5. General Formulation of Multichannel Extensions of NMF Variants

Blind source separation (BSS) is generally a mathematically ill-posed problem that involves separating out individual source signals from microphone array inputs. The frequency domain BSS approach is particularly notable in that it provides the flexibility needed to exploit various models for the time-frequency representations of source signals and/or array responses. Many frequency domain BSS approaches can be categorized according to the way in which the source power spectrograms and/or the mixing process are modeled. For source power spectrogram modeling, the non-negative matrix factorization (NMF) model and its variants have recently proved very powerful. For mixing process modeling, one reasonable way involves introducing a plane wave assumption so that the spatial covariances of each source can be described explicitly using the direction of arrival (DOA). This chapter provides a general formulation of the frequency domain BSS that makes it possible to incorporate the models for the source power spectrogram and the source spatial covariance matrix. Through this formulation, we reveal the relationship between the state-of-the-art BSS approaches. We further show that combining these models allows us to solve the problems of source separation, DOA estimation, dereverberation, and voice activity detection in a unified manner.