Introduction

Sound source separation refers to the problem of synthesizing $N$ source signals given an $M$ channel mixture of those source signals. When there are fewer input mixtures than sources to be separated ($M < N$), we have the degenerate case. In the degenerate case, it is necessary to use prior information about the source signals to perform demixing, because of the ill-posed nature of the inverse mathematical problem. We presently consider the two mixture degenerate case. In digital audio, we frequently encounter this case, as many or most currently available commercial digital recordings contain two channels (stereo) but more than two instruments, voices, or other sounds. A variety of approaches to this and other degenerate problems have been tried [3]. Each method exploits one or more features of the sound sources, as they must do in order to be successful. Such features include the sources' time-frequency sparsity, their time-frequency independence, and their distinct amplitude and delay characteristics between the mixtures. A brief review of these techniques for the two source case is included in [1]. We find that the DUET system [4,5,1] has achieved particularly convincing results, but can still be improved. Specifically, we note that the system only works as intended when in fact the sources are distinct in time-frequency space. This is referred to as ``source sparsity'' although non-overlap of sources is also required. This is because co-occurring sparse sources cannot be separated. In performance of tonal Western music, sources are in general sparse because instrumental ranges are finite and most compositions do not require constant playing or singing throughout time. The sources, however, are not in general independent, unless the ensemble is without skill or the music requires that players sound notes in a deliberately random fashion. The harmonic nature of Western music exacerbates the problem, because harmonics whose fundamental frequencies are in (possibly imperfectly) consonant relations will overlap. Even in the case of dissonant or deliberately random music, pitches are in general discretized to the 12-tone Western scale, leading to overlap of some harmonics. Given these facts, it is necessary that the DUET system be modified if it is to deal with non-independent sources such as those seen in music. Presently, we consider a method for the case when exactly two unknown sources are present. This means that two instruments or voices are sounding though we do not know a priori if it is, for example, the bass and cello or cello and flute. Clearly, this case is only an incremental improvement of the current one-source-at-a-time system. However, in the cases of musical trios or four speaker examples, the two-source assumption is of great benefit. To consider the benefit in the current approach, we first review the DUET system and the related delay and scale subtraction scoring (DASSS) [2], and explore how these models are affected when two sources are present at the same point in time-frequency space. In the third section, we consider how to exploit the two-source system response in a Bayesian context. Specifically, we develop a method for scoring the probability that two particular sources are active given DASSS data. We conclude with a musical example showing the efficacy of using Bayesian Modeling of DASSS data rather than DUET for determining and demixing two active sources.