**5. Metrics of spatial compatibility**

The sum rate maximization problem (4) can be solved by fixing the precoder structure and power allocation method. Under ZF-based precoding the performance strongly depends on the spatial correlation between the components of **H**(*S*). The more correlated the channels, the higher the power penalty imposed by ZF schemes which yields a degradation of the achievable SINRs and a poor system performance. For this reason, problem (4) has been tackled in the literature by optimizing the spatial compatibility between scheduled users. This is accom‐ plished by optimizing a specific metric over the channel matrix which can be related or can provide indirect information of the achievable sum rate for a given set of users and channel realization. A metric of spatial compatibility is a function of the CSI at the transmitter so that *f* :ℂ |*S* |×*Nt* ↦ℝ+ where the mapping quantifies how profitable is to select *S* for transmission. Different metrics for spatial compatibility have been proposed in the literature and this section presents a unified treatment of the most common metrics used by several algorithms that solve (4) sub-optimally. It is worth mentioning that the optimization over a given metric may bring some advantages in terms of computational complexity, for instance, iterative evaluation of *f* (**H**(*S*)) does not require the computation of the optimum power allocation. Some metrics are given by simple relations between the row vectors in **H**(*S*) which avoids matrix operations. Under certain SNR constraints described in Section 6, the user set that solves problem (4) achieves maximum multiplexing diversity, i.e., its cardinality is equal to *Nt* [5]. In such a SNR regime, the search space of the solution of problem (4) is reduced from *L* to *L Nt* = *K* ! /(*Nt* !(*K* - *Nt*)!) and optimization over *f* (**H**(*S*)) can simplify the search of the optimum user set *S*. In this section we introduce the channel metrics, its main properties, and the relations between them.
