On the Reconstruction of Multiple Sinusoidal Signals from Compressed Measurements

The introduction of compressive sensing in wireless smart transducers can substantially reduce the high impact of sampling rate on their overall power consumption. Such systems are often dealing with signals that can be expressed as a sum of multiple sinusoids, having a frequency-sparse representation. Although the reconstruction of frequency-sparse signals has been widely studied and solutions based on greedy and relaxation methods exist, their performance is degraded in presence of spectral leakage, which affects the sparse representation of the signal and consequently, its estimation accuracy. In this paper, a two-stage optimization approach, named Opti2, is presented for the reconstruction of frequency-sparse signals that can be expressed as a sum of multiple real-valued sinusoidal waveforms. The estimation provided by basis pursuit denoising (BPDN) sparse optimization is computed in the first stage and used as initial guess for the second stage, where a non-linear least squares (NLLS) problem is formulated to improve the estimation of the signal parameters from undersampled data. Simulation results demonstrate that the proposed approach outperforms existing methods in terms of accuracy, showing its robustness to noise and compression rate.