Advanced Signal Processing Techniques on a Riemannian Manifold

黎曼流形上的先进信号处理技术

• 批准号: RGPIN-2019-05415
• 负责人: Wong, Kon
• 金额: $ 2.04万
• 依托单位: McMaster University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=682065
• 关键词: Advanced; Signal; Processing; Techniques; Riemannian

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. **

黎曼流形上的高级信号处理******传统信号处理技术基于线性向量空间。应用统计学和线性代数，在检测、估计和设计方面已经开发出复杂的理论和算法，并应用于语音、雷达、声纳、通信等。随着信号处理的进步，信号特征也需要进行处理。其中一项特征是功率谱密度 (PSD) 矩阵。然而，PSD的结构限制使其形成流形，并且必须使用黎曼距离（RD）沿流形表面进行测量。使用提升和投影理论，申请人导出了 RD 的封闭式表达式，该表达式在信号处理应用上进行了测试，产生了显着改进的结果。本研究计划描述了旨在开发基于流形几何形状和 RD 的加工技术和应用的研究。通过这些基于新概念的技术，可能会出现更好的信号处理算法。选择以下研究课题进行探索。*****1． PSD 矩阵的概率分布及其在信号检测中的应用 信号处理的一个主要必要条件是信号的统计特性。该项目探索 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。最佳信号设计 许多信号设计问题都可以归结为 PSD 矩阵的设计。我们使用 RD 作为理想 PSD 矩阵与信号 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示为流形上 PSD 矩阵之间 RD 的优化。我们应用申请人开发的交替升力和投影技术来解决此类问题。通信中的 MIMO 雷达信号设计和脉冲整形是可以从该项目中受益的两个应用。******3。最小黎曼距离 (MRD) 估计和过滤 我们寻求 PSD 矩阵的最佳组合，并建立流形上的 MRD 估计原理，与 MMSE 过滤的概念平行。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 kth 均值和阶次统计 我们将为 PSD 矩阵的这些统计量建立数学上严格的 RD 定义，并开发算法来定位它们。我们还将得出这些统计数据的最佳组合，以便可以促进信号 PSD 矩阵的鲁棒估计。****** 上述提出的项目代表了与传统信号处理方法的根本背离，从向量空间转向流形。随着 RD 测量的更加准确，如此开发的算法可能会更加准确。 **