Modulated measurement matrix design for compressed sensing

In this paper, we extend the idea of the seeding matrix design and introduce the modulated matrix framework for compressed sensing. The 1-D state evolution equation is derived to track the sample distortion performance as a function of the signal distribution and the rescaling matrix. A special example, the two-block matrix, is presented as a generalization of the hybrid zeroing matrix. The first order phase transition is further studied to better understand the dynamics. With the two-block matrix, exact recovery can be achieved in the region where the homogeneous Gaussian matrix is not optimal for the sparse signals. For compressible signals, the reconstruction quality can also be effectively improved.