Research On The Frames Theory Application In Compressive Sampling

In recent years a new kind of sampling technology--compressive sampling theory has attractedthe attention of domestic and foreign researchers. The basis of it is various branches of appliedmathematics, especially probability theory. Compressive sampling which is an efficient samplingmechanism takes advantage of sparse or compressible signals and uses a variety of efficientalgorithms from less observed data to recover the original signal. From the global perspectivecompressive sample theory get a random measurement matrix to multiple observations of the targetsignal. This can both capture the signal characteristics of sparse and make sure that betweenobservational datas can also enable greater information redundancy. At present the application ofcompressive sampling technology has entered many areas such as the beam forming, signalclassification, ultra wideband channel estimation, super resolution radar and medical imaging. Howto design the measurement matrix and look for more efficient signal reconstruction algorithmbecomes the hot issue of compressive sampling.In compressive sampling technology, signals are no longer decomposed in a completeorthogonal base, but rather an over complete atomic database used to represent the signal, this leadsto the study the application of framework theory in compressive sampling. Framework theory as themain tool of wavelet analysis has been widely used in signal analysis, image processing, numericalcomputing and other fields. The frame is similar to the sequence of a set of linear correlationorthogonal sequence, with a certain degree of redundancy and inherits many good properties oforthogonal basis, these works out decomposition coefficients in the low-precision, and reconstructsthe signal in the high-precision. Many special frameworks (such as Grassmannian frame which hasthe lowest correlation between the framework of vectors) has been widespread concerned incommunications field. Framework theory is applied to the sparse signal detection and the design ofmeasurement matrix in compressive sampling.This paper focus on the application of framework theory mainly including the measurementmatrix design and sparse signal detection in compressive sampling. There are four parts in the paper.First we introduce the details of compressive sampling theory, including the basic principles,measurement matrix of the design requirements, several signal reconstruction algorithms theory,algorithm simulation and the basic knowledge of analog to information converter. Secondly, weintroduce the framework theory, give basic knowledge of space and the orthogonal basis and brieflyintroduce the concepts of frame. We discuss the details of several specific framework andframework-related property. Then we research a very special kind of equiangular tight frame--Grassmannian framework, including the construction of the framework and severalstructure theorems and inference algorithms, and the feasibility of the algorithm through simulation.At last the paper gives a design method of measurement matrix, which purpose is to optimize thesparse signal detection performance, and obtained optimal value of signal-to-noise ratio when signalsparse degree different.