Basis Pursuit (BP) and Basis Pursuit Denoising (BPDN), well established techniques for computing sparse representations, minimize an ℓ² data fidelity term, subject to an ℓ¹ sparsity constraint or regularization term, by mapping the problem to a linear or quadratic program. BPDN with an ℓ¹ data fidelity term has recently been proposed, also implemented via a mapping to a linear program. We introduce an alternative approach via an Iteratively Reweighted Least Squares algorithm, providing computational advantages and greater flexibility in the choice of data fidelity term norm.