CGSS: A New Framework of Compressed Sensing Based on Geometric Sequential Representation Against Insufficient Observations

In this article, we introduce a novel compressed sensing (CS) scheme for sparse signal recovery in an effective method, namely compressed geometric sequential sensing (CGSS). This comes from the fact that an observation vector in CS can be interpreted as a superposition of multiple geometric sequences if the sensing matrix is a partial discrete Fourier transform (DFT) matrix. The main idea is based on the mathematical property that the nonorthogonally superposed geometric sequences can be decomposed, without loss of information, into the original geometric sequences in specific patterned ways. With this method, a K-sparse vector can be perfectly reconstructed through only
$2K$
observations in the ideal case (i.e., noise-free observations) regardless of the length of the original K-sparse vector. To verify the robustness of our proposed scheme, it is compared with existing CS techniques under two environments with noisy observations, which are the additive white Gaussian noise (AWGN) and the impulsive noise. In the simulation part, we show that the performance of CGSS can be improved through an appropriate denoising technique in AWGN cases. Notably, in impulsive noisy cases, the proposed scheme enables the perfect reconstruction of the sparse signal within the given condition.