Affect on $p(D\vert u,v,r)$ for $D$: DASSS Data

To consider the affect of two active sources on DASS scores, we recall the definition of the DASS functions $Y_i$ in equation 4:

$$\begin{eqnarray*} Y_i &\equiv& X_1 - \ensuremath{\frac{1}{a_i}}e^{+j\omega\delta_i} X_2 \end{eqnarray*}$$

where the presence of exactly one source $S_g$ leads to model values:

$$\begin{eqnarray*} \hat{Y}_{i=g} &=& 0 \\ \hat{Y}_{i\neq g} &=& \alpha_{j,i} S_j \\ &=& \alpha_{j,i} X_1. \end{eqnarray*}$$

and we recall the definition

$$\alpha_{u,v} \equiv (1-\ensuremath{\frac{a_v}{a_u}}e^{j\omega(\delta_u - \delta_v)}).$$

It may be shown that when two sources $S_u$ and $S_v$ are active, the $Y_i$ values will not be as specified in the one source model case, but rather

$$\hat{Y}_{i=u} = \alpha_{uv} S_v = \alpha_{uv} r e^{j\theta} S_u$$ (20)

$$\hat{Y}_{i=v} = \alpha_{vu} S_u$$ (21)

$$\hat{Y}_{i \neq (u\vert v)} = \alpha_{iu} S_u + \alpha_{iv} S_v = \alpha_{iu} S_u + \alpha_{iv} r e^{j\theta} S_u$$ (22)

$$= S_u(\alpha_{iu} + r e^{j\theta}\alpha_{iv}).$$ (23)

We now make several observations. All other things being equal, the distributions on $\hat{Y}_{i=u}$ and $\hat{Y}_{i=v}$ will tend to have smaller magnitude than that on $\hat{Y}_{i \neq (u\vert v)}$. This by itself may suggest which two sources are active, and we pursue that angle in upcoming research. There is, however, more information to exploit, and we consider in more detail the data $D$. For a given frequency $\omega$, the magnitudes of $\hat{Y}_{i=u}$ and $\hat{Y}_{i=v}$ will be constant versus the phase offset $\theta$ between the two sources. In fact, they will be equal to the magnitude of the appropriate $\alpha$ value multiplied by the appropriate source amplitude (see equations 20 and 21). The values $\hat{Y}_{i \neq (u\vert v)}$, however will vary based on $\theta$ for a given frequency. An example using the same mixing parameters as before is shown in figure 3. We see that for this particular $(u,v,r)$ combination, $\hat{Y}_{i=u}$ and $\hat{Y}_{i=v}$ yield exactly one value as $\theta$ varies, but that $\hat{Y}_{i \neq (u\vert v)}$ yields a variety. A closed form expression for the extreme values of $\hat{Y}_{i \neq (u\vert v)}$ can be obtained by inspection of equation 22 or 23.

Figure 3: Top: DUET mixing parameter data when sources $u=1$ and $v=2$ are active, $r = 0.6$, and $\omega = 3888$ rad/s. Middle: DASS data $\vert Y_1\vert$ (yellow), $\vert Y_2\vert$ (green), and $\vert Y_3\vert$ (red). We see that $\vert Y_3\vert$ varies with $\theta$ but $\vert Y_1\vert$ and $\vert Y_2\vert$ do not. Bottom: Zoom-in of the histogram distributions on $\vert Y_i\vert$ corresponding to $P(D\vert u,v,r)$ (true values for $\vert Y_1\vert$ and $\vert Y_2\vert$ extend to 1000 as expected). In these plots, all $\vert Y_i\vert$ are unnormalized, and $\vert S_u\vert = 1$.

Due to these properties, we do not need $N$ full sets of points to store the distribution of $Y$ values for a particular $(u,v,r)$ combination. For $\hat{Y}_{i=u}$ and $\hat{Y}_{i=v}$, we must only record values of $\alpha$, because $\hat{Y}_{i=u}$ and $\hat{Y}_{i=v}$ are purely functions of $\alpha$ and $r$. The $\hat{Y}_{i \neq (u\vert v)}$ values, however, must be recorded as distributions due to the variation with $\theta$.

In order for these values to be meaningful, however, we must consider that the magnitude of $\vert S_u\vert$ affects all the $\vert\hat{Y}_i\vert$ values. Fortunately, we see by inspection of equations 20 through 23 that this magnitude multiplies all $\hat{Y}_i$ values equally; the ratios of the various $Y$ values to each other hold under a change in the magnitude $\vert S_u\vert$. Thus, when we store $Y$ values for $P(D\vert u,v,r)$, we need only store ratios. We arbitrarily choose to normalize the stored data such that $\hat{Y}_{i=v} = 1$.

As a final note, we observe that as the magnitude of $S_u$ or $S_v$ goes to zero, $r$ goes to $\infty$ or zero, respecitvely. We see by inspection of equations  20 through 23 that when this occurs, the two model values converge to the appropriate one source model values. We may understand the current model then, as a superset of the one source DUET system. We may set some arbitrary (or better, perceptually motivated) threshold on the minimum or maximum $r$ values at which we consider only one source active. At that point, we may allow $u$ or $v$ to be set to ``NULL,'' leaving only one active source in a two-source representation.