| The extrapolation of a band-limited signal is to reconstruct the unknown part of the signal form its known segment in the time interval[-T,T],and is a classical signal reconstruction problem,and has a wide range of applications.The studies of the extrapolation of band-limited signals have theoretical significance and practical applications.The main contributions and innovations of this thesis are as follows.(1)The extrapolation of band-limited signals in frequency domain can be represented as solving the integral equation(32)F(28)g which has a unique solution.Discretize(32)F(28)g into the linear system Ax(28)b which is assumed to have a unique solution.When T is small,the ill-posedness of the extrapolation of band-limited signals in theory leads to Ax(28)b ill-posed,so it is difficult to obtain effective extrapolation results by solving Ax(28)b.We prove that with T increasing,whether uniform samples or uniform samples at random in the interval[-T,T]being used,the condition number of A~*A is gradually improved,and thereby the posedness of Ax(28)b is improved;when T is appropriately small,considering the equationA~*Ax(28)A~*b equivalent with Ax(28)b,with A~*A being a symmetric positive definite matrix,we present a method of successively reweighting and its corresponding reweighted Landweber iterative scheme.After the process of reweighting is conducted k times,the condition number of the matrix of the equivalent linear system is improved.The simulations show that with an appropriately small T,compared with directly reconstructing,using the reweighted Landweber iterative scheme to extrapolate the band-limited signal has obviously better reconstruction results.(2)For the equation(32)~*(28)(32)F(28)(32)~*(28)g equivalent with the integral equation(32)F(28)g of the band-limited signals extrapolation in frequency domain,a method of successively reweighting is presented,and the equation after being reweighted k times is equivalent with(32)~*(28)(32)F(28)(32)~*(28)g.For every positive integer m,the m th condition number of the equation after being reweighted is obviously smaller than the one of(32)~*(28)(32)F(28)(32)~*(28)g,and a reweighted Landweber iterative scheme for the extrapolation of band-limited signals is presented correspondingly.When implementing extrapolation,a high accuracy numerical method is presented.The simulation results show that,by using the proposed method,we can obtain effective extrapolation results from the finite samples which are in a much smaller interval[-T,T].(3)We extend the method in(2)to the extrapolation of two-dimensional band-limited signals.When implementing the iteratively reconstructing,we present a method of successively reweighting the operator equation for reconstruction,and the direct method and the decomposition method for finding the solution.(4)Blind multiband signal reconstruction is to reconstruct the multiband signal from the sampling points in time domain.The signal is discretized over an appropriately large frequency interval containing all of its bands,and then frequency domain signal reconstruction can be solved by using the sparse signal recovery.Based on the facts that fewer sampling points are needed for compressed sensing recovery and the RIC(Restricted Isometry Constant)of the observation matrix is required to be small enough to recover the sparse signal,a reweighted method is proposed to improve the condition number of the observation matrix so as to improve its RIC,as well as a corresponding reweighted OMP(Orthogonal Matching Pursuit)method for blind multiband signal reconstruction.The proposed method is also applicable to the recovery of general sparse signals.In simulations,for an appropriately large frequency interval segment,an appropriately small discrete interval satisfying the reconstruction error range is taken.The simulation results show that for blind multiband signal reconstruction and the recovery of general sparse signals,the reweighted OMP method has a higher efficient reconstruction rate than the OMP algorithm directly used in the same conditions. |
