One of the main impairments in wireless communications is small-scale channel fading. This refers to random fluctuations in the channel gain, which are caused by microscopic changes in the propagation environments. The fluctuations make the channel unreliable, since occasionally the channel gain is very small and the transmitted data is then received in error.

The diversity achieved by sending a signal over multiple channels with independent realizations is key to combating small-scale fading. Spatial diversity is particularly attractive, since it can be obtained by simply having multiple antennas at the transmitter or the receiver. Suppose the probability of a bad channel gain realization is p. If we have M antennas with independent channel gains, then the risk that all of them are bad is p^M. For example, with p=0.1, there is a 10% risk of getting a bad channel in a single-antenna system and a 0.000001% risk in an 8-antenna system. This shows that just a few antennas can be sufficient to greatly improve reliability.

In Massive MIMO systems, with a “massive” number of antennas at the base station, the spatial diversity also leads to something called “channel hardening”. This terminology was used already in a paper from 2004:

M. Hochwald, T. L. Marzetta, and V. Tarokh, “Multiple-antenna channel hardening and its implications for rate feedback and scheduling,” IEEE Transactions on Information Theory, vol. 50, no. 9, pp. 1893–1909, 2004.

In short, channel hardening means that a fading channel behaves as if it was a non-fading channel. The randomness is still there but its impact on the communication is negligible. In the 2004 paper, the hardening is measured by dividing the instantaneous supported data rate with the fading-averaged data rate. If the relative fluctuations are small, then the channel has hardened.

Since Massive MIMO systems contain random interference, it is usually the hardening of the channel that the desired signal propagates over that is studied. If the channel is described by a random M-dimensional vector h, then the ratio ||h||²/E{||h||²} between the instantaneous channel gain and its average is considered. If the fluctuations of the ratio are small, then there is channel hardening. With an independent Rayleigh fading channel, the variance of the ratio reduces with the number of antennas as 1/M. The intuition is that the channel fluctuations average out over the antennas. A detailed analysis is available in a recent paper.

The figure above shows how the variance of ||h||²/E{||h||²} decays with the number of antennas. The convergence towards zero is gradual and so is the channel hardening effect. I personally think that you need at least M=50 to truly benefit from channel hardening.

Channel hardening has several practical implications. One is the improved reliability of having a nearly deterministic channel, which results in lower latency. Another is the lack of scheduling diversity; that is, one cannot schedule users when their ||h||² are unusually large, since the fluctuations are small. There is also little to gain from estimating the current realization of ||h||², since it is relatively close to its average value. This can alleviate the need for downlink pilots in Massive MIMO.

One word that is tightly connected with Massive MIMO is pilot contamination. This is a phenomenon that can appear in any communication system that operates under interference, but in this post, I will describe its basic properties in Massive MIMO.

The base station wants to know the channel responses of its user terminals and these are estimated in the uplink by sending pilot signals. Each pilot signal is corrupted by inter-cell interference and noise when received at the base station. For example, consider the scenario illustrated below where two terminals are transmitting simultaneously, so that the base station receives a superposition of their signals—that is, the desired pilot signal is contaminated.

When estimating the channel from the desired terminal, the base station cannot easily separate the signals from the two terminals. This has two key implications:

First, the interfering signal acts as colored noise that reduces the channel estimation accuracy.

Second, the base station unintentionally estimates a superposition of the channel from the desired terminal and from the interferer. Later, the desired terminal sends payload data and the base station wishes to coherently combine the received signal, using the channel estimate. It will then unintentionally and coherently combine part of the interfering signal as well. This is particularly poisonous when the base station has M antennas, since the array gain from the receive combining increases both the signal power and the interference power proportionally to M. Similarly, when the base station transmits a beamformed downlink signal towards its terminal, it will unintentionally direct some of the signal towards to interferer. This is illustrated below.

In the academic literature, pilot contamination is often studied under the assumption that the interfering terminal sends the same pilot signal as the desired terminal, but in practice any non-orthogonal interfering signal will cause the two effects described above.