Phase constrained complex NMF: Separating overlapping partials in mixtures of harmonic musical sources

This paper examines complex non-negative matrix factorization (CMF) as a tool for separating overlapping partials in mixtures of harmonic musical sources. Unlike non-negative matrix factorization (NMF), CMF allows for the development of source separation procedures founded on a mixture model rooted in the complex-spectrum domain (in which the superposition of overlapping sources is preserved). This paper introduces a physically motivated phase constraint based on the assumption that the source´s pitch is sufficient in specifying the phase evolution of the harmonics over time, uniting sinusoidal modelling of acoustic sources with the CMF analysis of their spectral representations. The CMF-based separation procedure, armed with this novel phase constraint, is demonstrated to offer a superior performance to NMF when employed as a tool for separating overlapping partials in the acoustic test cases considered.