Many empirical studies suggest that samples of continuous-time signals taken at locations randomly deviated from an equispaced grid (i.e., off-the-grid) can benefit signal acquisition, e.g., undersampling and anti-aliasing. However, explicit statements of such advantages and their respective conditions are scarce in the literature. This paper provides some insight on this topic when the sampling positions are known, with grid deviations generated i.i.d. from a variety of distributions. By solving a square-root LASSO decoder with an interpolation kernel we demonstrate the capabilities of nonuniform samples for compressive sampling, an effective paradigm for undersampling and anti-aliasing. For functions in the Wiener algebra that admit a discrete $s$-sparse representation in some transform domain, we show that $\mathcal{O}(s\log N)$ random off-the-grid samples are sufficient to recover an accurate $\frac{N}{2}$-bandlimited approximation of the signal. For sparse signals (i.e., $s \ll N$), this sampling complexity is a great reduction in comparison to equispaced sampling where $\mathcal{O}(N)$ measurements are needed for the same quality of reconstruction (Nyquist-Shannon sampling theorem). We further consider noise attenuation via oversampling (relative to a desired bandwidth), a standard technique with limited theoretical understanding when the sampling positions are non-equispaced. By solving a least squares problem, we show that $\mathcal{O}(N\log N)$ i.i.d. randomly deviated samples provide an accurate $\frac{N}{2}$-bandlimited approximation of the signal with suppression of the noise energy by a factor $\sim\frac{1}{\sqrt{\log N}}$.

翻译：大量实证研究表明，在均匀网格位置随机偏移（即非网格）处获取的连续时间信号样本，能够有益于信号获取，例如欠采样和抗混叠。然而，文献中关于此类优势及其相应条件的明确阐述较为罕见。本文在采样位置已知、网格偏移由多种分布独立同分布生成的情况下，对该问题提供了深入见解。通过求解带有插值核的平方根LASSO解码器，我们展示了非均匀样本在压缩采样（一种有效的欠采样与抗混叠范式）中的能力。对于在某种变换域中具有离散$s$稀疏表示的Wiener代数函数，我们证明：$\mathcal{O}(s\log N)$个随机非网格样本足以恢复信号的高精度$\frac{N}{2}$带限近似。对于稀疏信号（即$s \ll N$），该采样复杂度相较于等间隔采样（根据奈奎斯特-香农采样定理，需要$\mathcal{O}(N)$个测量值才能达到相同重构质量）实现了显著降低。我们进一步考虑了通过过采样（相对于目标带宽）进行噪声抑制的技术——当采样位置非等间隔时，该标准方法的理论理解仍十分有限。通过求解最小二乘问题，我们证明：$\mathcal{O}(N\log N)$个独立同分布随机偏移样本能够提供信号的高精度$\frac{N}{2}$带限近似，同时噪声能量被抑制约$\sim\frac{1}{\sqrt{\log N}}$倍。