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.