Fast and Guaranteed Blind Multichannel Deconvolution Under a Bilinear System Model
Fast and Guaranteed Blind Multichannel Deconvolution Under a Bilinear System Model
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DOI：
10.1109/tit.2018.2840711
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发表时间：
2016-10
影响因子：
2.5
通讯作者：
Kiryung Lee;Ning Tian;J. Romberg
中科院分区：
文献类型：
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作者：
Kiryung Lee;Ning Tian;J. Romberg
我们考虑了多通道盲反卷积问题，我们观察了多个通道的输出，这些通道都是由相同的未知输入激发的。从这些观察中，我们希望估计每个通道的脉冲响应。我们表明，如果通道遵循双线性模型，则该问题是适定的，其中通道响应的集合被建模为位于低维子空间中，但每个通道由独立增益调制。在这个模型下，我们展示了如何通过最小化非凸集上的二次函数来找到信道估计。分析了求解该非凸规划的两种方法，并给出了各自的性能保证。第一种是交替特征向量的方法，它将程序分解为一系列特征值问题。第二种是截断幂迭代，它可以被粗略地解释为一种寻找对称矩阵的最大特征向量的方法，它遵循我们的双线性模型的附加约束。与大多数非凸优化算法一样，这两种算法的性能高度依赖于有一个好的起点。我们将展示如何从通道测量中构造这样一个起点。我们的性能保证是非渐近的，并提供了每个信道观察到的样本数量的充分条件，以保证信道估计具有一定的准确性。我们的分析使用了一个带有随机绘制的“通用”子空间的模型，并且我们以高概率显示了性能界限。在数学上，关键估计是通过量化某些随机矩阵的特征向量近似其平均值的特征向量的程度而得出的。我们还提出了一系列的数值结果，证明了经验性能与所提出的理论是一致的。
We consider the multichannel blind deconvolution problem where we observe the output of multiple channels that are all excited with the same unknown input. From these observations, we wish to estimate the impulse responses of each of the channels. We show that this problem is well-posed if the channels follow a bilinear model where the ensemble of channel responses is modeled as lying in a low-dimensional subspace but with each channel modulated by an independent gain. Under this model, we show how the channel estimates can be found by minimizing a quadratic function over a non-convex set. We analyze two methods for solving this non-convex program, and provide performance guarantees for each. The first is a method of alternating eigenvectors that breaks the program down into a series of eigenvalue problems. The second is a truncated power iteration, which can roughly be interpreted as a method for finding the largest eigenvector of a symmetric matrix with the additional constraint that it adheres to our bilinear model. As with most non-convex optimization algorithms, the performance of both of these algorithms is highly dependent on having a good starting point. We show how such a starting point can be constructed from the channel measurements. Our performance guarantees are non-asymptotic, and provide a sufficient condition on the number of samples observed per channel in order to guarantee channel estimates of certain accuracy. Our analysis uses a model with a “generic” subspace that is drawn at random, and we show the performance bounds hold with high probability. Mathematically, the key estimates are derived by quantifying how well the eigenvectors of certain random matrices approximate the eigenvectors of their mean. We also present a series of numerical results demonstrating that the empirical performance is consistent with the presented theory.