Short communicationSparse-based estimation performance for partially known overcomplete large-systems

We assume the direct sum ⟨A⟩⊕⟨B⟩ for the signal subspace. As a result of post-measurement, a number of operational contexts presuppose the a priori knowledge of the LB-dimensional “interfering” subspace ⟨B⟩ and the goal is to estimate the LA amplitudes corresponding to subspace ⟨A⟩. Taking into account the knowledge of the orthogonal “interfering” subspace ⟨B⟩⊥, the Bayesian estimation lower bound is derived for the LA-sparse vector in the doubly asymptotic scenario, i.e. N, LA, LB → ∞ with a finite asymptotic ratio. By jointly exploiting the Compressed Sensing (CS) and the Random Matrix Theory (RMT) frameworks, closed-form expressions for the lower bound on the estimation of the non-zero entries of a sparse vector of interest are derived and studied. The derived closed-form expressions enjoy several interesting features: (i) a simple interpretable expression, (ii) a very low computational cost especially in the doubly asymptotic scenario, (iii) an accurate prediction of the mean-square-error (MSE) of popular sparse-based estimators and (iv) the lower bound remains true for any amplitudes vector priors. Finally, several idealized scenarios are compared to the derived bound for a common output signal-to-noise-ratio (SNR) which shows the interest of the joint estimation/rejection methodology derived herein.