Conjecture on optimal weighted beamforming vectors for the two-user broadcast channel

Let ${\bf\Sigma}_1$ and ${\bf\Sigma}_2$ denote the channel covariance matrices, let $\rho$ be the signal-to-noise ratio, and let $\mathcal R_{\sf weighted}$ be the weighted ergodic sum-rate. Write ${\sf Dom.\ eig}(\bullet)$ for the unit-norm dominant eigenvector operation, and let $\alpha^{\star}(\rho),\beta^{\star}(\rho),\gamma^{\star}(\rho),\delta^{\star}(\rho)$ be the parameters maximizing the ergodic sum-rate under the weighted candidate beamforming vectors specified in the source. Weighted optimal beamforming conjecture. In the intermediate-$\mathsf{SNR}$ regime, the weighted ergodic sum-rate is maximized by

$${\bf w}_{1}=e^{j\nu_1}{\sf Dom.\ eig}\left(\left(\alpha^{\star}(\rho){\bf\Sigma}_2+{\bf I}\right)^{-1}\left(\gamma^{\star}(\rho){\bf\Sigma}_1+{\bf I}\right)\right),$$ $${\bf w}_{2}=e^{j\nu_2}{\sf Dom.\ eig}\left(\left(\beta^{\star}(\rho){\bf\Sigma}_1+{\bf I}\right)^{-1}\left(\delta^{\star}(\rho){\bf\Sigma}_2+{\bf I}\right)\right),$$

for some $\nu_i\in[0,2\pi)$, $i=1,2$. The conjecture is motivated by numerical studies suggesting similar performance for all channel covariance matrices and all weight choices, but its optimality is not proved in the supplied text.