Constructions Of Deterministic Sparse Measurement Matrices In Compressed Sensing

Compressed sensing is a signal acquisition method to simultaneously acquire and reduce dimension of signals that admit sparse representations.This technique is a process of collecting linear,non-adaptive measurements of a signal,which can be formalized as multiplying the signal with a measurement matrix.The measurement matrix affects the two important processes of signal sampling and signal recovery,so it plays a key role in compressed sensing.This thesis mainly studies the construction methods of deterministic sparse measurement matrices.Aiming at the problem that the random measurement matrix with good reconstruction performance always needs high storage requirement and is difficult to implement in hardware,this thesis constructs deterministic sparse measurement matrices with low storage requirement,low computational complexity in both encoding and recovery,easy incremental updates.A construction method of sparse bipolar measurement matrices is proposed by nesting a Hadamard matrix into the base matrix,and two construction methods of complex sparse measurement matrices are proposed by nesting DFT matrices into different base matrices.The coherence of sparse measurement matrices constructed by these types of methods meets the Welch bound asymptotically.It means that the performance meets near-optimal.These construction methods enrich the composition of the existing deterministic measurement matrices and expand the matrix dimensions.Aiming at the problem that the dimensions of most deterministic measurement matrices are limited,this thesis constructs deterministic sparse measurement matrices with flexible dimensions.More concretely,when the compression ratio is fixed,the dimensions of the matrix can be flexibly changed and the recovery performance of signals can be guaranteed.Firstly,a basic matrix with a cyclic structure is constructed based on finite fields,and it is sparsely represented by a binary function.Secondly,the sparse matrix is intercepted by introducing the variable L,which further expands the matrix dimensions.Finally,a Hadamard matrix is nested into the binary matrix to obtain the deterministic sparse measurement matrix with flexible dimensions.In this construction method,multiple variables are introduced to ensure the dimensional variability of the measurement matrix when the compression ratio is fixed.At the same time,the sparse measurement matrix with flexible dimensions has sparseness and cyclic structure,thereby ensuring lower computational complexity,saving time and storage costs.