The tedious, time-consuming, and buggy nature of system-level simulations is exacerbated with massive MIMO. This post offers some relieve in the form of analytical expressions for downlink conjugate beamforming . These expressions enable the testing and calibration of simulators—say to determine how many cells are needed to represent an infinitely large network with some desired accuracy. The trick that makes the analysis feasible is to let the shadowing grow strong, yet the ensuing expressions capture very well the behaviors with practical shadowings.
The setting is an infinitely large cellular network where each -antenna base station (BS) serves single-antenna users. The large-scale channel gains include pathloss with exponent and shadowing having log-scale standard deviation , with the gain between the th BS and the th user served by a BS of interest denoted by .With conjugate beamforming and receivers reliant on channel hardening, the signal-to-interference ratio (SIR) at such user is 
where is the gain from the serving BS and is the share of that BS’s power allocated to user . Two power allocations can be analyzed:
SIR-equalizing : , with the proportionality constant ensuring that . This makes . Moreover, as and grow large,
The analysis is conducted for , which makes it valid for arbitrary BS locations.
For notational compactness, let . Define as the solution to where is the lower incomplete gamma function. For , in particular, . Under a uniform power allocation, the CDF of is available in an explicit form involving the Gauss hypergeometric function (available in MATLAB and Mathematica):
where “” indicates asymptotic () equality, is such that the CDF is continuous, and
Alternatively, the CDF can be obtained by solving (e.g., with Mathematica) a single integral involving the Kummer function :
This latter solution can be modified for the SIR-equalizing power allocation as
The spectral efficiency of user is with CDF readily characterizable from the expressions given earlier. From , the sum spectral efficiency at the BS of interest can be found as Expressions for the averages and are further available in the form of single integrals.
With a uniform power allocation,
and . For the special case of , the Kummer function simplifies giving
With an equal-SIR power allocation
Application to Relevant Networks
Let us now contrast the analytical expressions (computable instantaneously and exactly, and valid for arbitrary topologies, but asymptotic in the shadowing strength) with some Monte-Carlo simulations (lengthy, noisy, and bug-prone, but for precise shadowing strengths and topologies).
First, we simulate a 500-cell hexagonal lattice with , and . Figs. 1a-1b compare the simulations for – dB with the analysis. The behaviors with these typical outdoor values of are well represented by the analysis and, as it turns out, in rigidly homogeneous networks such as this one is where the gap is largest.
For a more irregular deployment, let us next consider a network whose BSs are uniformly distributed. BSs (500 on average) are dropped around a central one of interest. For each network snapshot, users are then uniformly dropped until of them are served by the central BS. As before, , and . Figs. 2a-2b compare the simulations for dB with the analysis, and the agreement is now complete. The simulated average spectral efficiency with a uniform power allocation is b/s/Hz/user while (2) gives b/s/Hz/user.
The analysis presented in this post is not without limitations, chiefly the absence of noise and pilot contamination. However, as argued in , there is a broad operating range (– with very conservative premises) where these effects are rather minor, and the analysis is hence applicable.
Drones could shape the future of technology, especially if provided with reliable command and control (C&C) channels for safe and autonomous flying, and high-throughput links for multi-purpose live video streaming. Some months ago, Prabhu Chandhar’s guest post discussed the advantages of using massive MIMO to serve drone – or unmanned aerial vehicle (UAV) – users. More recently, our Paper 1 and Paper 2 have quantified such advantages under the realistic network conditions specified by the 3GPP. While demonstrating that massive MIMO is instrumental in enabling support for UAV users, our works also show that merely upgrading existing base stations (BS) with massive MIMO might not be enough to provide a reliable service at all UAV flying heights. Indeed, hardware solutions need to be complemented with signal processing enhancements through all communications phases, namely, 1) UAV cell selection and association, 2) downlink BS-to-UAV transmissions, and 3) uplink UAV-to-BS transmissions. These are outlined below.
1. UAV cell selection and association
As depicted in Figure 1(a), most existing cellular BSs create a fixed beampattern towards the ground. Thanks to this, ground users tend to perceive a strong signal strength from nearby BSs, which they use for connecting to the network. Instead, aerial users such as the red drone in Figure 1(a) only receive weak sidelobe-generated signals from a nearby BS when flying above it. This results in a deployment planning issue as illustrated in Figure 1(b), where due to the radiation of a strong sidelobe, the tri-sector BSs located in the origin can be the preferred server for far-flung UAVs (red spots). Consequently, these UAVs might experience strong interference, since they perceive signals from a multiplicity of BSs with similar power.
On the other hand, thanks to their capability of beamforming the synchronization signals used for user association, massive MIMO systems ensure that aerial users generally connect to a nearby BS. This optimized association enhances the robustness of the mobility procedures, as well as the downlink and uplink data phases.
2. Downlink BS-to-UAV transmissions
During the downlink data phase, UAV users are very sensitive to the strong inter-cell interference generated from a plurality of BSs, which are likely to be in line-of-sight. This may result in performance degradation, preventing UAVs from receiving critical C&C information, which has an approximate rate requirement of 60-100 kbps. Indeed, Figure 2 shows how conventional cellular networks (‘SU’) can only guarantee 100 kbps to a mere 6% of the UAVs flying at 150 meters. A conventional massive MIMO system (‘mMIMO’) enhances the data rates, albeit only 33% of the UAVs reach 100 kbps when they fly at 300 meters. This is due to a well-known effect: pilot contamination. Such an effect is particularly severe in scenarios with UAV users, since they can create strong uplink interference to many line-of-sight BSs simultaneously. In contrast, the pilot contamination decays much faster with distance for ground UEs.
In a nutshell, Figure 2 tells us that complementing conventional massive MIMO with explicit inter-cell interference suppression (‘mMIMO w/ nulls’) is essential when supporting high UAVs. In a ‘mMIMO w/ nulls’ system, BSs incorporate additional signal processing features that enable them to perform a twofold task. First, leveraging channel directionality, BSs can spatially separate non-orthogonal pilots transmitted by different UAVs. Second, by dedicating a certain number of spatial degrees of freedom to place radiation nulls, BSs can mitigate interference on the directions corresponding to users in other cells that are most vulnerable to the BS’s interference. Indeed, these additional capabilities dramatically increase the percentage of UAVs that meet the 100 kbps requirement when these are flying at 300 m, from 33% (‘mMIMO’) to a whopping 98% (‘mMIMO w/ nulls’).
3. Uplink UAV-to-BS transmissions
Unlike the downlink, where UAVs should be protected to prevent a significant performance degradation, it is the ground users who we should care about in the uplink. This is because line-of-sight UAVs can generate strong interference towards many BSs, therefore overwhelming the weaker signals transmitted by non-line-of-sight ground users. The consequences of such a phenomenon are illustrated in Figure 3, where the uplink rates of ground users plummet as the number of UAVs increases.
Again, ‘mMIMO w/nulls’ – incorporating additional space-domain inter-cell interference suppression capabilities – can solve the above issue and guarantee a better performance for legacy ground users.
Overall, the efforts towards realizing aerial wireless networks are just commencing, and massive MIMO will likely play a key role. In the exciting era of fly-and-connect, we must revisit our understanding of cellular networks and develop novel architectures and techniques, catering not only for roads and buildings, but also for the sky.
Three massive-MIMO-related highlights from IEEE ICC in Kansas City, MO, USA, this week:
J. H. Thompson from Qualcomm gave a keynote on 5G, relaying several important insights. He stressed the fundamental role of Massive MIMO, utilizing reciprocity (which in turn, of course, implies TDD). This is a message we have been preaching for years now, and it is reassuring to hear a main industry leader echo it at such an important event. He pointed to distributed Massive MIMO (that we know of as “cell-free massive MIMO“) as a forthcoming technology, not only because of the macro-diversity but also because of the improved channel rank it offers to multiple-antenna terminals. This new technology may enable AR/VR/XR, wireless connectivity in factories and much more… where conventional massive MIMO might not be sufficient.
In the exhibition hall Nokia showcased a 64×2=128 Massive MIMO array, with fully digital transceiver chains, small dual-polarized path antennas, operating at 2.5 GHz and utilizing reciprocity – though it wasn’t clear exactly what algorithmic technology that went inside. (See photographs below.) Sprint already has deployed this product commercially, if I understood well, with an LTE TDD protocol. Ericsson had a similar product, but it was not opened, so difficult to tell exactly what the actual array looked like. The Nokia base station was only slightly larger, physically, than the flat-screen-base-station vision I have been talking about for many years now, and along the lines that T. Marzetta from Bell Labs had already back in 2006. Now that cellular Massive MIMO is a commercial reality… what should the research community do? Granted there are still lots of algorithmic innovation possible (and needed), but …. Cell-free massive MIMO with RF over fiber is the probably the obvious next step.
T. Marzetta from NYU gave an industry distinguished talk, speculating about the future of wireless beyond Massive MIMO. What, if anything at all, could give us another 10x or 100x gain? A key point of the talk was that we have to go back to (wave propagation) physics and electromagnetics, a message that I very much subscribe to: the “y=Hx+w” models we typically use in information and communication theory are in many situations rather oversimplified. Speculations included the use of super-directivity, antenna coupling and more… It will be interesting to see where this leads, but at any rate, it is interesting fundamental physics.
There were also lots of other (non-Massive MIMO) interesting things: UAV connectivity, sparsity… and a great deal of questions and discussion on how machine learning could be leveraged, more about that at a later point in time.
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.
To read the preface of Fundamentals of Massive MIMO, click here. You can also purchase the book here.
Our recent guest post about the combination of Massive MIMO and drones has received a lot of interest on social media. The use of unmanned aerial vehicles (UAVs) for wireless communications is certainly an emerging topic that deserves further attention!
While the previous blog post focused on Massive MIMO aspects of UAV communications, other theoretical research findings are reviewed in this tutorial by Walid Saad and Mehdi Bennis:
Furthermore, the team of the ERC Advanced PERFUME project, lead by Prof. David Gesbert, has recently demonstrated what appears to be the world’s first autonomous flying base station relays. This exciting achievement is demonstrated in the following video:
Recently, there has been a hype on the use of drones (also called unmanned aerial vehicles (UAVs)) for civilian and military applications. Especially, in the coming decades, lightweight miniature drones are expected to play a major role in the society. Nowadays, small drones are available in toy shops so that an individual could buy it for personal uses such as aerial videography. However, due to security reasons, the personal use of drones is limited to low altitudes (up to 120 m in most countries) and visible line-of-sight. On the other hand, it is most likely that, in many countries, government agencies and commercial firms will be allowed to use drones for a variety of services (See: link 1 and link 2.)
There are many foreseen applications that involve a large number of drones in a limited area such as disaster management, traffic monitoring, crowd management, and crop monitoring. The major communication requirements of most of the drone networks are: several tens of Mbps throughput for streaming high-resolution video, low latency for command and control, highly reliable connectivity in a three-dimensional coverage area, high-mobility support, and simultaneous support for a large number of drones.
The existing wireless systems are unsuitable for communicating with a large number of drones in long-range, high throughput, and high-altitude applications for the following reasons:
In many drone communication scenarios, the mobility and traffic patterns of drones are different from the ground users. For example, in aerial surveillance applications, the uplink traffic is much higher than the downlink traffic. Depending on the application, the drones will fly at high speed (10-50 m/s) in a 3D space.
The propagation environment in drone communication scenarios will be line-of-sight, even under high mobility.
The terrestrial wireless communication networks are optimized for indoor, short range, low mobility (e.g. WiFi), and low altitude (e.g. LTE).
In LTE, since the base station antennas are tilted towards the ground, coverage is possible only if the drones fly below 100 m altitude. Apart from coverage, the co-channel interference generated from the neighboring cells will be a major problem in satisfying the high throughput requirements of drones.
The MAC layer protocols of the existing systems have to be redesigned according to the drones’ requirements, especially regarding the re-transmission protocols which are related to latency and crucial for drone control.
Since the existing wireless systems are connected to the power grids, they might not be available during emergency situations such as earth-quake, massive flooding, and tsunami. Further, in mountain and sea environments, cellular networks are not widely available. This problem can be overcome by deploying flying UAV base stations over the sky.
For the above-mentioned reasons, instead of borrowing from existing wireless technologies, it would be better to develop a new technology, considering the specific drone networks’ requirements and propagation characteristics. As of now, spectrum allocation and standardization efforts for drone communication networks are in the initial stage of development. This is where Massive MIMO can play a key role. The attractive features of Massive MIMO, such as spatial multiplexing and range extension, can be exploited to design flexible and efficient drone communication systems. 5G is based on the concept of network slicing, where the network can be configured differently depending on the use case. Therefore, it is possible to deploy a variation of 5G for drone communications along with appropriately tilted antenna arrays to provide connectivity to the drones flying at high altitudes.
In our recent papers (1 and 2), we illustrated the use for Massive MIMO for drone communications. From these papers, we make the following observations:
The Massive MIMO performance in rich scattering is well understood by the use of ergodic rate bounds that are available in closed form. In line-of-sight, the ergodic rate performance depends on the relative positions of the drones as they move very quickly in 3D space. Interestingly, in case of line-of-sight, the uplink ergodic rate bounds (with MRC receiver) are available in closed form for some specific cases, for example, for the uniformly distributed drone positions within a spherical volume. However, more work is needed to understand the ergodic rate performance with arbitrary drone distributions.
The element-spacing in the ground station array affects the rate performance depending on the distribution of the drones. For a given distribution of the drone positions, ground station array has to be optimized to maximize the ergodic rate.
The probability of outage due to polarization mismatch can be made negligible by appropriately selecting the orientation and polarization of the individual array elements. For example, circularly polarized cross-dipole antenna elements perform much better when compared to linearly polarized dipoles. (For more details, see this paper.) This means that the use of simple antenna elements, such as cross-dipoles, reduce the concerns of
antenna pattern designs. Further, the drones can be equipped with a single cross-dipole.
The range extension due to the increased number of antennas can eliminate the need for multi-hop solutions in many drone communication scenarios.
TDD based Massive MIMO can be used for simultaneously supporting several tens of drones both at μ-wave and mm-wave frequencies.
TDD based Massive MIMO can support high-mobility drone communications. In some scenarios (e.g., deterministic trajectories), the channel can be extrapolated without sending pilot symbols.
Below are some examples of use cases of Massive MIMO enabled drone communication systems. The technical details of Massive MIMO based system design can be found in this paper. The Massive MIMO design parameters for some of the use cases can be found in this paper.
Drone racing: In recent years, drone racing, also called “the sport of the future”, is becoming popular around the world. In drone racing, low latency is important for drone control, because even a few tens of milliseconds delay might crash the drone when it moves at the speed of 40-50 m/s. Interestingly, in our digital world, analog transmission is used for sending videos from racing drones to the pilots. The reason is that, unlike digital transmission, an analog transmission does not incur any processing delay and the overall latency is about only 15 ms. Currently, the 5.8 GHz band (5650 MHz to 5925 MHz) is used for drone racing. The transmitter and receiver use frequency modulation and it requires 40 MHz frequency separation to avoid cross-talks between neighboring channels. As a result, the number of simultaneous drones in a contest is limited to eight. The video quality is also poor. By using Massive MIMO, several tens of drones can simultaneously participate in a contest and the pilots can enjoy latency-free high-quality video transmission.
Sports streaming: Utilizing drones for sports streaming will change the way we view the sports events. High resolution 4K 360-degree videos taken by multiple drones at different angles can be broadcasted to enable the viewers to have an entirely a new experience. If there are 20 drones covering a sports event, the required sum throughput will be in the order of 10 Gbps. Massive MIMO in the mm-wave frequency band can be used to achieve this high throughput. This can become reality as already there are signs towards the use of drones for covering sports events. For instance, during the 2018 Winter Olympics, drones will be extensively used.
Surveillance/ Search and Rescue/Disaster management: During natural disasters, a network of drones can be quickly deployed to enable the rescue teams to assess the situation in real-time via high-resolution video streaming. Depending on the area to be covered and desired video quality, the sum throughput requirement will be in the order of Gbps. A Massive MIMO array deployed over a ground vehicle or a large aerial vehicle can be used for serving a swarm of drones.
Aerial survey: A swarm of drones can be used for high-resolution aerial imagery of several kilometers of landscape. There are many uses of aerial survey, including state governance, city planning, 3D cartography, and crop monitoring. Massive MIMO can be an enabler for such high throughput and long-range applications.
Backhaul for flying base stations: During emergency situations and heavy traffic conditions, UAVs could be used as flying base stations to provide wireless connectivity to the cellular users. A Massive MIMO base station can act as a high-capacity backhaul to a large number of flying base stations.
Space exploration: Currently, it takes several hours to receive a photo taken by the Curiosity Mars rover. It is possible to use Massive MIMO to reduce the overall transmission delay. For example, by using a massive antenna array deployed in an orbiter (see the above figure), a swarm of drones and rovers roaming on the surface of another planet can send videos and images to earth. The array can be used to spatially multiplex the uplink transmission from the drones (and possibly the rovers) to the orbiter. Note that the distance between the Mars surface and the orbiter is about 400 km. If the drones fly at an altitude of a few hundred meters and spread out over the region with a few hundred kilometers of radius, the angular resolution of the array is sufficient for spatial multiplexing. The array can be used to transmit the collected images and videos to earth by exploiting the array gain. This might sound like a science fiction, but NASA is already developing a 256 element antenna array for future Mars rovers to enable direct communication with the earth.