Type of Document Dissertation Author Jiang, Jiayi Author's Email Address jiayi.jiang@vanderbilt.edu URN etd-03252016-123616 Title Quantization in Signal Processing with Frame Theory Degree PhD Department Mathematics Advisory Committee

Advisor Name Title Alexander Powell Committee Chair Akram Aldroubi Committee Member Brett Byram Committee Member Doug Hardin Committee Member Keywords

- Frame
- Fusion Frame
- Sigma-Delta Quantization
- Sobolev Dual
Date of Defense 2016-03-25 Availability unrestricted AbstractQuantization is an important part of signal processing. Several issues influence the performanceof a quantization algorithm. One is the “basis” we choose to represent the signal, another is how

we quantize the “basis” coefficients. We desire to explore these two things in this thesis. In the first chapter, we review frame theory and fusion frame theory. In the second chapter, we introduce a

popular quantization algorithm known as Sigma-Delta quantization and show how to apply it

to finite frames. Then we give the definition and properties of Sobolev duals which are optimized

duals associated to Sigma-Delta quantization. The contraction Sobolev duals depends on the frame, and in chapter 3, we prove that for any finite unit-norm frame, the best error bound that can be achieved

from the reconstruction with Sobolev duals in rth Sigma-Delta quantization is equal to order O(N^-r), where the error bound can be related to both operator norm and Frobenius norm. In the final chapter, we develop Sigma-Delta quantization for fusion frames. We construct stable first-order and high-order Sigma-Delta algorithms for quantizing fusion frame projections of f onto W_n, where W_n is an M_n dimension subspace of R^d. Our stable 1st-order quantizer uses only log2(Mn+1) bits per subspace. Besides, we give an algorithm to calculate the Kashin representations for fusion frames to improve the performance of the high-order Sigma-Delta quantization algorithm. Then by defining the left inverse and the canonical left inverse for fusion frames, we prove the property that the canonical left inverse has the minimal operator norm and Frobenius norm. Based on this property, we give the idea of Sobolev left inverses for fusion

frames and prove it leads to minimal squared error.

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