Section: New Results

Application of the Continuous Exact ${\ell }_0$ relaxation to Channel and DOA sparse estimation problems

Participants : Emmanuel Soubies, Laure Blanc-Féraud.

This work is made in collaboration with Adilson Chinatto, Cynthia Junqueira, João M. T. Romano (University of Campinas, Brazil) and Pascal Larzabal, Jean-Pierre Barbot (ENS Cachan, SATIE Lab).

This work is devoted to two classical sparse problems in array processing: Channel estimation and DOA (Direction Of Arrivals) estimation. We show how our results on ${\ell }_0$ optimization [1] , [14] , [17] can be used, at the same computational cost, in order to obtain improvement in comparison with ${\ell }_1$ optimization (usually used) for sparse estimation. Moreover, for the DOA case, we show that our analysis conducted in the Single Measurement Vector (SMV) case [1] can be generalized to the Multiple Measurement Vectors (MMV) case. In that case, the variable $x$ is not a vector of ${ℝ}^N$ but a matrix of ${ℝ}^{N\times K}$ where $N$ is the signal length and $K$ the number of snapshots. Hence, one wants to apply sparsity to the rows of $x$, i.e. $x$ must have a small number of nonzero rows, instead of applying the sparsity on all the components of $x$. This results in a row-structured sparsity penalty which is modelled using a mixed ${\ell }_2$-${\ell }_0$ norm.

Finally, numerical experiments demonstrate the efficiency of the proposed approach compared to classical methods as ${\ell }_1$ relaxation, Iterative Hard Thresholding or MUSIC algorithms and that it can reach the Cramer Rao Bound in some cases [4] .