Compressed Sensing of Complex Sinusoids: An Approach Based on Dictionary Refinement

In the existing compressed sensing (CS) theory, the accurate reconstruction of an unknown signal lies in the awareness of its sparsifying dictionary. For the signal represented by a finite sum of complex sinusoids, however, it is impractical to set a fixed sparsifying Fourier dictionary prior to signal reconstruction due to our ignorance of the signal's component frequencies. To address this, we model the sparsifying Fourier dictionary as a parameterized dictionary, with the sampled frequency grid points treated as the underlying parameters. Consequently, the sparsifying dictionary is refinable during the signal reconstruction process, and its refinement can be accomplished via the adjustment of the frequency grid. Furthermore, based on the philosophy of the variational expectation-maximization (EM) algorithm, we develop a novel recovery algorithm for CS of complex sinusoids. The algorithm achieves joint sparse representation recovery and sparsifying dictionary refinement by successively executing steps of signal coefficients estimation and dictionary parameters optimization. Simulation results under different conditions demonstrate that compared to the state-of-the-art CS recovery methods, the proposed algorithm achieves much higher signal reconstruction accuracy, and yields superior performance both in suppressing additive noise in measurements and in reconstructing signals with closely-spaced component frequencies.