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.

数字数据是我们现代技术背后的驱动力。 蜂窝电话和光盘是需要准确、有效和鲁棒地处理信息的无处不在的例子。 本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，以便在噪声环境和机器缺陷的情况下最大限度地减少信息丢失，并确保数值稳定性。 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和它们的量化版本之间产生小的能量误差差，所述量化版本由高阶Sigma-Delta递归方案构造。 所提出的一般理论的例外导致了大量的算术几何问题。 该理论的技术要求仔细研究与欧几里得空间的精细拼接密切相关的不变集。 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，在这项研究中产生的误差估计将分析的情况下，多维离散傅立叶变换，这是一个主力在任何频谱分析。 在这种情况下，数论问题，与剩余数系统处理器，出现。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（如互联网）提供可靠的数据传输。 我们正在开发的方法也将用于多天线编码设计，这将保证在移动的无线通信中清晰、稳定地接收消息。 此外，我们提出的有限帧Σ-Δ量化方法是一个自然的组成部分，在新兴的多功能环境技术与雷达和通信的统一处理。 一个自然的应用是在敌对环境中减少船只的信号。 其他应用的方法开发，特别是研究直接处理有限框架，是一个几何和数论性质。 例如，柏拉图立体的顶点是有限框架，具有与球填充问题和球码相关的非常理想的性质。