英语翻译In the CS theory framework,the knowledge of the signal s

英语翻译
In the CS theory framework,the knowledge of the signal sparsity allows signal reconstruction from a small number of measurements.In the array signal processing framework,this means that the knowledge of the spatial signal sparsity allows to achieve a high bearing angle resolution using short sensor array.Our spatial interpretation of the compressive sensing relates to the pioneering results in [27]-[29],where the compressive beamforming was proposed for the problem of direction of arrival estimation.However,these works,similarly to the majority of the published works exploit the temporal sparsity of the received signals.
The paper is organized as follows.Results from the com- pressive sensing theory are summarized in Section II.The addressed problem is formulated in Section III.The spatial compressive sampling approach for the field directionality estimation is presented in Section IV.Estimation performance of the proposed spatial CS-based approach is evaluated via simulations in Section V.Our conclusions are summarized in Section VI.
II.COMPRESSIVE SENSING
This section summarizes notations and that main results from the CS theory [15]-[26].The CS theory addresses the following underdetermined and noisy problem:
where xJ is a pure J-sparse signal,and where constants c0 and c1 are well behaved and small.Note that this results suggest that when the signal x is J-sparse,the estimation error is bounded by the energy of the noise w only.
This framework provides an opportunity for the sensing matrix Φ design [17].One should find the sensing matrix that obeys the RIP and allows to recover as many elements of the signal x from M measurements,as possible.The RIP that was proposed in [18] and [19] is closely related to the incoherency between the sparsity and the measurements basses,providing an efficient way to obtain the sensing matrix that satisfy it.In [23] it was shown that the incoherency property allows exact reconstruction of the signal that is sparse in one basis using the sensing matrix from the second incoherent basis.It was shown in [23] that the time-frequency pair of orhtonormal basses that are related via the Fourier transform is highly incoherent.Moreover it was shown that the time-frequency pair of spikes and complex sinusoids yields the most mutually incoherent pair providing the best sparsity conditions.This property was used in [19] and [23] to show that considering that the signal x is sparse in the basis of spikes,the minimal number of the measurements that is required for its recovery,using the partial Fourier matrix of complex sinusoids with M uniformly selected rows,is:M ≥ c2 J/(log N)4 .

在CS理论框架中,知道信号稀疏度可以允许由数量少的测量值进行信号重构.在阵列信号处理框架中,这意味着,知道空间信号的稀疏度就可用短的传感器阵列达到高的象限角分辨率.我们对压缩传感的空间解释与文献[27]-[29]中的先导结果有关,其中,为了到达估计的方向问题提出了压缩波束形成.不过这些研究和大多数已发表的论文一样,研究的是被接收信号的时间稀疏度.
本文的组织如下.由压缩传感理论得到的结果被归纳在第2节中.所着重解决的问题在第3节中用公式加以描述.用于场方向性估计的空间压缩取样方法在第4节中介绍.所提出的基于空间CS的方法在第5节中通过仿真进行了评价.我们的结论则归纳在第6节中.
2 压缩传感
本小节归纳了CS理论的符号表示及其主要结果[15]-[26].CS理论着重解决这在解决中的问题和噪声问题：
式中Φ为一个已知的尺度为M×N（M＜N）的已知传感矩阵.CS理论的主要目标是恢复长度为N的信号 ,形成长度为M的,受到白零均值高斯噪声w(具有协方差矩阵Γw= )污染的测量值 .这一提法不当的问题的解只有在信号x的某些性质可得到考虑时才有可能[17].CS理论认为信号x在某种高阶稀疏基础 中是“稀疏的”或“可压缩的”,提供了以下的表示式：
式中,稀疏度基础矩阵Ψ的列是来自稀疏度基础的矢量,而尺度N×1的稀疏度矢量的系数d只包含J＜＜N的非零元素.
传感矩阵Φ被认为以J-受约束等距常数δJ遵循RIP,这是对于任何J-稀疏信号满足式（3）的最小值：
对于式（1）的噪声情况,适用于任何信号x的通用估计量（不一定是稀疏的）在文献[20],[21]中做了介绍,其凸优化如下：
式中 .已经证明,假设 1,那么此估计量的性能被约束如下：
式中xj为一个纯的J-稀疏信号,这里的常数c0和c1表现良好而且较小.请注意,这一结果告诉我们,当信号x为j-稀疏时,估计误差只受噪声w的能量的约束.
这一框架为传感矩阵Φ的设计提供了一个机会[17].人们应该发现遵循RIP并允许从M个测量值恢复信号x尽可能多的元件的传感矩阵.在文献[18]和[19]中提出的RIP与稀疏度和测量低沉音之间的不相干性密切相关,从而为获得满足它的传感矩阵提供了一种高效的方法.在文献[23]中证明了,不相干性性质可以允许用传感矩阵从第二个不相干的基础准确地重构在一个基础上是稀疏的信号.文献[23]证明了,通过傅里叶变换相关的规范正交的低沉音的时间－频率对是高度不相干的.而且还证明了,尖峰和复杂正弦波的时间－频率对产生相互最不相干的对,从而提供了最好的稀疏度条件.这一性质被用到了文献[19]和[23]中,以证明,如果考虑信号x在尖峰基础上是稀疏的,那么用于这一恢复（用具有M行均匀选择行的复杂正弦波的部分傅立叶矩阵）所需的测量的最小数目应该为 M ≥ c2 J/(log N)4 ..