Shifted and convolutive source-filter non-negative matrix factorization for monaural audio source separation

This paper proposes an extension of non-negative matrix factorization (NMF), which combines the shifted NMF model with the source-filter model. Shifted NMF was proposed as a powerful approach for monaural source separation and multiple fundamental frequency (F0) estimation, which is particularly unique in that it takes account of the constant inter-harmonic spacings of a harmonic structure in log-frequency representations and uses a shifted copy of a spectrum template to represent the spectra of different F0s. However, for those sounds that follow the source-filter model, this assumption does not hold in reality, since the filter spectra are usually invariant under F0 changes. A more reasonable way to represent the spectrum of a different F0 is to use a shifted copy of a harmonic structure template as the excitation spectrum and keep the filter spectrum fixed. Thus, we can describe the spectrogram of a mixture signal as the sum of the products between the shifted copies of excitation spectrum templates and filter spectrum templates. Furthermore, the time course of filter spectra represents the dynamics of the timbre, which is important for characterizing the feature of an instrument sound. Thus, we further incorporate the non-negative matrix factor deconvolution (NMFD) model into the above model to describe the filter spectrogram. We derive a computationally efficient and convergence-guaranteed algorithm for estimating the unknown parameters of the constructed model based on the auxiliary function approach. Experimental results revealed that the proposed method outperformed shifted NMF in terms of the source separation accuracy.