This work investigates the parameter estimation performance of super-resolution line spectral estimation using atomic norm minimization. The focus is on analyzing the algorithm's accuracy of inferring the frequencies and complex magnitudes from noisy observations. When the Signal-to-Noise Ratio is reasonably high and the true frequencies are separated by $O(\frac{1}{n})$, the atomic norm estimator is shown to localize the correct number of frequencies, each within a neighborhood of size $O(\sqrt{\log n/n^3}\sigma )$ of one of the true frequencies. Here n is half the number of temporal samples and ${\sigma }^2$ is the Gaussian noise variance. The analysis is based on a primal–dual witness construction procedure. The obtained error bound matches the Cramér–Rao lower bound up to a logarithmic factor. The relationship between resolution (separation of frequencies) and precision or accuracy of the estimator is highlighted. Our analysis also reveals that the atomic norm minimization can be viewed as a convex way to solve a ${\ell }_1$-norm regularized, nonlinear and nonconvex least-squares problem to global optimality.

这项工作研究使用原子范数最小化的超分辨率线谱估计的参数估计性能。重点在于分析算法从噪声观测值推断频率和复杂幅度的准确性。当信噪比相当高且真实频率由$Ø（\frac{\mathrm{1个}}{ñ}）$，显示了原子范数估计器可以定位正确数量的频率，每个频率都在一个大小的邻域内 $Ø（\sqrt{\mathrm{日志}ñ/{ñ}^3}\sigma ）$真实频率之一。这里n是时间样本数量的一半，而${\sigma }^2$是高斯噪声方差。该分析基于原始-双重证人建设程序。所获得的误差范围与Cramér-Rao下限相匹配，直至对数因子。突出显示了分辨率（频率分离）与估计器的精度或准确性之间的关系。我们的分析还表明，原子范数最小化可以看作是解决原子力的凸面方法。${\ell }_{\mathrm{1个}}$-范数正则化，非线性和非凸最小二乘问题的全局最优性。