Time delay estimation arises in many applications in which a multipath medium has to be identified from pulses transmitted through the channel. Various approaches have been proposed in the literature to identify time delays introduced by multipath environments. However, these methods either operate on the analog received signal, or require high sampling rates in order to achieve reasonable time resolution. In this paper, our goal is to develop a unified approach to time delay estimation from low rate samples of the output of a multipath channel. Our methods result in perfect recovery of the multipath delays from samples of the channel output at the lowest possible rate, even in the presence of overlapping transmitted pulses. This rate depends only on the number of multipath components and the transmission rate, but not on the bandwidth of the probing signal. In addition, our development allows for a variety of different sampling methods. By properly manipulating the low-rate samples, we show that the time delays can be recovered using the well-known ESPRIT algorithm. Combining results from sampling theory with those obtained in the context of direction of arrival estimation methods, we develop necessary and sufficient conditions on the transmitted pulse and the sampling functions in order to ensure perfect recovery of the channel parameters at the minimal possible rate. Our results can be viewed in a broader context, as a sampling theorem for analog signals defined over an infinite union of subspaces.
在这项工作中，我们研究了鲁棒子空间跟踪问题，并获得了它的前两个可证明的保证之一。鲁棒子空间跟踪的目标是跟踪顺序到达的数据向量，位于一个缓慢变化的低维子空间，同时是由添加剂稀疏离群值的腐败鲁棒。它也可以被解释为鲁棒PCA（RPCA）的动态（时变）扩展，具有较小的差异，即RPCA也需要较短的跟踪延迟。我们开发了一种递归投影压缩感知算法，我们称之为近乎最优的基于ReProCS的压缩感知算法（ReProCS-NORST），因为它的跟踪延迟接近最优。我们证明，NORST解决了两个问题的非线性和动态RPCA下削弱标准RPCA的假设，两个简单的额外的假设（慢子空间变化和最离群值幅度下界），和一些小的假设。我们的保证表明，NORST享有接近最佳的跟踪延迟$O(r \log n \log(1/ε))$。其在子空间改变时间之间所需的延迟是相同的，并且其存储器复杂度是该值的$n$倍。因此，这两个也是接近最佳的。这里，$n$是环境空间维度，$r$是子空间的维度，并且$ε$是跟踪精度。NORST还具有最好的离群值容忍度与所有以前的RPCA或RPCA方法相比，无论是理论上还是经验上（包括真实视频），都不需要任何关于如何生成离群值支持的模型。这是可能的，因为它使用了额外的假设。
Robust Recovery of Subspace Structures by Low-Rank Representation
Shanghai Jiao Tong University（上海交通大学）；Microsoft Research Asia（微软亚洲研究院）；Peking University（北京大学）；National University of Singapore（新加坡国立大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Columbia University（哥伦比亚大学）
