Improved model-based spectral compressive sensing via nested least squares

This paper introduces a new algorithm for reconstructing signals with sparse spectrums from noisy compressive measurements. The proposed model-based algorithm takes the signal structure into account for estimating the unknown parameters which are the frequencies and amplitudes of linearly combined sinusoids. A high-resolution spectral estimation method is used to recover the frequencies of the signal elements, while the amplitudes of the signal components are estimated by minimizing the squared norm of the compressed estimation error using the least squares (LS) technique. The Cramer-Rao bound (CRB) for the given system model is also derived. It is shown that the proposed algorithm with properly selected step size of the LS algorithm achieves the CRB at high signal to noise ratio values.