Pilot Contamination: an Ultimate Limitation?

Many misconceptions float around about the pilot contamination phenomenon. While existent in any multi-cellular system, its effect tends to be particularly pronounced in Massive MIMO due to the presence of coherent interference, that scales proportionally to the coherent beamforming gain. (Chapter 4 in Fundamentals of Massive MIMO gives the details.)

A good system design definitely must not ignore pilot interference. While it is easily removed “on the average” through greater-than-one reuse, the randomness present in wireless communications – especially the shadow fading – will occasionally cause a few terminals to be severely hit by pilot contamination and bring down their performance. This is problematic whenever we are concerned about the provision of uniformly great service in the cell – and that is one of the principal selling arguments for Massive MIMO. Notwithstanding, the impact of pilot contamination can be reduced significantly in practice by appropriate pilot reuse and judicious power control. (Chapters 5-6 in Fundamentals of Massive MIMO gives many details.)

A more fundamental question is whether pilot contamination could be entirely overcome: Does there exist an upper bound on capacity that saturates as the number of antennas, M, is increased indefinitely? Some have speculated that it cannot; much in line with known capacity upper bounds for cellular base station cooperation. While this question may be of more academic than practical interest, it has long been open except for in some trivial special cases: If the channels of two terminals lie in non-overlapping subspaces and Bayesian channel estimation is used, the channel estimates will not be contaminated; capacity grows as log(M) when M increases without bound.

A much deeper result is established in this recent paper: the subspaces of the channel covariances may overlap, yet capacity grows as log(M). Technically, a Rayleigh fading with spatial correlation is assumed, and the correlation matrices for the contaminating terminals must only be linearly independent as M goes to infinity (exact conditions in the paper). In retrospect, this is not unreasonable given the substantial a priori knowledge exploited by the Bayesian channel estimator, but I found it amazing how weak the required conditions on the correlation matrices are. It remains unclear whether the result generalizes to the case of a growing number of interferers: letting the number of antennas go to infinity and then growing the network is not the same thing as taking an “infinite” (scalable) network and increasing the number of antennas. But this paper elegantly and rigorously answers a long-standing question that has been the subject of much debate in the community – and is a recommended read for anyone interested in the fundamental limits of Massive MIMO.

Pilot Contamination in a Nutshell

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.

Teaching a Course on Massive MIMO?

Our Cambridge book, Fundamentals of Massive MIMO, ships now from all major retailers.

Problem set: We have developed an extensive set of problems to go with the book. This problem set can be downloaded from the Cambridge resource page, www.cambridge.org/Marzetta, or from this direct link.

The difficulty level of the problem varies widely, rendering the material suitable for instruction at all levels. The problem set is very much a living document and may be extended or improved in the future. Many, though not all, of the problems have been tested on my students when I taught the subject last year. We appreciate, as always, comments or suggestions on the material.

A detailed solution manual is available to instructors who adopt the book.



List of errata: There is also a list of errata to the book – available via this direct link, or from the Cambridge resource page.

Have no fear of perfection — you’ll never reach it. — Salvador Dali

Which Technology Can Give Greater Value?

The IEEE GLOBECOM conference, held in Washington D.C. this week, featured many good presentations and exhibitions. One well-attended event was the industry panel “Millimeter Wave vs. Below 5 GHz Massive MIMO: Which Technology Can Give Greater Value?“, organized by Thomas Marzetta and Robert Heath. They invited one team of Millimeter Wave proponents (Theodore Rappaport, Kei Sakaguchi, Charlie Zhang) and one team of Massive MIMO proponents (Chih-Lin I, Erik G. Larsson, Liesbet Van der Perre) to debate the pros and cons of the two 5G technologies.


For millimeter wave, the huge bandwidth was identified as the key benefit. Rappaport predicted that 30 GHz of bandwidth would be available in 5 years time, while other panelists made a more conservative prediction of 15-20 GHz in 10 years time. With such a huge bandwidth, a spectral efficiency of 1 bit/s/Hz is sufficient for an access point to deliver tens of Gbit/s to a single user. The panelists agreed that much work remains on millimeter wave channel modeling and the design of circuits for that can deliver the theoretical performance without huge losses. The lack of robustness towards blockage and similar propagation phenomena is also a major challenge.

For Massive MIMO, the straightforward support of user mobility, multiplexing of many users, and wide-area coverage were mentioned as key benefits. A 10x-20x gain in per-cell spectral efficiency, with performance guarantees for every user, was another major factor. Since these gains come from spatial multiplexing of users, rather than increasing the spectral efficiency per user, a large number of users are required to achieve these gains in practice. With a small number of users, the Massive MIMO gains are modest, so it might not be a technology to deploy everywhere. Another drawback is the limited amount of spectrum in the range below 5 GHz, which limits the peak data rates that can be achieved per user. The technology can deliver tens of Mbit/s, but maybe not any Gbit/s per user.

Although the purpose of the panel was to debate the two 5G candidate technologies, I believe that the panelists agree that these technologies have complementary benefits. Today, you connect to WiFi when it is available and switch to cellular when the WiFi network cannot support you. Similarly, I imagine a future where you will enjoy the great data rates offered by millimeter wave, when you are covered by such an access point. Your device will then switch seamlessly to a Massive MIMO network, operating below 5 GHz, to guarantee ubiquitous connectivity when you are in motion or not covered by any millimeter wave access points.

Extreme Massive MIMO

Suppose extra antennas and RF chains came at no material cost. How large an array could eventually be useful, and would power consumption eventually render “extreme Massive MIMO” infeasible?

I have argued before that in a mobile access environment, no more than a few hundred of antennas per base station will be useful. In an environment without significant mobility, however, the answer is different. In [1, Sec. 6.1], one case study establishes the feasibility of providing (fixed) wireless broadband service to 3000 homes, using a single isolated base station with 3200 antennas (zero-forcing processing and max-min power control). The power consumption of the associated digital signal processing is estimated in [1, homework #6.6] to less than 500 Watt. The service of this many terminals is enabled by the long channel coherence (50 ms in the example).

Is this as massive as MIMO could ever get? Perhaps not. Conceivably, there will be environments with even larger channel coherence. Consider, for example, an outdoor city square with no cars or other traffic – hence no significant mobility. Eventually only measurements can determine the channel coherence, but assuming for the sake of argument 200 ms by 400 kHz, gives room for training of 40,000 terminals (assuming no more than 50% of resources are spent on training). Multiplexing these terminals would require at least 40,000 antennas, which would, at 3 GHz and half wavelength-spacing, occupy an area of 10 x 10 meters (say with a rectangular array for the sake of argument) – easily integrated onto the face of a skyscraper.

  • What gross rate would the base station offer? Assuming, conservatively, 1 bit/s/Hz spectral efficiency (with the usual uniform-service-for-all design), the gross rate in a 25 MHz bandwidth would amount to 1 Tbit/s.
  • How much power would the digital processing require? A back-of-the envelope calculation along the lines of the homework cited above suggests some 15 kW – the equivalent of a few domestic space heaters (I will return to the “energy efficiency” hype later on this blog).
  • How much transmit power is required? The exact value will depend on the coverage area, but to appreciate the order of magnitude, observe that when doubling the number of antennas, the array gain is doubled. If, simultaneously, the number of terminals is doubled, then the total radiated power will be independent of the array size. Hence, transmit power is small compared to the power required for processing.

Is this science fiction or will we be seeing this application in the future? The application is fully feasible, with today’s circuit technology, and does not violate known physical or information theoretic constraints. Machine-to-machine, IoT, or perhaps virtual-reality-type applications may eventually create the desirability, or need, to build extreme Massive MIMO.

[1] T. Marzetta, E. G. Larsson, H. Yang, H. Q. Ngo, Fundamentals of Massive MIMO, Cambridge University Press, 2016.


The Dense Urban Information Society

5G cellular networks are supposed to deal with many challenging communication scenarios where today’s cellular networks fall short.  In this post, we have a look at one such scenario, where Massive MIMO is key to overcome the challenges.

The METIS research project has identified twelve test cases for 5G connectivity. One of these is the “Dense urban information society”, which is

“…concerned with the connectivity required at any place and at any time by humans in dense urban environments. We here consider both the traffic between humans and the cloud, and also direct information exchange between humans or with their environment. The particular challenge lies in the fact that users expect the same quality of experience no matter whether they are at their workplace, enjoying leisure activities such as shopping, or being on the move on foot or in a vehicle.”

Source: METIS, deliverable D1.1 “Scenarios, requirements and KPIs for 5G mobile and wireless system

Hence, the challenge is to provide ubiquitous connectivity in urban areas, where there will be massive user loads in the future: up to  200,000 devices per km2 is predicted by METIS. In their test case, each device requests one data packet per minute, which should be transferred within one second. Hence, there is on average up to 200,000/60 = 3,333 users active per km2 at any given time.

Hexagonal cellular network, with adjacent cells having different colors for clarity.

This large number of users is a challenge that Massive MIMO is particularly well-suited for. One of the key benefits of the Massive MIMO technology is the high spectral efficiency that it achieves by spatial multiplexing of tens of user per cell. Suppose, for example, that the cells are deployed in a hexagonal pattern with a base station in each cell center, as illustrated in the figure. How many simultaneously active users will there be per cell in the dense urban information society? That depends on the area of a cell. An inter-site distance (ISD) of 0.25 km is common in contemporary urban deployments. In this case, one can show that the area covered by each cell is √3×ISD2/2 = 0.05 km2.


The number of active users per cell is then obtained by multiplying the cell area with the user density. Three examples are provided in the table below:

103 users/km2 104 users/km2 105 users/km2
Total number of users per cell 54 540 5400
Average active users per cell 0.9 9 90

Recall that 1/60 of the total number of users are active simultaneously, in the urban information society test case. This gives the numbers in the second row of the table.

From this table, notice that there will be tens of simultaneously active users per cell, when the user density is above 10,000 per km2. This is a number substantially smaller than the 200,000 per km2 predicted by the METIS project. Hence, there will likely be many future urban deployment scenarios with sufficiently many users to benefit from Massive MIMO.

A fraction of these users can (and probably will) be offloaded to WiFi-like networks, maybe operating at mmWave frequencies. But since local-area networks provide only patchy coverage, it is inevitable that many users and devices will rely on the cellular networks to achieve ubiquitous connectivity, with the uniform quality-of-service everywhere.

In summary, Massive MIMO is what we need to realize the dream of ubiquitous connectivity in the dense urban information society.

Macrocell Massive MIMO at 4.5 GHz: Field Trials in Japan

This impressive experiment serves 23 terminals with 64 base station antennas, at 4.5 GHz carrier, with a reported total spectral efficiency in the cell of nearly 80 bps/Hz. Several of the terminals are mobile, though it is not clear how fast.

Merouane Debbah, Vice-President of the Huawei France R&D center, confirms to the Massive MIMO blog that this spectral efficiency was achieved in the downlink, using TDD and exploiting channel reciprocity. This comes as no surprise, as it is not plausible that this performance could be sustained with FDD-style CSI feedback.

Another piece of evidence, that the theoretical predictions of Massive MIMO performance are for real.

News – commentary – mythbusting