Affect on $p(D\vert u,v,r)$ for $D$: DUET Amplitude and Delay Estimates

Given the notation used above for sources $S_u$ and $S_v$, we may express the magnitude estimator in equation 3 as

$$a_i = \ensuremath{\frac{\vert X_2(\omega_k,\tau)\vert}{\vert X_1(\omega_k,\tau)\vert}}$$ (15)

$$= \ensuremath{\frac{\vert a_u e^{-j\omega\delta_u}S_u + a_v re^{-j(\omega\delta_v-\theta)}S_u\vert}{\vert S_u + re^{j\theta}S_u\vert}}$$ (16)

$$= \ensuremath{\frac{\vert a_u e^{-j\omega\delta_u} + a_v re^{-j(\omega\delta_v-\theta)}\vert}{\vert 1 + re^{j\theta}\vert}}.$$ (17)

Several observations are in order. First, we see that if $S_u$ is much stronger than $S_v$ we have $r\rightarrow 0$, and thus $a_i \rightarrow a_u$, yielding the one-source estimate as it should. For $r\rightarrow \infty$, we have $a_i \rightarrow a_v$, also as expected. In other non-special cases, however, it is clear that the phase offset $\theta$ and frequency $\omega$ under consideration come into play in addition to the mixing parameters themselves. However, there is no cause to model $\theta$ as anything other than a uniformly distributed random variable. Given this, we may treat equation 17 as a function of a random variable, with its own distribution, specifically, $P(D\vert u,v,r)$ for each $\omega$, as we sought.

We plot an example showing numeric approximations to such distributions in figure 1. In this example, the mixing parameters are as in the table below, and sources $u=1$ and $v=2$ are active.

source	$a_i$	$\delta_i$
1	1.005	-6.5e-5
2	0.995	-2.5e-5
3	0.960	2.0e-5

Each ring seen in the plot corresponds to a particular value of $r$. When $r$ is near zero, source $u$ (source 1 in this example) is much stronger than source $v$ (source 2 in this case) and the mixing parameter estimates lie very near to the mixing parameter values for source 1, shown with a yellow star. As $r$ increases toward 1, the rings grow larger in this space. When $r>1$, the rings begin to center on the mixing parameter estimate for source $v$ (source 2 here), and as $r$ grows large, the rings converge there.

Each point within a given ring in the plot corresponds to a different value of $\theta$, the aforementioned phase offset between $S_u$ and $S_v$. Here we have used 1000 evenly spaced values on the range $[0, 2\pi)$ to simulate a uniform distribution. It can be seen in the large shape of the rings that $\theta$ has a great effect on the estimates generated. We still see general trends in parameter locations, such as the very high density of parameter estimates in the region between the two sources' mixing parameter values. In figure 2, we plot the same data as in figure 1, but while assuming a larger value of $\omega$. As expected from equation 17, the data is not the same, necessitating different distributions for each $\omega$.

Figure 1: Mixing parameter estimates when sources 1 and 2 are active and $\omega = 3888$ rad/s.

Figure 2: Mixing parameter estimates when sources 1 and 2 are active and $\omega = 9719$ rad/s.

Though we have obtained and plotted the desired data $D = (a_i,\delta_i)$ for one combination of sources, we must do more before we may fully represent the data $P(D\vert u,v,r)$ for each $\omega$. We must numerically calculate two dimensional $(a_i,\delta_i)$ histograms with many data points, by choosing grid blocks in $(a_i,\delta_i)$ space. We choose to partition the parameter space in figure 1 into $\Lambda$ amplitude partitions by $\Delta$ delay partitions. We count the number of blue circle data points in a given partition of figure 1 corresponding to a specific source combination $(u,v)$ and magnitude ratio $r$. Thus, for each frequency, $\omega$, we have a set of two dimensional histograms for each $(u,v,r)$ combination. The quantity $P(D\vert u,v,r)$ is then represented by a total of $\Lambda \times \Delta$ values for each $(u,v,r,\omega)$ combination.

Though the numeric approach incurs great computational expense, the mathematical intractability of obtaining closed form expressions from equation 3 makes such an approach necessary. (The phase estimator is as intractable as the magnitude estimator.) This expense can be prohibitive, however. As an illustration, if we have 4 sources and 2 are active, we have 6 possible combinations. For each, we may have 1024 distinct $\omega$ points in our STFT, and might want to consider 40 values of $r$ (as in the example plot). To smoothly represent $D$, we may want $\{\Lambda,\Delta\} \approx 100$. Thus we have $6\cdot 1024\cdot 40 \cdot 100 \cdot 100 = 2,457,600,000$ distribution values of $P(D\vert u,v,r,\omega)$ to calculate, each of which might use 1000 evenly spaced $\theta$ values. These are nearly prohibitive numbers with current processing speed. The good news is that once the distributions are calculated, it is simple to search the values $P(D\vert u,v,r)$ by matching a newly detected value of $D$ to the appropriate two dimensional histogram bin. Then we may search a space of 240 distributions (in our example) to find the optimal one in expression 14.

Given the computational issue (and a certain lack of elegance), we have considered two alternatives: ignoring delay information entirely, which shrinks the parameter space, and using DASSS data instead of the equation 3 DUET estimators. Each is considered below.