Some Results On Wave Packets Systems In Discrete Settings

In 1952,the frames were presented by Duffin and Schaeffer in the study of the nonharmonic Fourier analysis.Compared with the traditional orthogonal wavelets,the frames are generally redundant,which can lead to robustness.It means that redundancy makes the coefficients obtained in the low accuracy can recovery singals in a relatively high accuracy.It is an important analytical tool in signal processing,image processing,sampling theory,digital communication and so on.In addition,the frame theory also plays a more and more important role in the study of Optics and Banach space theory.The frames mainly include Gabor frames and wavelet frames.In practical applica-tions,the Gabor systems are more suitable for the processing of stationary signals,while the wavelet systems are more suitable for the treatment of the saltation signal.Therefore,the scholars at domestic and abroad hoped to find a new kind of function systems to unify the Gabor systems and the wavelet systems.This idea agreed with a class of functions called wave packets which is applied to the quantum mechanics in physics.In practical applications,due to the input/output datas and the filters are discrete,the implemen-tation of the algorithm based on the frames is done in the digital environment.In this paper,a frame of l2(Zd)will be given,and we will obtain the sufficient condition when it forms a Parseval frame,then we decompose and reconstruct the signal with this frame.Furthermore,we will get the notion of pth wave packet systems based on this frame,and apply it to multi-level decomposition of a signal.Finally,for the importance of the ratio of the frame bounds in the design of the filters,this paper will estimate the frame bounds of the given filter banks based on mutual-coherence of a matrix.This paper consists of five parts.The first chapter introduces the background of the frames and wave packets,as well as the main content and the related structure of this paper.The second chapter lists some concepts and properties for this paper.The third chapter begin with the frame operator,we will construct a Parseval frame for l2(Zd),and according to this frame we will establish a decomposition operator which is used to decompose the signal.Then we build a reconstruction operator to achieve the perfect reconstruction of the decomposition of the signal.The fourth chapter is the focus of this paper.We will get the structure of pth stage wave packet systems based on the last chapter,and apply it to multi-level decomposition of the signal.Finally,we analyse the algorithm of the decomposition and the computation.The fifth chapter estimate the frame bounds for the sequence space by employing the concept of the mutual-coherence of a matrix.