Inverse of Two-Source Model

In many cases where all single source models $\hat{Y}_i^g$ score poorly, we have observed that none of the actual $Y_i$ values are near zero. Our model indicates that this will in fact occur when more than one source is active for the bin under consideration. Given such data, we may hypothesize that two or more sources are present. If we claim that exactly two specific sources $u$ and $v$ are active, the problem of partitioning the input into those two sources may be solved via simply inverting the mixing matrix for those two sources:

$$\left[ \begin{array}{c} S_u \\ S_v \\ \end{array} \right] = \left[ \begin{array}{cc} 1 & 1 \\ a_u e^{-j\omega\delta_u} & a_v... ...end{array} \right]^{-1} \left[ \begin{array}{c} X_1 \\ X_2 \end{array} \right]$$ (7)

A confounding issue, though, is that it is impossible to determine which two sources $u$ and $v$ are present without making some assumptions. Because we may in general generate an exact solution from the equation above regardless of our guess of $u$ and $v$, our results will always appear to lead to zero error relative to the observed $Y_i$ values. Hence in the current implementation, we assume that the two sources whose fractional error $f(g)$ are lowest are the active sources. (It may be shown that this relies on the fact that the sum of two independent errors is likely to be greater than a single error.) In practice, this assumption appears to be accurate, as the results have been psychoacoustically convincing. A similar problem exists in regard to the number of sources that are active. We do not even know that only two sources are present. We may set up and solve a 3 or more source version of equation 7 and solve with a pseudoinverse operation. Again, however, we cannot claim a successful or unsuccessful result. The above highlight the need for using prior knowledge about the input sources in choosing an approach for multi-source sound source separation. The concept for one such model is outlined in the next subsection.