Bayesian Framework

As suggested above, the DASSS data produced by equation 4 may in fact reveal which two sources are active at a particular point in time-frequency space if exactly two sources are active. This is very useful information, because once the active sources are known, they may be demixed by solving for $S_u$ and $S_v$ in:

$$\left[ \begin{array}{c} S_u \\ S_v \\ \end{array} \right] = \left[ \begin{array}{cc} 1 & 1 \\ a_u e^{-j\omega\delta_u} & a_v... ...nd{array} \right]^{-1} \left[ \begin{array}{c} X_1 \\ X_2 \end{array} \right],$$ (12)

which follows directly from equations 1 and 2 when only two sources are active. Formally, we express the two most likely sources given some DUET or DASSS data $D$ as those maximizing $p(u,v\vert D)$. Applying Bayes' rule, we can express this as

$$p(u,v\vert D) = \ensuremath{\frac{p(D\vert u,v)p(u,v)}{p(D)}}.$$ (13)

We can see by inspection of equations 9 through 11 that the STFT frequency under consideration, $\omega$, affects DASSS data $D$ (the $Y_i$ values). So, we are mindful that the problem is a different one for each of the frequency values $\omega$ under consideration. Since $p(D)$ is not a function of $u$ or $v$, it is possible to discard it from the maximization (though we may wish to use it later if a confidence measure is sought). The quantity $p(u,v)$ is largely estimated with musical knowledge. For example we may know that the clarinet (source $u=1$ for example) tends to play at the same time as and less loudly than the violin (say, source $v=2$), whose frequency components tend to be at harmonics of frequencies above 200 Hz, and rarely throughout time. Though very useful, such information is not within our current signal processing interest, and is not considered now. (For now, we will treat all $p(u,v)$ as equally likely.) We are left, then, to consider $p(D\vert u,v)$ for each $\omega$, the probability that particular DASSS data is produced when sources $v$ and $u$ (but no others) are active at frequency $\omega$. To this end, we next explicitly return to the distributions suggested by equations 9 through 11. By doing so, we identify the necessary values of $p(D\vert u,v)$ where $D$ represents DASS scores $Y_i$.