使用随机探针的稀疏信道分离
Sparse Channel Separation using Random Probes
AI总结：
本文提出通过同时激活多个源并使用随机探针信号，在信道响应稀疏的条件下，将叠加响应分离为单个信道响应，并利用压缩感知理论推导了保证高概率分离的探针信号长度下界。
摘要：
本文考虑估计多个源-接收器对之间的信道响应（或格林函数）的问题。通常，信道响应是逐个估计的：单个源发送已知的探针信号，接收器测量与信道响应卷积后的探针信号，并通过反卷积恢复响应。在本文中，我们证明如果信道响应是稀疏的且探针信号是随机的，那么通过同时激活所有源，我们可以显著减少探测信道所需的总时间。当所有源同时激活时，接收器测量所有信道响应与各自探针信号卷积的叠加。将这个累积响应分离为单个信道响应可以表述为一个线性逆问题。我们证明，即使探针信号相对较短，相应的线性方程组变得严重欠定，信道响应分离也是可能的（且稳定的）。我们推导了源信号长度的理论下界，保证以高概率实现这种分离。该下界通过将问题置于寻找欠定方程组稀疏解的背景下，并利用压缩感知理论的数学工具得到。最后，我们讨论了这些结果的一些实际应用，包括地震成像的正演模拟、多输入多输出通信中的信道均衡，以及通过编码孔径增加成像系统的视场。
英文摘要：
This paper considers the problem of estimating the channel response (or Green's function) between multiple source-receiver pairs. Typically, the channel responses are estimated one-at-a-time: a single source sends out a known probe signal, the receiver measures the probe signal convolved with the channel response, and the responses are recovered using deconvolution. In this paper, we show that if the channel responses are sparse and the probe signals are random, then we can significantly reduce the total amount of time required to probe the channels by activating all of the sources simultaneously. With all sources activated simultaneously, the receiver measures a superposition of all the channel responses convolved with the respective probe signals. Separating this cumulative response into individual channel responses can be posed as a linear inverse problem. We show that channel response separation is possible (and stable) even when the probing signals are relatively short in spite of the corresponding linear system of equations becoming severely underdetermined. We derive a theoretical lower bound on the length of the source signals that guarantees that this separation is possible with high probability. The bound is derived by putting the problem in the context of finding a sparse solution to an underdetermined system of equations, and then using mathematical tools from the theory of compressive sensing. Finally, we discuss some practical applications of these results, which include forward modeling for seismic imaging, channel equalization in multiple-input multiple-output communication, and increasing the field-of-view in an imaging system by using coded apertures.
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