Analog-to-Digital Conversion: Mathematics and Algorithms

计算机和数字信息技术的出现和发展极大地改变了我们生活的世界。今天，数字技术在我们的生活中无处不在。使所有这些技术成为可能的一个关键步骤是将模拟数据转换为数字数据，这是一个众所周知的模数转换或A/D转换过程。随着对更高精度和更尖端技术的要求，A/D转换算法的数学在这一追求中起着关键作用。本方案中提出的研究主要集中在A/D转换的数学和算法上。鉴于模拟设备本质上不同程度的不精确，如果需要高精度，稳健性是极其重要的。该提案将重点放在稳健的A/D转换算法上。在这项提议中，PI将解决两个主要的不精确度来源：不精确的量化和不精确的乘法。所提出的编码器是第一个对量化器和乘法器不精确具有完全健壮性的编码器。PI建议同时研究算法和从研究中产生的数学问题。PI还研究编码器可以扩展和改进的方法，以及许多相关的和经常具有挑战性的数学问题。本课题的核心是一类新颖的模数转换算法。这些算法的优点是它们对量化误差(不精确)和乘法不精确具有健壮性。这些A/D算法的新颖之处在于使用了基于黄金比的实数与斐波纳契数相关的展开。更一般的算法涉及一类代数整数和矩阵编码器。PI和他的合作者已经基于我们的算法之一构建了一个A/D转换器的电路，并取得了出色的结果。这一建议的一个主要目标是更深入地研究这些算法，特别是关于这些算法的稳定性和健壮性。同样重要的是，对这些A/D算法的研究提出了许多有趣和具有挑战性的数学问题。研究生将参与到这个项目中来。此外，拟议的研究自然有助于数学家和工程界之间的合作。

The advent of computer and digital information technologies and there development have greatly changed the world we live in. Today digital technologies are everywhere in our lives. A key step that makes all those technologies possible is to convert analog data into digital ones, a process known analog-to-digital conversion, or A/D conversion. With demand for higher precision and more cutting-edge technologies, the mathematics of A/D conversion algorithms plays a key role in this quest. The research proposed in this proposal focuses mainly on the mathematics and algorithms of A/D conversion. Given that analog devices are to different degrees imprecise by nature, robustness is extremely important if high precision is desired. The proposal will focus on robust A/D conversion algorithms. In this proposal the PI will address two of the main sources of imprecisions: imprecise quantizations and imprecise multiplications. The proposed encoders are the first encoders to be completely robust against quantizer and multiplier imprecisions. The PI proposes to study both the algorithms and the mathematical questions that arise from the study. The PI also studies ways the encoders can be extended and refined, along with many related and often challenging mathematical problems. The core of this project is a novel class of analog-to-digital conversion algorithms. These algorithms have the advantage that they are robust against quantization errors (imprecisions) and multiplication imprecisions. The novelty of these A/D algorithms comes from the use of Golden Ratio based expansions of real numbers related to the Fibonacci numbers. More general algorithms involve a class of algebraic integers and matrix encoders. The PI and his collaborators have already built a circuit of an A/D converter based on one of our algorithms, which has yielded outstanding result. A main objective of this proposal is to study these algorithms in greater depth, particularly concerning the stability and robustness of these algorithms. Equally important is that the study of these A/D algorithms has raised a number of interesting and challenging mathematical questions. Graduate students will be involved in the project. In addition, the proposed research lends naturally to collaborations between mathematicians and the engineering community.