Abstract:

The restricted isometry property (RIP) of the sensing matrix has recently received a lot of attention under the rubric of compressive sensing. Most studies have shown that a Gaussian random sensing matrix satisfies the RIP in theory. However, due to its high cost of storage and complex physical implementation, the Toeplitz sensing matrix is more in favored than the Gauss sensing matrix in actual. Because it can be implemented by the fast discrete Fourier transform. Proof of the RIP for the sensing matrix exploiting graph theory and the Gergorin’s disc theorem is given. The results show that the Toeplitz random matrix satisfies the RIP with high probability. And the least square (LS), linear minimum mean square error （LMMSE）, compressive sensing algorithm with the Gauss sensing matrix and compressive sensing algorithm with Toeplitz sensing matrix channel estimation algorithm are compared. The simulation results show that the Toeplitz sensing matrix has a superiority over the Gauss sensing matrix on computation complexity and the traditional channel estimation method on performance and computation complexity. It provides a theoretical and realistic foundation for reconstructing the original signal losslessly.