of the shifts, as long as these are nonzero This is an important improvement over [25], where we only showed that this bound could be achieved if shifts between the machines were spaced, which is not easy to achieve in practice. The bound we propose here generalizes to all shift configurations
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