Compressed sensing of superimposed chirps with adaptive dictionary refinement

Abstract

The compressed sensing (CS) theory shows that accurate signal reconstruction depends on presetting an appropriate signal sparsifying dictionary. For CS of superimposed chirps, this dictionary is typically taken to be a waveform-matched dictionary formed by blindly discretizing the frequency-chirp rate plane. However, since practical target parameters do not lie exactly on gridding points of the assumed dictionary, there is always mismatch between the assumed and the actual sparsifying dictionaries, which will cause the performance of conventional CS reconstruction methods to degrade considerably. To address this, we model the waveform-matched sparsifying dictionary as a parameterized one by treating its sampled frequency-chirp rate grid points as the underlying parameters. As a consequence, the sparsifying dictionary becomes refinable and its refinement can be achieved by optimizing the underlying parameters. Based on this, we develop a novel reconstruction algorithm for CS of superimposed chirps by utilizing the variational expectation-maximization (EM) algorithm. By alternating between steps of sparse coefficients estimation and dictionary parameters optimization, the algorithm integrates the process for dictionary refinement into that of signal reconstruction, and thus can achieve sparse reconstruction and dictionary optimization simultaneously. Experimental results demonstrate that the algorithm effectively deals with the performance degradation incurred by dictionary mismatch, and also outperforms the state-of-the-art CS reconstruction methods both in compressing the signal measurements and in suppressing the measurement noise.