Block Sparse Recovery With Redundant Measurement Matrices and Its Application in Frequency Agile Radar

Block compressed sensing (or sparse recovery) and its performance bound, i.e., conditions that guarantee reconstruction of the original sparse vector, have been widely studied. Most scenarios assume that the blocks in the measurement matrix are full-rank. In this setting, phase transition theory provides a precise performance bound on the exact reconstruction of the original vector. However, in many practical applications, the blocks of the measurement matrices may not have full rank, and it becomes impossible to recover the original vector elementwise. In this article, we consider the compressed sensing problem with such redundant measurement matrices and derive a performance bound using phase transition theory. We focus on reconstructing the contribution of each block in the original vector to the observed signal instead of the vector itself. We show that this method is equivalent to a traditional $\ell _{2,1}$ norm minimization after a certain linear transformation. We theoretically prove the transformed $\ell _{2,1}$ norm minimization has a phase transition phenomenon. Based on this result, we derive the closed-form phase transition curve of the method as a tight performance bound. We also apply this result to frequency agile radar for performance evaluation and waveform design. Simulations validate our theoretical conclusions.