# Pilot decontamination in angular domain

Massive MIMO is widely believed to be one of the key technologies for the 5th generation (5G) wireless networks. The realization of the high energy efficiency and throughput of massive MIMO, especially in the downlink transmission, relies on the accurate knowledge of the CSI. In practical wireless system, CSI is acquired based on training sequences sent by user terminals. Due to limited time and frequency resources, non-orthogonal pilot sequences are typically used by user terminals in neighboring cells, resulting in the so-called “pilot contamination”. In fact, it was believed that pilot contamination constituted a performance bottleneck of massive MIMO.

In this post, I will show a way to solve this problem.

Consider a base station equipped with $M$ antennas which form a uniform linear array (ULA). In the uplink training phase, users send pilot sequences to the base station. Assuming the statistics of the channel (channel covariance matrices) are known by the base station, we can construct an MMSE channel estimator $\widehat{{\mathbf{h}}}_j^{(j){\text{MMSE}}} = {\mathbf{R}}^{(j)}_j \left( \tau (\sum_{l=1}^{L}{\mathbf{R}}^{(j)}_l) + \sigma_n^2 {\mathbf{I}}_{M} \right)^{-1} {\mathbf{S}}^H \mathbf{y}^{(j)},$

where ${\mathbf{R}}^{(j)}_l$ is the channel covariance matrix, $\mathbf{S}$ is the training matrix, and $\mathbf{y}^{(j)}$ is the received signal at the base station. More details of this equation can be found in .

We have proved in  that with this MMSE estimator, pilot contamination can be made to vanish completely under a certain condition on the AoA distributions. This condition is what we call “non-overlapping” condition.
Theorem 1. If the AoA supports of the interference users are strictly non-overlapping with the AoA support of the desired user, then $\mathop {\lim }\limits_{M \to \infty } \widehat{{{\mathbf{h}}}}_j^{(j){\text{MMSE}}} = {\mathbf{\widehat h}}_j^{(j){\text{no int}}},$

where ${\mathbf{\widehat h}}_j^{(j){\text{no int}}}$ denotes the interference-free estimate.

An example of this non-overlapping condition is shown in Fig. 1. The multipath AoAs of user 1 are distributed within sector 1, while the AoAs of user 2 (interference user) are within sector 2, which does not overlap with sector 1.