Study Of Some Questions For Sigma-Delta (∑Δ) Quantization Of Finite Frames

The concept of frame was frstly introduced by Dufn and Schaefer in the1950swhen they were studying nonharmonic Fourier series. Every element in a Hilbert spacecan be represented by a frame, but diferent with a basis, the way of representation is notunique. It also has many extensive applications in signal processing, image processing,data compression, communication and so on, consequently it has very broad prospect ofapplication.Information is the objective refection of the state of things, and the unity of themeaning and symbol. Information is the specifc content of the signal, and signal isthe form of expression of information. In order to use the signal to obtain information,process information and transmit information efectively, we must choose reasonable signalexpression which is very important. The frame is just a way of expressing signal.This thesis is mainly to discuss some questions for Sigma-Delta quantization of fniteframes, and consists of fve chapters.Chapter1states the origin of Sigma-Delta quantization, and sketches the structureand works of the thesis.The basic concept, character, and some basic facts of fnite frame and PCM quanti-zation are listed in chapter2.Chapter3is one of the main contents. It introduces the related concepts and char-acter of frst order and second order Sigma-Delta quantization, and generalizes the ap-proximate error of arbitrary order Sigma-Delta quantization, and discusses the variationof error of non-unit norm tight frames.Chapter4is another main content. It presents the defnition and the variation oferror about second order Sigma-Delta quantization of harmonic frame, and generalizesthe approximate error of arbitrary order Sigma-Delta quantization of harmonic frame.The fnal part is in chapter5. First, the defnition of generalized harmonic frame isgiven. Secondly, the approximate error of second order Sigma-Delta quantization of gen-eralized harmonic frame is discussed. Finally, the approximate error of arbitrary order Sigma-Delta quantization of generalized harmonic frame is obtained.