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

We may simplify the estimator and shrink the search space by considering the delay mixing parameters to be irrelevant, which removes the frequency dependence of $p(D\vert u,v,r)$, yielding only one such set of distributions rather than $\omega$ of them. This is in fact valid for studio recordings in which a simple delay-free cross-fade is used to create stereo mixing. Formally, $\delta_u = \delta_v = 0$, and equation 17 becomes

$$a_i = \ensuremath{\frac{\vert a_u + a_v re^{j\theta}\vert}{\vert 1 + re^{j\theta}\vert}}. %%\label{ampest}$$ (18)

Now it is possible to proceed by considering the geometry of the numerator and denominator individually. Doing so for each and applying the law of cosines, we obtain

$$a_i = \sqrt{\ensuremath{\frac{a_u^2 + r^2 a_v^2 - 2 r a_u a_v \cos(\pi - \theta)}{1+ r^2 - 2 r \cos(\pi - \theta)}}}. %%\label{ampest}$$ (19)

Again treating $\theta$ as uniformly distributed, we may do a systematic fine grain search to numerically create a pdf for $a_i$ as before. Now, however, the number of distributions we must calculate is only $N$ choose 2, multiplied by the number of gradations in $r$, because the dependence on $\omega$ has been removed. Also, the histograms are one- rather than two-dimensional, because there is no delay parameter to consider.

This is still non-ideal because the discarded delay information may in fact be valid and useful. Though this may pursued in the future, we next consider an alternative, using DASSS to more efficiently represent the distributions $P(D\vert u,v,r,\omega)$ and include both amplitude and delay information.