Sampling theory in fractional Fourier Transform (FrFT) domain has been studied extensively in the last decades. This interest stems from the ability of the FrFT to generalize the traditional Fourier Transform, broadening the traditional concept of bandwidth and accommodating a wider range of functions that may not be bandlimited in the Fourier sense. Beyond bandlimited functions, sampling and recovery of sparse signals has also been studied in the FrFT domain. Existing methods for sparse recovery typically operate in the transform domain, capitalizing on the spectral features of spikes in the FrFT domain. Our paper contributes two new theoretical advancements in this area. First, we introduce a novel time-domain sparse recovery method that avoids the typical bottlenecks of transform domain methods, such as spectral leakage. This method is backed by a sparse sampling theorem applicable to arbitrary FrFT-bandlimited kernels and is validated through a hardware experiment. Second, we present Cram\'er-Rao Bounds for the sparse sampling problem, addressing a gap in existing literature.

翻译：分数阶傅里叶变换（FrFT）域的采样理论在过去几十年中得到了广泛研究。这一研究兴趣源于FrFT对传统傅里叶变换的推广能力，它拓宽了传统带宽的概念，能够容纳更多在傅里叶意义上可能不具备带限性质的函数。除带限函数外，稀疏信号在FrFT域中的采样与恢复问题也得到了研究。现有的稀疏恢复方法通常在变换域中运作，利用FrFT域中尖峰信号的频谱特征。本文在该领域贡献了两项新的理论进展。首先，我们提出了一种新颖的时域稀疏恢复方法，避免了变换域方法中典型的瓶颈（如频谱泄漏）。该方法基于适用于任意FrFT带限核的稀疏采样定理，并通过硬件实验验证。其次，针对稀疏采样问题，我们给出了克拉美-拉奥界，填补了现有文献的空白。