We study recovery of amplitudes and nodes of a finite impulse train from noisy frequency samples. This problem is known as super-resolution under sparsity constraints and has numerous applications. An especially challenging scenario occurs when the separation between Dirac pulses is smaller than the Nyquist-Shannon-Rayleigh limit. Despite large volumes of research and well-established worst-case recovery bounds, there is currently no known computationally efficient method which achieves these bounds in practice. In this work we combine the well-known Prony's method for exponential fitting with a recently established decimation technique for analyzing the super-resolution problem in the above mentioned regime. We show that our approach attains optimal asymptotic stability in the presence of noise, and has lower computational complexity than the current state of the art methods.

翻译：我们研究从含噪频率样本中恢复有限脉冲串的幅度和节点位置的问题。该问题在稀疏约束下被称为超分辨，具有广泛的应用。当狄拉克脉冲之间的间隔小于奈奎斯特-香农-瑞利极限时，会出现尤为具有挑战性的场景。尽管已有大量研究且最坏情况下的恢复界限已完善建立，但目前尚无已知的计算高效方法能在实际中达到这些界限。本文我们将经典的指数拟合Prony方法与近期建立的用于分析上述条件下超分辨问题的降采样技术相结合。我们证明该方法在存在噪声时能达到最优渐近稳定性，且计算复杂度低于当前最先进的方法。