The problem of multiple emitters direction finding using an array of sensors is addressed. We describe a sparsity-based covariance-matrix fitting method. The procedure consists of finding a sparse representation of the sample covariance matrix, using an over-complete basis obtained from array manifold samples. Sparsity is encouraged by an ℓ1-norm penalty function. The penalty function is minimized efficiently by linear programming. The proposed method is simple enough to provide useful insight and it does not require the identification of the signal and noise subspaces. Therefore, the method does not rely on a good estimate of the number of emitters. Some of the approach properties are super-resolution, robustness to noise, robustness to emitter correlation, and no sensitivity to initialization. Special emphasis is given to uncorrelated sources and uniform linear arrays.