A joint robust estimation and random matrix framework with application to array processing

An original interface between robust estimation theory and random matrix theory for the estimation of population covariance matrices is proposed. Consider a random vector x = A_Ny ∈ C^N with y ∈ C^M made of M ≥ N independent entries, E[y] = 0, and E[yy*] = I_N. It is shown that a class of robust estimators Ĉ_N of C_N = A_NA*_N, obtained from n independent copies of x, is (N, n)-consistent with the traditional sample covariance matrix r̂_N in the sense that ∥Ĉ_N - αr̂_N∥ → 0 in spectral norm for some α > 0, almost surely, as N, n → ∞ with N/n and M/N bounded. This result, in general not valid in the fixed N regime, is used to propose improved subspace estimation techniques, among which an enhanced direction-of-arrival estimator called robust G-MUSIC.