Under-determined audio source separation using the convolutive narrowband approximation and flexible ℓp regularizer

Convolutive narrowband approximation (CNA) model is widely utilized in audio source separation, particularly in strongly reverberant scenarios. However, in under-determined audio source separation, the CNA model often faces a serious ill-conditioned inverse filtering problem due to its mixing matrices containing columns that approach zero. To mitigate this issue, a ${\ell }_p$ regularizer-based approach is proposed in this paper, assuming mixing filters of CNA model are known or pre-estimated. First, the STFT mixture components of all frames at each frequency bin are concatenated into a long vector, and the CNA system is accordingly expanded to a block-circulant structured linear mixing model. Next, a maximum likelihood estimation, constrained by the proposed mixing model, is introduced to exploit the sparsity of STFT source components with the ${\ell }_p$ regularizer, where these components are assumed to independently follow super-Gaussian distributions. Finally, an augmented Lagrange multiplier with efficient iterative strategy is developed to search for the suitable sparse solution. The proposed strategy with regularizer p can be flexibly selected in a range of $(0,1]$ to achieve robustness and accuracy in performance. Experimental results in various under-determined cases demonstrate the superior performance of the proposed algorithm over state-of-the-art approaches.