Phase retrieval is in general a non-convex and non-linear task and the corresponding algorithms struggle with the issue of local minima. We consider the case where the measurement samples within typically very small and disconnected subsets are coherently linked to each other - which is a reasonable assumption for our objective of antenna measurements. Two classes of measurement setups are discussed which can provide this kind of extra information: multi-probe systems and holographic measurements with multiple reference signals. We propose several formulations of the corresponding phase retrieval problem. The simplest of these formulations poses a linear system of equations similar to an eigenvalue problem where a unique non-trivial null-space vector needs to be found. Accurate phase reconstruction for partially coherent observations is, thus, possible by a reliable solution process and with judgment of the solution quality. Under ideal, noise-free conditions, the required sampling density is less than two times the number of unknowns. Noise and other observation errors increase this value slightly. Simulations for Gaussian random matrices and for antenna measurement scenarios demonstrate that reliable phase reconstruction is possible with the presented approach.
翻译:阶段检索一般是非隐形和非线性任务,对应的算法与当地微型问题相对应。我们考虑的情况是,通常非常小和不相连接的子集中的测量样品相互连贯地连接,这是我们天线测量目标的合理假设。讨论的测量组别分为两类,可以提供这种额外信息:多谱系统和带有多个参考信号的全息测量。我们建议了相应的阶段检索问题的若干配方。这些配方最简单的配方构成一个线性方程系统,类似于需要找到独特的非三角性无空间矢量的等式问题。因此,通过可靠的解决方案进程和对溶性质量的判断,可以对部分一致的观测进行准确的阶段重建。在理想情况下,无噪音条件下,所需的取样密度比未知数少两倍。噪音和其他观测错误会增加这一数值。高斯随机矩阵和天线测量假设情景的模拟显示,可以按所述方法进行可靠的阶段重建。