Separating source signals from observed signals poses significant challenges, particularly in underdetermined scenarios. Analyzing the sparse relationships among Time-Frequency (TF) vectors within their subspaces has shown promise. However, these approaches often lead to the NP-hard \(\ell ^0\) -norm minimization problem, especially when sparsity is insufficient. To overcome this limitation, we introduce a novel algorithm, UBSS-SAF, which substitutes the \(\ell ^0\) -norm with Smooth Approximation Functions (SAF). Our method effectively identifies the dominant vector within the one-dimensional subspace under sparse constraints and accurately estimates the mixing matrix. Compared to existing algorithms for underdetermined blind signal separation, the proposed method demonstrates superior performance in resolving the sparse representation relationships among all TF vectors of observed signals. Theoretical analysis and experimental evaluations have been conducted to validate the effectiveness of the proposed algorithm.
