Research On Frame Theory And Its Application In The Signal Transmission

Frame theory is the product of the combination of functional analysis,nonlinear approximation theory,operator theory and signal theory,which is a new research direction gradually developed after wavelet theory.The development of frame theory has contributed significantly to the combination of the pure mathematics and the engineering applications,so the application prospects are wide.Now,the theory of frames has been extensively applied in image processing,signal processing,sampling theory,data compression,system modelling,code and communication and so on.With the rapid development and widespread application of modern information technology,people have paid more attention to the exploitation and utilization of information resources.Although frame theory has been developed very quickly and well,as a new research direction,there are still many questions which need to be studied.This thesis mainly studies the fundamental theory of frame,and focuses on problems where frame theory can be applied to signal reconstructions where the erasures occur in the data transmission process.The main contents are as follows.1.We investigate frame design problem based on matrices.By using singular value decomposition of matrices,we obtain a method constructing some specific frames.Moreover,we also construct some tight frames by using the Unitary matrices.These methods solve the problem of the computational complexity of the inverse frame operator.The procedure of construction is easy,which extends the frame in the actual application.2.We investigate some equalities and inequalities for fusion frames.By using the theory and method of the bounded linear operator,we build some equalities and inequalities for fusion frames in the Hilbert space.The results help to solve the relevant issues of the fusion frame in the parallel processing and high energy physics experiment.3.Since g-frame is the generalization of the frame.We investigate the relevant conclusions of the g-frame.We first introduce a bounded linear operator relating to the gframe operator of a given g-frame and study the stability of g-frames.Further,by using the reconstruction error,namely the(normalized)worst-case error,we study the dual g-frames that are optimal for the erasures of g-frames,and discuss the sufficient and necessary conditions under which the canonical dual g-frame is the unique optimal dual g-frame with erasures.Finally,we give the new characterization of approximately dual g-frames with respect to local frames by using given g-frames and bounded operators,and prove that if two g-frames are close to each other,then we can find approximately dual g-frames of them which are close to each other.4.We investigate the problems where frame theory can be applied to signal reconstructions where the erasures occur in the data transmission process.Based on method of optimal directions(MOD),a new approach is proposed.The approach can search adaptively optimal dual frames in the process of signal reconstruction,and deals with the problem that it may not be the optimal for a particular class of input signals.The optimal dual frame searched by the new method,can minimizes the error between the reconstruction signal and the original signal.In addition,the new method solves the reconstruction problem during the signal transmission process.Numerical experiments demonstrate the effectiveness of the new method.