Research On Measurement Matrix Of Compressed Sensing

It is well known that the Shannon-Nyquist sampling theorem is one of the most important theorems in the traditional signal processing theory framework.In order to recover the original signal from the discrete sampling data,the sampling rate must be more than the double of the band of the signal.This theorem only uses the band of the signal,but doesn’t notice some other characteristics of the signal itself.The Compressed Sensing theory pays an attention to the sparsity of the signal itself and puts forward a new mechanism and method of signal sampling.It notes that: when the signal in a domain satisfies a certain sparsity,we can go to sample by the rate which is much lower than the Nyquist sampling rate,then the sampling rate no longer depends on the band of the signal,but the information rate contained by the signal.Measurement Matrix is an important research content of Compressed Sensing theory.Firstly,Measurement Matrix ensures that the signal from the higher dimensional space to lower dimensional space to carry the sufficient information.Secondly,Measurement Matrix has an important impact on the complexity and accuracy of the back-end reconstruction.Currently,Compressed Sensing theory model stills need to be further improved.Some problems exist in the research of Measurement Matrix.This paper mainly focuses on the research on Measurement Matrix,and has done the following work.1.The basic model of the Compressed Sensing theory is studied in this paper.Some extensions of the sparse signal model and the common reconstruction algorithms are also studied.On the foundation of the basic model,the discrimination theory and the construction method of the Measurement Matrix are studied.The performance of some common Measurement Matrices is analyzed in detail,and the performance difference of different Measurement Matrices is compared by experiments.2.Essentially,the basic model of the Compressed Sensing theory is a solution of the underdetermined equations,which solution space satisfies certain sparsity.On the basic model of the Compressed Censing theory,the paper analyses the significance of matrix coherence in the sparse solution of the linear equations,avoids computing the inverse matrix directly from the point of the coherence,and devires a relation between the column correlation and the degree of sparsity for the binary sparse signal.3.Following the idea of constructing the Measurement Matrix based on the linear representation of the orthogonal basis,a selection strategy for the corresponding coefficient vector is proposed in the paper.According to this strategy,a kind of Measurement Matrix based on the pseudo random sequence with excellent auto-correlation is constructed,which has a certain cyclic structure,and the kind of the element is relatively single.This method is more conducive to the hardware implementation of the Measurement Matrix.It is proved that the matrix constructed by this method satisfies RIP characteristics of a certain order,as the Measurement Matrix is feasible.Simulation results also showes that its performance is better than the same size of other Measurement Matrixs such as Gause Measurement Matrix and avoids the uncertainty of random Measurement Matrix.4.According to the Measurement Matrix is constructed by using the linear representation of the orthogonal basis,which compression rate is restricted,a kind of Measurement Matrix is restructured by using the transform of Kronecker Product,which compression ratio is higher.The Measurement Matrix constructed by this method has some characters,such as retention cycle characteristics,the single kinds of the elements and a certain sparseness.The column coherence of this matrix almost closes to Welch Bound.The simulation results also showes that its performance is better than Gause Measurement Matrix and other Measurement Matrix under the same conditions.