Underdetermined inverse sparse signal reconstruction problems in the presence of non-Gaussian noise interference are often encountered in high-mobility wireless communications and signal processing. These problems can be solved by finding the minimizer of a suitable objective function which consists of a data-fitting term and a regularization term with different mixed-norms. Based on the Gaussian-noise assumption, two mixed norms (i.e. ℓ2/ℓ1 and ℓ∞/ℓ1) were confirmed as effective as well as stable algorithms for reconstructing sparse signals. However, the two algorithms are unable to reconstruct signal stable under non-Gaussian noise environments. In this paper, we propose a stable least absolute deviation (LAD) algorithm (i.e., ℓ1/ℓ1) for achieving two aspects: exploiting signal sparse structure information as well as mitigating the non-Gaussian noise interference. First of all, regularization parameter of the proposed algorithm is selected via Monte Carlo simulations. Then, experimental results in different non-Gaussian environments are used to demonstrate the effectiveness of the proposed algorithm.