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 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 $\mathbf {2\Omega }$-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.

中文翻译：

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多通道时间编码限带信号的采样与重构

典型的采样是通过记录给定时刻输入信号的幅度来执行的；然而，当设备的时钟发生未知偏移时，使用多个设备并行采样和重建信号成为一个更难解决的问题。或者，可以记录信号（或其积分）穿过给定阈值的时间。例如，这可以模拟集成和激发神经元，并已被 Lazar 和 Tóth 以“时间编码机器”的名义进行了研究。这种采样方法更接近自然界中发现的方法。在本文中，我们表明，当使用时间编码机器时，从多通道重建具有更直观的解决方案，并且不需要机器之间转换的知识。我们表明，$\mathbf {2\Omega }$-带限信号，然后 $\mathbf {M}$-带有移位积分器的通道时间编码可以采样并完美地重建信号 $\mathbf {M}$倍带宽。此外，我们提出了一种算法来执行这种重建，并证明它在无噪声情况下收敛到正确的唯一解，而无需了解机器积分器之间的相对偏移。这与经典的多通道采样完全不同，在经典多通道采样中，采样设备之间的未知偏移会给完美重建带来问题。