One of the most read posts on this blog is Six differences between Massive MIMO for sub-6 GHz and mmWave, where we briefly outlined the key differences between how the Massive MIMO technology would be implemented and utilized in different frequency bands. Motivated by the great feedback and interest in this topic, we joined forces with Liesbet Van der Perre and Stefano Buzzi to write a full-length magazine article. It has recently been submitted to IEEE Wireless Communications and a pre-print can be found on ArXiv.org:
Several people have recently asked me for a simple way to quantify the spectral efficiency gains that we can expect from Massive MIMO. In theory, going from 4 to 64 antennas is just a matter of changing a parameter value. However, many practical issues need be solved to bring the technology into reality and the solutions might only be developed if we can convince ourselves that the gains are sufficiently large.
While there is no theoretical upper limit on how spectrally efficient Massive MIMO can become when adding more antennas, we need to set some reasonable first goals. Currently, many companies are trying to implement analog beamforming in a cost-efficient manner. That will allow for narrow beamforming, but not spatial multiplexing.
By following the methodology in Section 3.3.3 in Fundamentals of Massive MIMO, a simple formula for the downlink spectral efficiency is:
where is the number of base-station antennas, is the number of spatially multiplexed users, is the quality of the channel estimates, and is the number of channel uses per channel coherence block. For simplicity, I have assumed the same pathloss for all the users. The variable is the nominal signal-to-noise ratio (SNR) of a user, achieved when . Eq. (1) is a rigorous lower bound on the sum capacity, achieved under the assumptions of maximum ratio precoding, i.i.d. Rayleigh fading channels, and equal power allocation. With better processing schemes, one can achieve substantially higher performance.
To get an even simpler formula, let us approximate (1) as
by assuming a large channel coherence and negligible noise.
What does the formula tell us?
If we increase while is fixed , we will observe a logarithmic improvement in spectral efficiency. This is what analog beamforming can achieve and, hence, I am a bit concerned that the industry will be disappointed with the gains that they will obtain from such beamforming in 5G.
If we instead increase and jointly, so that stays constant, then the spectral efficiency will grow linearly with the number of users. Note that the same transmit power is divided between the users, but the power-reduction per user is compensated by increasing the array gain so that the performance per user remains the same.
The largest gains come from spatial multiplexing
To give some quantitative numbers, consider a baseline system with and that achieves 2 bit/s/Hz. If we increase the number of antennas to , the spectral efficiency will become 5.6 bit/s/Hz. This is the gain from beamforming. If we also increase the number of users to users, we will get 32 bit/s/Hz. This is the gain from spatial multiplexing. Clearly, the largest gains come from spatial multiplexing and adding many antennas is a necessary way to facilitate such multiplexing.
This analysis has implicitly assumed full digital beamforming. An analog or hybrid beamforming approach may achieve most of the array gain for , but I believe that the gains from spatial multiplexing will be substantially smaller.
Last year, the startup company Pivotal Commware secured venture capital (e.g., from Bill Gates) to bring its holographic beamforming technology to commercial products. Despite the word “holographic”, this is not a technology focused on visual-light communications. Instead, the company uses passive electronically steered antennas (PESAs) that are designed for radio-frequencies (RFs) in the micro- and millimeter-wave bands. It is the impedance pattern created in the distribution network over the array that is called a “hologram” and different holograms lead to beamforming in different spatial directions. The company reportedly aims at having commercial products ready this year.
Will the futuristic-sounding holographic beamforming make Massive MIMO obsolete? Not at all, because this is a new implementation architecture, not a new beamforming scheme or spatial multiplexing method. According to the company’s own white paper, the goal is to deliver “a new dynamic beamforming technique using a Software Defined Antenna (SDA) that employs the lowest C-SWaP (Cost, Size, Weight, and Power)“. Simply speaking, it is a way to implement a phased array in a thin, conformable, and affordable way. The PESAs are constructed using high volume commercial off-the-shelf components. Each PESA has a single RF-input and a distribution network that is used to vary the directivity of the beamforming. With a single RF-input, only single-user single-stream beamforming is possible. As explained in Section 1.3 in my recent book, such single-user beamforming can improve the SINR, but the rate only grows logarithmically with the number of antennas. Nevertheless, cost-efficient single-stream beamforming from massive arrays is one of the first issues that the industry tries to solve, in preparation for a full-blown Massive MIMO deployment.
The largest gains from multiple antenna technologies come from spatial multiplexing of many users, using a Massive MIMO topology where the inter-user interference is reduced by making the beams narrower as more users are to be multiplexed. The capacity then grows linearly with the number of users, as also explained in Section 1.3 of my book.
Can holographic beamforming be used to implement Massive MIMO with spatial multiplexing of tens of users? Yes, similar to hybrid beamforming, one could deploy an array of PESAs, where each PESA is used to transmit to one user. Eric J. Black, CTO and founder of Pivotal Commware, refers to this as “sub-aperture based SDMA“. If you want the capability of serving ten users simultaneously, you will need ten PESAs.
If the C-SWaP of holographic beamforming is as low as claimed, the technology might have the key to cost-efficient deployment of Massive MIMO. The thin and conformable form factor also makes me think about the recent concept of Distributed Large Intelligent Surface, where rooms are decorated with small antenna arrays to provide seamless connectivity.
“A dear child has many names” is a Swedish saying and it certainly applies to Massive MIMO. It is commonly claimed that the Massive MIMO concept originates from the seminal paper “Noncooperative Cellular Wireless with Unlimited Numbers of Base Station Antennas” by Thomas Marzetta, published in 2010. This is basically correct, except for the fact that the paper only talks about “Multi-user MIMO systems with very large antenna arrays“. Marzetta then published several papers using the large‐scale antenna systems (LSAS) terminology, before switching to calling it Massive MIMO in more recent years. Over the years, various papers have also called it “very large multiuser MIMO” and “large-scale MIMO“. Nowadays, Massive MIMO is used by almost everyone in the research community, and even by marketing people.
If you search at IEEEXplore, the origin of the name remains puzzling. The earliest papers are “Massive MIMO: How many antennas do we need?” by Hoydis/ten Brink/Debbah and “Achieving Large Spectral Efficiency with TDD and Not-so-Many Base-Station Antennas” by Huh/Giuseppe Caire/Papadopoulos/Ramprashad, both from 2011. However, these papers are referring to Marzetta’s seminal paper, which doesn’t call it “Massive MIMO”.
If you instead read the news reports by ZDNet and Silicon from the 2010 Bell Labs Open Days in Paris, the origin of “Massive MIMO” becomes clearer. Marzetta presented his concept and reportedly said that “We haven’t been able to come up with a catchy name”, but told ZDNet that “massive MIMO” and “large-scale MIMO” were two candidates. To the Massive MIMO blog, Marzetta now explains why he initially abandoned these potential names, in favor for LSAS:
When I explained the concept to the Bell Labs Director of Research, he commented that it didn’t sound at all like MIMO to him. He recommended strongly that I think of a name that didn’t contain the acronym “MIMO”, hence, LSAS. Eventually (after everyone else called it Massive MIMO) I abandoned “LSAS” and started to call it “Massive MIMO”.
In conclusion, the Massive MIMO name came originally from Marzetta, who used it when first describing the concept to the public, but the name was popularized by other researchers.
The term “Massive MIMO” has become synonymous with providing massive data rates in wireless networks, but this is not the technology’s only good trait. In this video presentation, which has also been given as an IEEE 5G webinar, I explain how Massive MIMO can enhance future cellular networks from many different perspectives.
Many radio resource allocation tasks are combinatorial in nature. It might be to associate a user equipment (UE) to a base station (BS) from a set of BSs, to select a set of time-frequency resources for transmission to a particular UE, or to assign pilot sequences to a set of users. The unfortunate thing with discrete combinatorial optimization problems is that the number of combinations grows very rapidly with the number of UEs and the number of discrete options that can be made for each of them. For example, suppose there are K UEs and you have to pick one out of D options for each of them, then there are DK different combinations. Hence, the worst-case computational complexity grows exponentially with K.
Interestingly, some radio resource allocation problems that appear to have exponential complexity can be relaxed to a form that is much easier to solve – this is what I call “relax and conquer”. In optimization theory, relaxation means that you widen the set of permissible solutions to the problem, which in this context means that the discrete optimization variables are replaced with continuous optimization variables. In many cases, it is easier to solve optimization problems with variables that take values in continuous sets than problems with a mix of continuous and discrete variables.
A basic example of this principle arises when communicating over a single-user MIMO channel. To maximize the achievable rate, you first need to select how many data streams to spatially multiplex and then determine the precoding and power allocation for these data streams. This appears to be a mixed-integer optimization problem, but Telatar showed in his seminal paper that it can be solved by the water-filling algorithm. More precisely, you relax the problem by assuming that the maximum number of data streams are transmitted and then you let the solution to a convex optimization problem determine how many of the data streams that are assigned non-zero power; this is the optimal number of data streams. Despite the relaxation, the global optimum to the original problem is obtained.
There are other, less known examples of the “relax and conquer” method. Some years ago, I came across the paper “Jointly optimal downlink beamforming and base station assignment“, which has received much less attention than it deserves. The UE-BS association problem, considered in this paper, is non-trivial since some BSs might have many more UEs in their vicinity than other BSs. Nevertheless, the paper shows that one can solve the problem by first relaxing it so that all BSs transmit to all the UEs. The author formulates a relaxed optimization problem where the beamforming vectors (including power allocation) are selected to satisfy each UEs’ SINR constraint, while minimizing the total transmit power. This problem is solved by convex optimization and, importantly, the optimal solution is always such that each UE only receives a non-zero signal power from one of the BSs. Hence, the seemingly difficult combinatorial UE-BS association problem is relaxed to a convex optimization problem, which provides the optimal solution to the original problem!
I have reused this idea in several papers. The first example is “Massive MIMO and Small Cells: Improving Energy Efficiency by Optimal Soft-cell Coordination“, which considers a similar setup but with a maximum transmit power per BS. The consequence of including this practical constraint is that it might happen that some UEs are served by multiple BSs at the optimal solution. These BSs send different messages to the UE, which decode them by successive interference cancelation, thus the solution is still practically achievable.
One practical weakness with the two aforementioned papers is that they take small-scale fading realizations into account in the optimization, thus the problem must be solved once per coherence interval, requiring extremely high computational power. More recently, in the paper “Joint Power Allocation and User Association Optimization for Massive MIMO Systems“, we applied the same “relax and conquer” method to Massive MIMO, but targeting lower bounds on the downlink ergodic capacity. Since the capacity bounds are valid as long as the channel statistics are fixed (and the same UEs are active), our optimized BS-UE association can be utilized for a relatively long time period. This makes the proposed algorithm practically relevant, in contrast to the prior works that are more of academic interest.
Another example of the “relax and conquer” method is found in the paper “Joint Pilot Design and Uplink Power Allocation in Multi-Cell Massive MIMO Systems”. We consider the assignment of orthogonal pilot sequences to users, which appears to be a combinatorial problem. Instead of assigning a pilot sequence to each UE and then allocate power, we relax the problem by allowing each user to design its own pilot sequence, which is a linear combination of the original orthogonal sequences. Hence, a pair of UEs might have partially overlapping sequences, instead of either identical or orthogonal sequences (as in the original problem). The relaxed problem even allows for pilot contamination within a cell. The sequences are then optimized to maximize the max-min performance. The resulting problem is non-convex, but the combinatorial structure has been relaxed so that there are only optimization variables from continuous sets. A local optimum to the joint pilot assignment and power control problem is found with polynomial complexity, using standard methods from the optimization literature. The optimization might not lead to a set of orthogonal pilot sequences, but the solution is practically implementable and gives better performance.
I never thought it would happen so fast. When I started to work on Massive MIMO in 2009, the general view was that fully digital, phase-coherent operation of so many antennas would be infeasible, and that power consumption of digital and analog circuitry would prohibit implementations for the foreseeable future. More seriously, reservations were voiced that reciprocity-based beamforming would not work, or that operation in mobile conditions would be impossible.
These arguments, it turned out, all proved to be wrong. In 2017, Massive MIMO was the main physical-layer technology under standardization for 5G, and it is unlikely that any serious future cellular wireless communications system would not have Massive MIMO as a main technology component.
But Massive MIMO is more than a groundbreaking technology for wireless communications: it is also an elegant and mathematically rigorous approach to teaching wireless communications. In the moderately-large number-of-antennas regime, our closed-form capacity bounds become convenient proxies for the link performance achievable with practical coding and modulation.
These expressions take into account the effects of all significant physical phenomena: small-scale and large-scale fading, intra- and inter-cell interference, channel estimation errors, pilot reuse (also known as pilot contamination) and power control. A comprehensive analytical understanding of these phenomena simply has not been possible before, as the corresponding information theory has too complicated for any practical use.
The intended audiences of Fundamentals of Massive MIMO are engineers and students. I anticipate that as graduate courses on the topic become commonplace, our extensive problem set (with solutions) available online will serve as a useful resource to instructors. While other books and monographs will likely appear down the road, focusing on trendier and more recent research, Fundamentals of Massive MIMO distills the theory and facts that will prevail for the foreseeable future. This, I hope, will become its most lasting impact.