Super-Resolution Limit of the ESPRIT Algorithm
Super-Resolution Limit of the ESPRIT Algorithm
复制标题
DOI：
10.1109/tit.2020.2974174
复制
发表时间：
2020-07-01
影响因子：
2.5
通讯作者：
Fannjiang, Albert
中科院分区：
文献类型：
--
作者：
Li, Weilin;Liao, Wenjing;Fannjiang, Albert
点目标的成像问题可以表述为从其M+1个连续的噪声傅立叶系数估计未知的原子测度。这个反问题的标准分辨率是1/M，超分辨率是指以更高的分辨率分辨原子的能力。当任何两个原子的距离小于1/M时，这个恢复问题是非常具有挑战性的，许多现有的算法要么不能处理这种情况，要么需要对测量的符号进行限制性假设。ESPRIT是一种不依赖于测度符号的有效方法。本文利用Vandermonde矩阵的最小奇异值给出了ESPRIT算法支撑匹配距离的一个显式误差界。当支持由多个分离良好的簇组成并且噪声足够小时，ESPRIT的支持误差像SRF（2 lambda-2）xNoise一样缩放，其中超分辨率因子（SRF）控制问题的难度，lambda是最大簇的基数。我们的误差界匹配的最小-最大速率的一个特殊的模型与一团紧密间隔的原子的一个因素M在小噪声制度，因此建立了ESPRIT的近最优性。数值实验验证了我们的理论。
The problem of imaging point objects can be formulated as estimation of an unknown atomic measure from its M+1 consecutive noisy Fourier coefficients. The standard resolution of this inverse problem is 1/M and super-resolution refers to the capability of resolving atoms at a higher resolution. When any two atoms are less than 1/M apart, this recovery problem is highly challenging and many existing algorithms either cannot deal with this situation or require restrictive assumptions on the sign of the measure. ESPRIT is an efficient method which does not depend on the sign of the measure. This paper provides an explicit error bound on the support matching distance of ESPRIT in terms of the minimum singular value of Vandermonde matrices. When the support consists of multiple well-separated clumps and noise is sufficiently small, the support error by ESPRIT scales like SRF(2 lambda-2)xNoise, where the Super-Resolution Factor (SRF) governs the difficulty of the problem and lambda is the cardinality of the largest clump. Our error bound matches the min-max rate of a special model with one clump of closely spaced atoms up to a factor of M in the small noise regime, and therefore establishes the near-optimality of ESPRIT. Our theory is validated by numerical experiments.