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.

If non-orthogonal pilot sequences also cause pilot contamination, how about using random pilots ? What is overall impact ?

Random pilots lead to pilot contamination as well. For each realization of the random pilots, you will get a set of pilot sequences that is either orthogonal or non-orthogonal. Then the pilot contamination effect follows in the same way as for deterministic pilots.

The potential benefit of having random pilots is that you can “share” the pilot contamination between the users, so that everyone get an equal share on average. However, since cell-edge users are more sensitive to pilot contamination than cell-center users, you might want to coordinate the pilot allocation across cells instead of randomize it. In that sense, random pilots is the baseline scheme that any “smart” scheme should be able to beat.

Thank you for the this interesting topic.

I would only like to inquire what is specific for Massive MIMO systems? Pilot contamination can occur in any cellular system. So, is the only difference the limited number of orthogonal pilot signal due to the massive antenna array?

Thank you

Massive MIMO is supposed to serve tens of users simultaneously, instead of one or a few as in conventional cellular networks. Hence, for a given number of orthogonal pilots, each pilot needs to be reused more frequently in space. That is why the pilot contamination interference caused by pilot reuse might have a greater impact in Massive MIMO.

The number of antennas does not affect the required number of pilots, if we use TDD and channel reciprocity.

I need some explanation on coherence interval and how to calculate it? I also wonder what is the covariance matrix?

The coherence interval is explained in many books and papers. One freely available source is the following book chapter:

http://www.diva-portal.org/smash/get/diva2:1049059/FULLTEXT02.pdf

Its dimensionality is the product of the coherence time and the coherence bandwidth. You can check out Chapter 2 in “Fundamentals “Fundamentals of Massive MIMO” or “Massive MIMO Networks” for details on how to compute these parameters in different scenarios. I think Fundamentals of Wireless Communications also contain discussions about this.

The covariance matrix of a random channel h is E{(h-E{h})(h-E{h})^H}, where E{} stands for expectation.

Thank you sir, really appreciate it.

What is the relationship between the length of a coherent interval and speed of a user equipment in massive MIMO? To be specific in Zhu et., al (2016) SPR and MBD technique! I want to validate my result with their work using the same base parameters but with my own developed model. Thank you.

The length of the coherence interval is the product of the coherence bandwidth and the coherence time. These parameters depend on the carrier frequency, user speed, and propagation environment, and is the same irrespective of single-antenna or massive MIMO communication.

You can find examples of how to compute the coherence time and coherence bandwidth in Fundamentals of Wireless Communications, which is freely available here: https://web.stanford.edu/~dntse/wireless_book.html

You can also find a detailed explanation in the book “Fundamentals of Massive MIMO” and a short description in the book “Massive MIMO Networks”.

“When estimating the channel from the desired terminal, the base station cannot easily separate the signals from the two terminals…”

Is it possible for each pilot signal to have a unique signature…..

Yes! This is called orthogonal pilot sequences and is normally assumed to be used within each cell in Massive MIMO.

So from your explanation pilot signals are affected by two factors:

1. Inter-cell interference

2. Noise in the channel

Does it also mean there is no way to know which pilot signal arrives first at the BS?

Just thinking about it.

The users are typically time synchronized with the base station in such a way that the signals are received at approximately the same time. There are still some small time differences, but you cannot exploit these to do something smart unless you know them, and you cannot know them without estimating them. It is a chicken-and-egg problem…

In a limited setting of a single cell:

Can pilot contamination be neglected, with orthogonal sequences allocated (feasible since there are only tens of active users)?

Or does it still need to be considered as an inter-user interference?

Yes! Most papers on single-cell Massive MIMO are considering a setup without pilot contamination, motivated in the way that you suggested.

Is there any need to do comparative analysis of fading models in massive MIMO before choosing a particular model? Or is the Rayleigh fading model generally adopted?

Many academic papers consider i.i.d. Rayleigh fading, since it leads to tractable and intuitive expressions. But practical channels will not be i.i.d. Rayleigh fading, so there is a need to evaluate other more realistic models to figure out if there are any substantial differences. In my recent book “Massive MIMO Networks”, we show how to analyze correlated Rayleigh fading. We also use 3GPP channel models to carry out realistic simulations.

Why do you assume that you have only one channel gain between the user and the antenna (flat-fading channel), but in the frequency selective case, there is an attenuation for each frequency, and we will have multiple channel gains for each frequency. Does this change the result of spectral efficiency for example in the presence of pilot contamination?

Most communication systems are indeed designed for frequency-selective channels, but the common approach is to use multi-carrier modulation to split the frequency-selective channel into many frequency-flat subchannels/subcarriers. Chapter 2 in Fundamentals of Massive MIMO provides a good explanation of that.

Thanks!

Thank you for the last suggestion. I read Chapter 2 in Fundamentals of Massive MIMO. Always concerning the OFDM associated to Massive MIMO and the fact that one complex gain is used between any pair of antennas. If I understood it correctly, the condition is that the slot duration is equal to the channel coherence time, then each coherence interval spans Nsmooth subcarriers and Nslot consecutive OFDM symbols. So the bandwidth are splitted into N subcarriers but only Nsmooth subcarriers are used. Is it true?

Concerning the question on Chapter 2: “So the bandwidth are splitted into N subcarriers but only Nsmooth subcarriers are used. Is it true?”

No. The entire system bandwidth, B0, is used. Effectively multiple consecutive coherence intervals are used simultaneously. But each coherence interval can be treated independently in the analysis. See figures 2.4 and 2.6.

Good day prof. Please I need a suggestion on what other matrices can be use to separate users sharing the same pilot sequence apart from Hadamard matrix.

Any unitary matrix will be equally good from a theoretical perspective, but there might be implementation aspects that may some more useful than others. An alternative to a Hadamard matrix is a DFT matrix or Zadoff-Chu sequences. This is briefly discussed in Section 3.1.1 of the book Massive MIMO Networks.

thank prof

Hello sir, I read in one of the journals on acquiring channel state information in massive MIMO that someone can reduce the computational complexity by whitening (projecting into signal subspace) the received signal vector. My question here is: how do you whiten such a signal? and is there any method someone can use to reduce the computational part from the stated above. I anticipate your kind response. thank you.

I don’t know which journal paper you are referring to. Whitening and projecting onto the signal space are actually two different things. I don’t think whitening by itself will reduce complexity – it is just an implementation detail. Projecting onto the signal space could potentially reduce the complexity of the remaining tasks, but you need to know the signal space for it to work.

I don’t think we need to worry much about the computational complexity, but first find the type of processing that performs the best and then find an efficient implementation of it.

Thank you prof. I really appreciate.

Does massive MIMO depend on spatial multiplexing or spatial diversity? If both, I need an explanation!

Strictly speaking, it doesn’t require any of these things, but:

a) The largest throughput gains from Massive MIMO comes from using it for spatial multiplexing. (http://ma-mimo.ellintech.se/2018/03/23/a-basic-way-to-quantify-the-massive-mimo-gain/)

b) The channel hardening property is a consequence of spatial diversity. (http://ma-mimo.ellintech.se/2017/01/25/channel-hardening-makes-fading-channels-behave-as-deterministic/)

Sir, what particular property of a unitary matrix makes it a better candidate for implementation purpose than another one?

That depends on what the unitary matrix is going to be used for. For example, when using a unitary matrix as pilot matrix, it is good if all the elements in the matrix have the same magnitude. You can read more about that in Section 3.1.1 of my book Massive MIMO Networks.