Accurate Digital Representations for Overcomplete Data Expansions

超完备数据扩展的准确数字表示

• 批准号: 0504924
• 负责人: John Benedetto
• 金额: $ 40.8万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-08-15 至 2009-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0504924&HistoricalAwards=false
• 关键词: Accurate; Digital; Representations; Overcomplete; Data

Digital data is the driving force behind much of our modern technology. Cellular phones and compact discs are ubiquitous examples of the need to handle information accurately, efficiently, and robustly. This research addresses these three criteria by introducing finite frames and coarse quantization ideas (providing redundancy and precision, respectively) so that together they minimize information loss in the case of noisy environments and machine imperfections, as well as ensuring numerical stability. The combination of frames and coarse quantization, as envisioned in this research, will generally produce small energy error differences between given signals and their quantized versions, which are constructed by higher order Sigma-Delta recursion schemes. The exceptions to the proposed general theory lead to a host of arithmetic-geometric problems. The technology for the theory requires a careful study of invariant sets closely connected with delicate tilings of Euclidean space. The first order Sigma-Delta analysis by the researchers proves its superiority over pulse code modulation methods for many applications. Further, the error estimates arising in this research will be analyzed in the case of the multidimensional discrete Fourier transform, which is a workhorse in any spectral analysis. In this setting, number theoretic problems, associated with residue number system processors, arise. A major goal of this research is to develop algorithms that provide reliable transmission of data over so-called erasure channels such as the internet. The methods that we are developing will also be used for multiple antenna code design that will guarantee clear, steady reception of messages in mobile wireless communications. Further, our proposed finite frame Sigma-Delta quantization methodology is a natural component in the emerging multifunction environment technology associated with the unified treatment of radar and communications. A natural application is to reduce a ship's signature in a hostile environment. Other applications of the methods to be developed, especially for the research dealing directly with finite frames, are of a geometric and number theoretic nature. For example, the vertices of the Platonic solids are finite frames with very desirable properties associated with sphere packing problems and spherical codes.

数字数据是我们许多现代技术背后的驱动力。 手机和光盘是准确、高效和稳健地处理信息的普遍需求的例子。 这项研究通过引入有限帧和粗量化思想（分别提供冗余和精度）来解决这三个标准，以便它们一起最大限度地减少噪声环境和机器缺陷情况下的信息丢失，并确保数值稳定性。 正如本研究中所设想的，帧和粗量化的组合通常会在给定信号与其量化版本之间产生较小的能量误差差异，这些信号是由高阶 Sigma-Delta 递归方案构建的。 所提出的一般理论的例外导致了许多算术几何问题。 该理论的技术需要仔细研究与欧几里得空间的微妙平铺密切相关的不变集。 研究人员的一阶 Sigma-Delta 分析证明了其在许多应用中优于脉冲编码调制方法。 此外，本研究中出现的误差估计将在多维离散傅立叶变换的情况下进行分析，多维离散傅立叶变换是任何频谱分析中的主力。 在这种情况下，会出现与余数系统处理器相关的数论问题。这项研究的一个主要目标是开发算法，通过所谓的擦除通道（例如互联网）提供可靠的数据传输。 我们正在开发的方法还将用于多天线代码设计，以保证移动无线通信中清晰、稳定的消息接收。 此外，我们提出的有限帧 Sigma-Delta 量化方法是与雷达和通信统一处理相关的新兴多功能环境技术的自然组成部分。 一个自然的应用是减少船舶在恶劣环境中的特征。 待开发方法的其他应用，特别是直接处理有限框架的研究，具有几何和数论性质。 例如，柏拉图固体的顶点是有限框架，具有与球体堆积问题和球体代码相关的非常理想的属性。