This paper develops a new mathematical framework for denoising in blind two-dimensional (2D) super-resolution upon using the atomic norm. The framework denoises a signal that consists of a weighted sum of an unknown number of time-delayed and frequency-shifted unknown waveforms from its noisy measurements. Moreover, the framework also provides an approach for estimating the unknown parameters in the signal. We prove that when the number of the observed samples satisfies certain lower bound that is a function of the system parameters, we can estimate the noise-free signal, with very high accuracy, upon solving a regularized least-squares atomic norm minimization problem. We derive the theoretical mean-squared error of the estimator, and we show that it depends on the noise variance, the number of unknown waveforms, the number of samples, and the dimension of the low-dimensional space where the unknown waveforms lie. Finally, we verify the theoretical findings of the paper by using extensive simulation experiments.

中文翻译：

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二维二维超分辨中原子范数降噪的数学理论

本文开发了一种新的数学框架，用于在使用原子范数的情况下以盲二维（2D）超分辨率进行去噪。框架对信号进行降噪处理，该信号由来自其噪声测量的未知数量的时间延迟和频移未知波形的加权和组成。此外，该框架还提供了一种用于估计信号中未知参数的方法。我们证明，当观察到的样本数量满足某个下限（取决于系统参数）时，我们可以在解决规则化的最小二乘原子范数最小化问题时，以非常高的精度估算无噪声信号。我们推导了估算器的理论均方误差，并表明它取决于噪声方差，未知波形的数量，样本的数量，以及未知波形所在的低维空间的维数。最后，我们通过广泛的模拟实验验证了本文的理论发现。