The recovery of Dirac impulses, or spikes, from filtered measurements is a classical problem in signal processing. As the spikes lie in the continuous domain while measurements are discrete, this task is known as super-resolution or off-the-grid sparse recovery. Despite significant theoretical and algorithmic advances over the past decade, these developments often overlook critical challenges at the analog-digital interface. In particular, when spikes exhibit strong-weak amplitude disparity, conventional digital acquisition may result in clipping of strong components or loss of weak ones beneath the quantization noise floor. This motivates a broader perspective: super-resolution must simultaneously resolve both amplitude and temporal structure. Under a fixed bit budget, such information loss is unavoidable. In contrast, the emerging theory and practice of the Unlimited Sensing Framework (USF) demonstrate that these fundamental limitations can be overcome. Building on this foundation, we demonstrate that modulo encoding within USF enables digital super-resolution by enhancing measurement precision, thereby unlocking temporal super-resolution beyond conventional limits. We develop new theoretical results that extend to non-bandlimited kernels commonly encountered in practice and introduce a robust algorithm for off-the-grid sparse recovery. To demonstrate practical impact, we instantiate our framework in the context of time-of-flight imaging. Both numerical simulations and hardware experiments validate the effectiveness of our approach under low-bit quantization, enabling super-resolution in amplitude and time.

翻译：从滤波测量中恢复狄拉克脉冲（或称尖峰）是信号处理中的一个经典问题。由于尖峰位于连续域而测量值是离散的，该任务被称为超分辨率或离网格稀疏恢复。尽管过去十年间在理论和算法上取得了显著进展，但这些发展往往忽视了模数接口处的关键挑战。具体而言，当尖峰呈现强弱幅度差异时，传统的数字采集可能导致强分量被削波或弱分量被淹没在量化噪声基底之下。这促使我们采取更广阔的视角：超分辨率必须同时解析幅度和时间结构。在固定比特预算下，此类信息损失是不可避免的。相比之下，新兴的无限制感知框架理论及实践表明，这些根本性限制是可以被克服的。基于此基础，我们证明USF中的模数编码能够通过提升测量精度来实现数字超分辨率，从而突破传统限制实现时间超分辨率。我们提出了适用于实践中常见非带限核的新理论结果，并引入了一种鲁棒的离网格稀疏恢复算法。为展示实际影响，我们在飞行时间成像场景中实例化了该框架。数值仿真与硬件实验均验证了本方法在低比特量化下实现幅度与时间超分辨率的有效性。