Accurate Digital Representations for Overcomplete Data Expansions

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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.