Analog-to-Digital Conversion: Mathematics and Algorithms

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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.