Sparse Bayesian learning for Compressed Sensing under Measurement Matrix Uncertainty

For Compressed Sensing (CS), the core problem is how to reconstruction the sparse unknown signal based on an underdetermined linear equation. Sparse Bayesian Learning (SBL) is an important algorithm for the above CS problem. In conventional research, the pre-designed measurement matrix is applied in the CS sensing system accurately. However, because of the non-ideality of the physical system, there exists some perturbation between the actual and the pre-designed measurement matrix. This kind of perturbation is named as Measurement Matrix Uncertainty (MMU). In this paper, we propose a new algorithm named as Matrix-Uncertain SBL (MU-SBL) in order to extend SBL into CS signal reconstruction under MMU. In MU-SBL, MMU effects are absorbed into an independent non-identically distributed (non-i.i.d.) Gaussian noise vector, whose variances are estimated based on the variances of the perturbation matrix and the reconstruction results from SBL. In general, MU-SBL iterates between non-i.i.d. noise variances estimation and SBL sparse signal reconstruction. Finally, MU-SBL is also applied to multipath sparse Single-Input-Single-Output Orthogonal Frequency-Division Multiplexing (SISO-OFDM) channel estimation based on CS, in which MMU comes from the nonlinearity of Power Amplifier (PA). MU-SBL is shown to outperform conventional SBL by the simulation results on both the artificial Gaussian sparse signal reconstruction and OFDM sparse channel estimation.