Frame And Application In Signal Processing

Wavelet analysis made milestone-like progress in the Fourier analysishistory, it is an emerging discipline which develpoed rapidly in the recent years.it'sapplication domain is very extensive,including many matematics disciplines,signalanalysis ,picture processing, quantummechanics and so on many aspects.Now it is amain tool in wavelet analysis .Fame theory derives from the signal processing. In 1952,Duffin and Shaffer introduced the concept of frame for Hilbert spaces in order to studysome deep problem in nonharmonic Fourier series. When wavelet theory is boomingDaubechies, Grossman and Meyer connected continuous wavelet transforms withframes theory and introduced wavelet frames. Today frames theory have been widelyused in wavelet analysis, signal analysis, image processing, numerical analysis, Banachspaces theory, etc.This paper mainly talks about the theory of frame and purtabation theroy offrame.The results that are quoted in this paper are mostly classical conclusions or thenewest conclusions which show the research level and the developing direction. On thebasis of the results, this paper generalizes some results and gives some new results.This paper is composed of four parts: The chapter 1 is an introduction whichsummarizes the emergence, development of wavelet analysis and frames theory.The chapter 2 presents the basic properties of frame. The frame is a special Besselsequence, which extends the Parseval equality of orthonormal basis in Hilbert space tothe two-sided inequality in general sequence. For every (?)f∈H, wehave f＝sum from i∈I(-1f_i>+c_i)f_i, here {c_i}_i∈I∈R(T)^⊥is the noise data. This shows theframe decomposition have some resistance to the noise. With the redundancy of frameincreasing, the resistance will become stronger. Therefore it is necessary to discuss theframe properties.The first part in the chapter 3 studies stabality of frame and wavelet frame in L²(R).The wavelet frame is a special sequence which is a frame for L²(R). The sequence isacquired by a special function acted by two transforms: dilation operator D_aⁿ with realparameter a＞1, n∈Z and translation operator T_mb with real parameter b＞0, m∈Z.The fourth chapter divides into two parts,the first part mainly introduced MRA of wavelet and Mallat alogrithm,the second part haven the frame multi-resolution analysisand The second part has given the frame multi-resolution analysis,And explained theframe multi-resolution analysis through the concrete example in the signal processingapplication.