Optimal precoding matrices conjecture for slow-fading MIMO channels

Let $\Gamma(\mathbf{Q},R)$ denote the goodput-to-power ratio, let $P_{\mathrm{out}}(\mathbf{Q},R)$ denote the outage probability, and let $\overline{P}$ be the available transmit-power constraint. Let $n_t$ be the number of transmit antennas and let $\mathbf{I}_{n_t}$ be the $n_t\times n_t$ identity matrix. Optimal precoding matrices conjecture. There exists a power threshold $\overline{P}_0$ such that:

• if $\overline{P}\leq\overline{P}_0$, then

$$\mathbf{Q}^*\in\arg\min_{\mathbf{Q}}P_{\mathrm{out}}(\mathbf{Q},R)\quad\Rightarrow\quad\mathbf{Q}^*\in\arg\max_{\mathbf{Q}}\Gamma(\mathbf{Q},R);$$

• if $\overline{P}>\overline{P}_0$, then $\Gamma(\mathbf{Q},R)$ has a unique maximum at

$$\mathbf{Q}^*=\frac{p^*}{n_t}\mathbf{I}_{n_t},\qquad p^*\leq\overline{P}.$$

The conjecture has been validated for all the special cases solved in the paper. It says that below the threshold, maximizing the goodput-to-power ratio is equivalent to minimizing outage probability, whereas above the threshold uniform power allocation is optimal and using all available power is generally suboptimal for energy efficiency.