Shannon's sampling theorem is one of the cornerstone topics that is well understood and explored, both mathematically and algorithmically. That said, practical realization of this theorem still suffers from a severe bottleneck due to the fundamental assumption that the samples can span an arbitrary range of amplitudes. In practice, the theorem is realized using so-called analog-to-digital converters (ADCs) which clip or saturate whenever the signal amplitude exceeds the maximum recordable ADC voltage thus leading to a significant information loss. In this paper, we develop an alternative paradigm for sensing and recovery, called the Unlimited Sampling Framework. It is based on the observation that when a signal is mapped to an appropriate bounded interval via a modulo operation before entering the ADC, the saturation problem no longer exists, but one rather encounters a different type of information loss due to the modulo operation. Such an alternative setup can be implemented, for example, via so-called folding or self-reset ADCs, as they have been proposed in various contexts in the circuit design literature. The key task that we need to accomplish in order to cope with this new type of information loss is to recover a bandlimited signal from its modulo samples. In this paper we derive conditions when perfect recovery is possible and complement them with a stable recovery algorithm. The sampling density required to guarantee recovery is independent of the maximum recordable ADC voltage and depends on the signal bandwidth only. Our recovery guarantees extend to measurements affected by bounded noise, which includes the case of round-off quantization. Numerical experiments validate our approach. Applications of the unlimited sampling paradigm can be found in a number of fields such as signal processing, communication and imaging.


翻译:Shannon的抽样理论是数学和算法两方面都非常理解和探索的奠基主题之一。 也就是说, 实际实现该理论仍然受到严重瓶颈的影响, 因为从根本上假设样本可以跨越任意的振幅范围。 实际上, 当信号振动超过可记录的最大振动 ADC 电压时, 就会闪烁或饱和, 从而导致大量信息损失。 在本文中, 我们为感测和复原开发开发了一种替代模式, 称之为不限量的取样框架。 它基于这样的观察: 当一个信号在进入 ADC 之前, 可以通过一个随机的振动操作绘制出一个适当的固定间隔时, 饱和问题不再存在, 但有一个相反的情况是, 当信号振动时, 信号会发生不同类型的信息损失。 例如, 只能通过所谓的软化或自设的电压电压电压电压电路路路段 。 在电路段的回收过程中, 我们发现一个最精确的电路路路路路段的恢复模型, 需要一个最精确的电路路段回路段的电路路路路路段的电路路路路路。 路路路路路路路路路路路路路段的电路路路路路路段的电路路路路路路路路路路流的电路由, 路路路路段的电路由, 路段的电路路路路段路由, 路路路路由, 路段路路路路路路路段路段路由的电路路路路路由, 路路路路路路路路路路由, 路路路由, 路由, 路由的电路由,我们需要, 路由一个关键路由, 路由, 路路路路路路由,要路由, 路由,要路由路路由路路由,我们需要,要路由,我们需要,要路路路路段路段路段路段路由,要路由,要路路路路由,要路由,要路由,要路,要路,我们,要路,要路,要路,要路,要路,要路由路由路,要路,我们,我们,我们,我们,我们,要路,

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