This paper revisits the CHAMPAGNE algorithm within the Sparse Bayesian Learning (SBL) framework and establishes its connection to reweighted sparse coding. We demonstrate that the SBL objective can be reformulated as a reweighted $\ell_{21}$-minimization problem, providing a more straightforward interpretation of the sparsity mechanism and enabling the design of an efficient iterative algorithm. Additionally, we analyze the behavior of this reformulation in the low signal-to-noise ratio (SNR) regime, showing that it simplifies to a weighted $\ell_{21}$-regularized least squares problem. Numerical experiments validate the proposed approach, highlighting its improved computational efficiency and ability to produce exact sparse solutions, particularly in simulated MEG source localization tasks.