Robustness of Compressive Sensing under multiplicative perturbations: The challenge of fading channels

We investigate the robustness of Compressive Sensing (CS) as a direct signal acquisition and reconstruction method at the wireless receiver in fading channels. The wireless channel introduces additive as well as multiplicative random perturbations to the received signal. The original CS theory considers only additive and bounded perturbations for signal reconstruction with an a priori known basis. However, the impact of multiplicative perturbations, which manifests itself as basis mismatch, on the CS reconstruction performance is largely unknown. In this paper we first formulate such multiplicative perturbations due to wireless fading channel in the CS acquisition and reconstruction problem. We then show that these perturbations can result in significant errors in signal reconstruction if the basis is not properly adjusted. Furthermore, we will propose a method for adjusting the elements of basis to the fading channel coefficients and discuss possible improvements in the signal reconstruction.