Performance analysis of spatial smoothing schemes in the context of large arrays

This paper adresses the statistical behaviour of spatial smoothing subspace DoA estimation schemes using a sensor array in the case where the number of observations $N$ is significantly smaller than the number of sensors $M$, and that the smoothing parameter $L$ is such that $M$ and $NL$ are of the same order of magnitude. This context is modelled by an asymptotic regime in which $NL$ and $M$ both converge towards $\infty$ at the same rate. As in recent works devoted to the study of (unsmoothed) subspace methods in the case where $M$ and $N$ are of the same order of magnitude, it is shown that it is still possible to derive improved DoA estimators termed as Generalized-MUSIC with spatial smoothing (G-MUSIC SS). The key ingredient of this work is a technical result showing that the largest singular values and corresponding singular vectors of low rank deterministic perturbation of certain Gaussian block-Hankel large random matrices behave as if the entries of the latter random matrices were independent identically distributed. This allows to conclude that when the number of sources and their DoA do not scale with $M,N,L,$ a situation modelling widely spaced DoA scenarios, then both traditional and Generalized spatial smoothing subspace methods provide consistent DoA estimators whose convergence speed is faster than $\frac{1}{M}$. The case of DoA that are spaced of the order of a beamwidth, which models closely spaced sources, is also considered. It is shown that the convergence speed of G-MUSIC SS estimates is unchanged, but that it is no longer the case for MUSIC SS ones.