Research On Deterministic Measurement Matrix Design Based On Codebooks

Compressed sensing theory is a kind of sampling theory emerging in recent years.Compressed sensing theory uses a non-adaptive,linear projection way to measure a sparse signal and recovers it through a corresponding recovery algorithm.It breaks through the bottleneck of the traditional sampling theorem and realizes the unity of sampling and compression..So it has received wide attention from scholars.How to construct a measurement matrix with good performance in compressed sensing is the focus of research.Because the random measurement matrices have obvious shortcomings such as large storage overhead and limited hardware applications,this paper conducts research around constructing deterministic measurement matrices.Based on the relationship between the measurement matrices and the codebooks,new deterministic measurement matrices are obtained by studying the construction method of the new codebooks.Firstly,a method for constructing deterministic measurement matrices based on the characters sums over a finite field is proposed.The method of constructing codebook based on Gaussian over finite field is used to construct the deterministic measurement matrices.Then a construction method of deterministic measurement matrices based on Jacobi sums over finite fields is proposed.The correlation of the deterministic measurement matrices constructed by the method asymptotically approach Levenstein bound and Welch bound,and their performance is better than random measurement matrices.And It is easy to implement in hardware,does not waste storage resources,and has good use value.Secondly,the basic idea of constructing codebooks based on binary sets is further extended,and the restrictions on the transformation matrix are relaxed..It isproposed to use the Zadoff-Chu matrix as a new transformation matrix to construct deterministic measurement matrices.A class of deterministic measurement matrices with different parameters were obtained by using the difference set,almost difference set,the cyclotomic classes of order,and Eisenstein sum respectively with the Zadoff-Chu matrix.Finally,according to the characteristics of the Zadoff-Chu matrix,a class of deterministic measurement matrices with partial cyclic structure is obtained by using binary Golay complementary sequences with even lengths.The correlation of such deterministic measurement matrices nearly approach the Welch bound.Its performance is significantly better than the random measurement matrices,and it can be applied to convolutional compressed sensing.