We propose a new algorithm for the adaptive identification of sparse systems. The algorithm is based on the minimization of the RLS cost function when regularized by adding a sparsity inducing ℓ₁ norm penalty. The resulting recursive update equations for the system impulse response estimate are in a similar form to the regular RLS. However, they include novel terms which account for the sparsity prior. The proposed, ℓ₁ relaxation based RLS algorithm emphasizes sparsity during the adaptive filtering process and allows for faster convergence when the system under consideration is sparse. Computer simulations comparing the performance of the proposed algorithm to conventional RLS and other adaptive algorithms are provided. Simulations demonstrate that the new algorithm exploits the inherent sparse structure effectively.