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推动力。 蜂窝电话和紧凑型椎间盘是无处不在的例子，即需要准确，有效，稳健地处理信息。 这项研究通过引入有限的框架和粗量化思想（分别提供冗余和精确度）来解决这三个标准，从而使它们在嘈杂的环境和机器缺陷的情况下一起将信息丢失最小化，并确保数值稳定性。 如本研究所述，框架和粗量化的组合通常会在给定信号及其量化版本之间产生较小的能量误差差异，这些版本由高阶Sigma-Delta递归方案构成。 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微妙的瓷砖密切相关的不变式集合。 研究人员的一阶Sigma-Delta分析证明了其优于脉冲代码调制方法，用于许多应用程序。 此外，在多维离散傅立叶变换的情况下，将分析本研究中产生的错误估计，这在任何光谱分析中都是主力。 在这种情况下，出现了与残基号系统处理器相关的数字理论问题。这项研究的主要目的是开发算法，这些算法可在所谓的擦除渠道（例如互联网）上提供可靠的数据传输。 我们正在开发的方法也将用于多个天线代码设计，该设计将保证在移动无线通信中清晰，稳定地接受消息。 此外，我们提出的有限框架Sigma-Delta量化方法是与统一处理雷达和通信相关的新兴多功能环境技术中的自然组成部分。 自然的应用是在敌对的环境中减少船舶的签名。 要开发的方法的其他应用，尤其是直接处理有限帧的研究，具有几何和数字理论性质。 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形代码相关的非常理想的特性。