准Toeplitz测量矩阵的有限等距性质分析

摘要

CS can be widely utilized in linear time-invariant(LTl) system identification problems through the use of Toeplitz matrices. However, measurements were usually need to be undersampling in the applications which signal frequency was high to reduce the requirement of sampling fequency, this led the measurement matrices turn into quasi-Toeplitz matrices which formed by extracting a part of row vectors from Toeplitz matrices. This paper considered quasi-Toeplitz matrices as sensing matrices. Theoretical derivation showed that the quasi-Toeplitz matrices satisfy the restricted isometry property(RIP). Compared the performance of quasi-Toeplitz matrices and other compressed sensing matrices in simulations. It shows that the quasiToeplitz matrices perform similar to other compressed sensing matrices and can be used as CS measurement matrices.%Toeplitz测量矩阵的卷积特性使压缩感知理论在线性时不变系统辨识问题中得到广泛应用.但在信号频率较高的场合往往需要对测量结果进行欠采样,以利用压缩感知理论降低系统对采样频率的要求,这导致测量模型中的测量矩阵变为由Toeplitz矩阵中等间隔抽取若干行组成的子矩阵(准Toeplitz矩阵).为此讨论了准Toeplitz矩阵作为测量矩阵的可行性.通过理论推导证明了准Toeplitz矩阵的有限等距性质,在仿真中比较了使用准Toeplitz矩阵与其他测量矩阵的重构效果.结果表明,准Toeplitz满足有限等距性质,使用准Toeplitz矩阵的重构效果与其他测量矩阵相近,可以作为压缩感知测量矩阵.