In this paper, a coordinate descent algorithm for robust sparse signal representation in redundant dictionaries is proposed. Under the coordinate descent framework, each target coefficient is robustly estimated applying the weighted median to a scaled-and-shifted version of the input signal weighted by the magnitude of an atom associated to the underlying coefficient. Sparsity is induced by appending, in the weighted median operation, a zero-valued sample weighted by an adaptive parameter. This leads to a generalized thresholding function over each target coefficient minimizing, thus, both the bias on the nonzero-value estimates and the sensitivity to small levels of noise. Furthermore, a continuation approach is included in order to set a suitable value of the regularization parameter that leads to the best representation at a current noise level. Numerical simulations are presented, in the context of compressive sensing, to compare the performance of the proposed algorithm to those yielded by state-ofthe-art methods.