Advanced Signal Processing Techniques on a Riemannian Manifold

基于黎曼流形的高级信号处理传统的信号处理技术是基于线性向量空间的。应用统计学和线性代数，在检测、估计和设计方面开发了复杂的理论和算法，并应用于语音、雷达、声纳、通信等。随着信号处理的发展，信号的特征也需要处理。其中一个特征是功率谱密度（PSD）矩阵。然而，PSD具有结构约束使其形成流形，必须使用黎曼距离（riemanan distance， RD）沿着流形表面进行测量。利用提升和投射理论，申请人推导出了封闭形式的RD表达式，并在信号处理应用中进行了测试，结果显著改善。目前的研究计划描述的调查，旨在开发加工技术和应用的基础上的几何和流形的RD。通过这些建立在新概念上的技术，可能会出现更好的信号处理算法。选择以下研究主题进行探索。1. PSD矩阵的概率分布及其在信号检测中的应用在信号处理中一个重要的必要条件是信号的统计特性。这个项目探讨了PSD矩阵在流形上的概率分布。然后，这些将用于建立似然比测试（LRT），以检测噪声中的信号。2. 优化信号设计-许多信号设计问题可以简化为PSD矩阵的设计。我们使用RD作为理想和信号PSD矩阵之间差异的度量。因此，信号和滤波器组的设计可以表述为流形上PSD矩阵之间RD的优化。我们采用申请人开发的交替升降和工程技术来解决这些问题。MIMO雷达信号设计和通信中的脉冲整形是两个可能受益于该项目的应用。3. 最小黎曼距离（MRD）估计和滤波-我们寻求PSD矩阵的最佳组合，并在流形上建立MRD估计原理，在概念上与MMSE滤波平行。我们将把这种新的MRD估计技术应用于自适应滤波、自适应阵列处理和自适应波束形成。4. PSD矩阵的均值和阶统计量-我们将根据RD为PSD矩阵的这些统计量建立数学上严格的定义，并开发算法来定位它们。我们还将推导这些统计的最佳组合，以便于对信号PSD矩阵进行稳健估计。上述提议的项目代表了从传统信号处理方法的根本背离，从向量空间转移到流形。随着对RD的测量更加精确，所开发的算法也可能更加准确。

Advanced Signal Processing On a Riemannian Manifold Traditional signal processing techniques are based on linear vector space. Applying statistics and linear algebra, sophisticated theory and algorithms in detection, estimation, and design have been developed and applied to speech, radar, sonar, communications, etc. As signal processing advances, signal features are subject to processing. One such feature is the power spectral density (PSD) matrix. However, PSD have structural constraints making them form a manifold, and measurement must be carried out along the surface of manifold using Riemannian distance (RD). Using a theory of lifting and projecting, the applicant derived closed-form expressions of RD which were tested on signal processing applications yielding results of dramatic improvement. The present research proposal describes investigations aiming at developing processing techniques and applications based on the geometry and RD of the manifold. Through these techniques founded on new concepts, superior algorithms for signal processing may emerge. The following topics of research are chosen for exploration. 1. Probability Distribution of the PSD Matrices and Application to Signal Detection - A major necessity in signal processing is the statistical properties of the signals. This project explores the probability distributions of the PSD matrices on the manifold. These will then be used to establish likelihood ratio tests (LRT) for detection of signals in noise. 2. Optimum Signal Designs - Many signal design problems can be reduced to the design of the PSD matrix. We use RD as the measure for discrepancies between the ideal and the signal PSD matrices. Thus, the design of signals and filter banks can be formulated as an optimization of the RD between PSD matrices on the manifold. We apply the alternating lift and project technique developed by the applicant to solve such problems. MIMO radar signal design and pulse shaping in communications are two applications that may benefit from this project. 3. Minimum Riemannian Distance (MRD) Estimation and Filtering - We seek an optimum combination of PSD matrices and establish the principle of MRD estimation on the manifold, parallel in concepts to MMSE filtering. We will apply this new MRD estimation technique to adaptive filtering, adaptive array processing, and adaptive beamforming. 4. kth Mean and Order Statistics of PSD Matrices - We will established mathematically rigorous definitions in terms of RD for these statistics of PSD matrices and develop algorithms to locate them. We will also derive optimum combinations of these statistics so that robust estimation of signal PSD matrices can be facilitated. The above proposed projects represent a fundamental departure from the traditional signal processing approach, shifting from the vector space to the manifold. With the more accurate measurement of RD, the algorithms so developed are likely to be more accurate.