This is supposedly a simple question to answer; an antenna is a device that emits radio waves. However, it is easy to get confused when comparing wireless communication systems with different number of transmit antennas, because these systems might use antennas with different physical sizes and properties. In fact, you can seldom find fair comparisons between contemporary single-antenna systems and Massive MIMO in the research literature.

Each antenna type has a predefined radiation pattern, which describes its inherent directivity; that is, how the gain of the emitted signal differs in different angular directions. An ideal isotropic antenna has no directivity, but a practical antenna always has a certain directivity, measured in dBi. For example, a half-wavelength dipole antenna has 2.15 dBi, which means that there is one angular direction in which the emitted signal is 2.15 dB stronger than it would be with a corresponding isotropic antenna. On the other hand, there are other angular directions in which the emitted signal is weaker. This is not a problem as long as there will not be any receivers in those directions.

In cellular communications, we are used to deploying large vertical antenna panels that cover a 120 degree horizontal sector and have a strong directivity of 15 dBi or more. Such a panel is made up of many small radiating elements, each having a directivity of a few dBi. By feeding them with the same input signal, a higher dBi is achieved for the panel. For example, if the panel consists of 8 patch antenna elements, each having 7 dBi, then you get a 7+10·log₁₀(8) = 16 dBi antenna.

The picture above shows a real LTE site that I found in Nanjing, China, a couple of years ago. Looking at it from above, the site is structured as illustrated to the right. The site consists of three sectors, each containing a base station with four vertical panels. If you would look inside one of the panels, you will (probably) find 8 cross-polarized vertically stacked radiating elements, as illustrated in Figure 1. There are two RF input signals per panel, one per polarization, thus each panel acts as two antennas. This is how LTE with 8TX-sectors is deployed: 4 panels with dual polarization per base station.

At the exemplified LTE site, there is a total of 8·8·3 =192 radiating elements, but only 8·3 = 24 antennas. This disparity can lead to a lot of confusion. The Massive MIMO version of the exemplified LTE site may have the same form factor, but instead of 24 antennas with 16 dBi, you would have 192 antennas with 7 dBi. More precisely, you would connect each of the existing radiating elements to a separate RF input signal to create a larger number of antennas. Therefore, I suggest to use the following antenna definition from the book Massive MIMO Networks:

Note that, with this definition, an array that uses analog beamforming (e.g., a phased array) only constitutes one antenna. It is usually called an adaptive antenna since the radiation pattern can be changed over time, but it is nevertheless a single antenna. Massive MIMO for sub-6 GHz frequencies is all about adding RF chains (also known as antenna ports), while not necessarily adding more radiating elements than in a contemporary system.

What is the purpose of having more RF chains?

With more RF chains, you have more degrees of freedom to modify the radiation pattern of the transmitted signal based on where the receiver is located. When transmitting a precoded signal to a single user, you adjust the phases of the RF input signals to make them all combine constructively at the intended receiver.

The maximum antenna/array gain is the same when using one 16 dBi antenna and when using 8 antennas with 7 dBi. In the first case, the radiation pattern is usually static and thus only a line-of-sight user located in the center of the cell sector will obtain this gain. However, if the antenna is adaptive (i.e., supports analog beamforming), the main lobe of the radiation pattern can be also steered towards line-of-sight users located in other angular directions. This feature might be sufficient for supporting the intended single-user use-cases of mm-wave technology (see Figure 4 in this paper).

In contrast, in the second case, we can adjust the radiation pattern by 8-antenna precoding to deliver the maximum gain to any user in the sector. This feature is particularly important for non-line-of-sight users (e.g., indoor use-cases), for which the signals from the different radiating elements will likely be received with “random” phase shifts and therefore add non-constructively, unless we compensate for the phases by digital precoding.

Note that most papers on Massive MIMO keep the antenna gain constant when comparing systems with different number of antennas. There is nothing wrong with doing that, but one cannot interpret the single-antenna case in such a study as a contemporary system.

Another, perhaps more important, feature of having multiple RF chains is that we can spatially multiplex several users when having multiple antennas. For this you need at least as many RF inputs as there are users. Each of them can get the full array gain and the digital precoding can be also used to avoid inter-user interference.

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:

Massive MIMO in Sub-6 GHz and mmWave: Physical, Practical, and Use-Case Differences

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:

(1)  

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

(2)  

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 for 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 . However, although hybrid beamforming allows for spatial multiplexing, I believe that the gains will be substantially smaller than with full digital beamforming.

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 D^K 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.