Accurate Digital Representations for Overcomplete Data Expansions

数字数据是我们许多现代技术背后的驱动力。移动电话和光盘是需要准确、高效和可靠地处理信息的无处不在的例子。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，以便它们一起在噪声环境和机器缺陷的情况下将信息损失降至最低，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，这些量化版本是由更高阶Sigma-Delta递归方案构造的。提出的一般理论的例外情况导致了一系列算术几何问题。这一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分析证明了它在许多应用方面优于脉冲编码调制方法。此外，在多维离散傅立叶变换的情况下，将分析在这项研究中产生的误差估计，这是任何频谱分析中的主力。在这种情况下，出现了与剩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，在互联网等所谓的擦除通道上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编码设计，以确保在移动无线通信中清晰、稳定地接收消息。此外，我们提出的有限帧Sigma-Delta量化方法是与雷达和通信的统一处理相关的新兴多功能环境技术的自然组成部分。一种自然的应用是在恶劣的环境中减少船只的签名。待开发的方法的其他应用，特别是直接涉及有限框架的研究，具有几何和数论性质。例如，柏拉图固体的顶点是有限框架，具有与球包装问题和球码相关的非常理想的性质。

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.