# The Common SINR Mistake

We are used to measuring performance in terms of the signal-to-interference-and-noise ratio (SINR), but this is seldom the actual performance metric in communication systems. In practice, we might be interested in a function of the SINR, such as the data rate (a.k.a. spectral efficiency), bit-error-rate, or mean-squared error in the data detection. When the receiver has perfect channel state information (CSI), the aforementioned metrics are all functions of the same SINR expression, where the power of the received signal is divided by the power of the interference plus noise. Details can be found in Examples 1.6-1.8 of the book Optimal Resource Allocation in Coordinated Multi-Cell Systems.

In most cases, the receiver only has imperfect CSI and then it is harder to measure the performance. In fact, it took me years to understand this properly. To explain the complications, consider the uplink of a single-cell Massive MIMO system with single-antenna users and antennas at the base station. The received -dimensional signal is

where  is the unit-power information signal from user is the fading channel from this user, and is unit-power additive Gaussian noise. In general, the base station will only have access to an imperfect estimate  of , for

Suppose the base station uses   to select a receive combining vector  for user . The base station then multiplies it with to form a scalar that is supposed to resemble the information signal :

From this expression, a common mistake is to directly say that the SINR is

which is obtained by computing the power of each of the terms (averaged over the signal and noise), and then claim that is an achievable rate (where the expectation is with respect to the random channels). You can find this type of arguments in many papers, without proof of the information-theoretic achievability of this rate value. Clearly,  is an SINR, in the sense that the numerator contains the total signal power and the denominator contains the interference power plus noise power. However, this doesn’t mean that you can plug into “Shannon’s capacity formula” and get something sensible. This will only yield a correct result when the receiver has perfect CSI.

A basic (but non-conclusive) test of the correctness of a rate expression is to check that the receiver can compute the expression based on its available information (i.e., estimates of random variables and deterministic quantities). Any expression containing fails this basic test since you need to know the exact channel realizations to compute it, although the receiver only has access to the estimates.

What is the right approach?

Remember that the SINR is not important by itself, but we should start from the performance metric of interest and then we might eventually interpret a part of the expression as an effective SINR. In Massive MIMO, we are usually interested in the ergodic capacity. Since the exact capacity is unknown, we look for rigorous lower bounds on the capacity. There are several bounding techniques to choose between, whereof I will describe the two most common ones.

The first lower bound on the uplink capacity can be applied when  the channels are Gaussian distributed and are the MMSE estimates with the corresponding estimation error covariance matrices . The ergodic capacity of user is then lower bounded by

Note that this expression can be computed at the receiver using only the available channel estimates (and deterministic quantities). The ratio inside the logarithm can be interpreted as an effective SINR, in the sense that the rate is equivalent to that of a fading channel where the receiver has perfect CSI and an SNR equal to this effective SINR. A key difference from  is that only the part of the desired signal that is received along the estimated channel appears in the numerator of the SINR, while the rest of the desired signal appears as in the denominator. This is the price to pay for having imperfect CSI at the receiver, according to this capacity bound, which has been used by Hoydis et al. and Ngo et al., among others.

The second lower bound on the uplink capacity is

which can be applied for any channel fading distribution. This bound provides a value close to  when there is substantial channel hardening in the system, while  will greatly underestimate the capacity when varies a lot between channel realizations. The reason is that to obtain this bound, the receiver detects the signal as if it is received over a non-fading channel with gain (which is deterministic and thus known in theory, and easy to measure in practice), but there are no approximations involved so is always a valid bound.

Since all the terms in are deterministic, the receiver can clearly compute it using its available information. The main merit of  is that the expectations in the numerator and denominator can sometimes be computed in closed form; for example, when using maximum-ratio and zero-forcing combining with i.i.d. Rayleigh fading channels or maximum-ratio combining with correlated Rayleigh fading. Two early works that used this bound are by Marzetta and by Jose et al..

The two uplink rate expressions can be proved using capacity bounding techniques that have been floating around in the literature for more than a decade; the main principle for computing capacity bounds for the case when the receiver has imperfect CSI is found in a paper by Medard from 2000. The first concise description of both bounds (including all the necessary conditions for using them) is found in Fundamentals of Massive MIMO. The expressions that are presented above can be found in Section 4 of the new book Massive MIMO Networks. In these two books, you can also find the right ways to compute rigorous lower bounds on the downlink capacity in Massive MIMO.

In conclusion, to avoid mistakes, always start with rigorously computing the performance metric of interest. If you are interested in the ergodic capacity, then you start from one of the canonical capacity bounds in the above-mentioned books and verify that all the required conditions are satisfied. Then you may interpret part of the expression as an SINR.

# Achieving Spectral Efficiency, Link Reliability, and Low-Power Operation

On January 17, I will give a 1-hour webinar in the IEEE 5G Webinar Series. I was asked to talk about “Massive MIMO for 5G below 6 GHz” since there has been a lot of focus on mmWave frequencies in the 5G discussions, although the primary band for 5G seems to be in the range 3.4-3.8 GHz, according to Ericsson.

The full title of my webinar is Massive MIMO for 5G below 6 GHz: Achieving Spectral Efficiency, Link Reliability, and Low-Power Operation. I will cover the basics of Massive MIMO and explain how the technology is not only great for enhancing the broadband access, but also for delivering the link reliability and low-power operation required by the internet of things. I have made sure that the overlap with my previous webinar is small.

If you watch the webinar live, you will have the chance to ask questions. Otherwise, you can view the recording of the webinar afterward. All the webinars in the IEEE 5G Webinar Series are available for anyone to view.

As a final note, I wrote a guest blog post at IEEE ComSoc Technology News in late December. It follows up and my previous blog post about GLOBECOM and is called: The Birth of 5G: What to do next?

# Wireless Communications with UAVs: Theory and Practice

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:

You can also check out this tutorial by Rui Zhang.

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:

# Challenges on the Path to Deployment

I attended GLOBECOM in Singapore earlier this week. Since more and more preprints are posted online before conferences, one of the unique features of conferences is to meet other researchers and attend the invited talks and interactive panel discussions. This year I attended the panel “Massive MIMO – Challenges on the Path to Deployment”, which was organized by Ian Wong (National Instruments). The panelists were Amitava Ghosh (Nokia), Erik G. Larsson (Linköping University), Ali Yazdan (Facebook), Raghu Rao (Xilinx), and Shugong Xu (Shanghai University).

No common definition

The first discussion item was the definition of Massive MIMO. While everyone agreed that the main characteristic is that the number of controllable antenna elements is much larger than the number of spatially multiplexed users, the panelists put forward different additional requirements. The industry prefers to call everything with at least 32 antennas for Massive MIMO, irrespective of whether the beamforming is constructed from codebook-based feedback, grid-of-beams, or by exploiting uplink pilots and TDD reciprocity. This demonstrates that Massive MIMO is becoming a marketing term, rather than a well-defined technology. In contrast, academic researchers often have more restrictive definitions; Larsson suggested to specifically include the TDD reciprocity approach in the definition. This is because it is the robust and overhead-efficient way to acquire channel state information (CSI), particularly for non-line-of-sight users; see Myth 3 in our magazine paper. This narrow definition clearly rules out FDD operation, as pointed out by a member of the audience. Personally, I think that any multi-user MIMO implementation that provides performance similar to the TDD-reciprocity-based approach deserves the Massive MIMO branding, but we should not let marketing people use the name for any implementation just because it has many antennas.

Important use cases

The primary use cases for Massive MIMO in sub-6 GHz bands are to improve coverage and spectral efficiency, according to the panel. Great improvements in spectral efficiency have been demonstrated by prototyping, but the panelist agreed that these should be seen as upper bounds. We should not expect to see more than 4x improvements over LTE in the first deployments, according to Ghosh. Larger gains are expected in later releases, but there will continue to be a substantial gap between the average spectral efficiency observed in real cellular networks and the peak spectral efficiency demonstrated by prototypes. Since Massive MIMO achieves its main spectral efficiency gains by multiplexing of users, we might not need a full-blown Massive MIMO implementation today, when there are only one or two simultaneously active users in most cells. However, the networks need to evolve over time as the number of active users per cell grows.

In mmWave bands, the panel agreed that Massive MIMO is mainly for extending coverage. The first large-scale deployments of Massive MIMO will likely aim at delivering fixed wireless broadband access and this must be done in the mmWave bands; there is too little bandwidth in sub-6 GHz bands to deliver data rates that can compete with wired DSL technology.

Initial cost considerations

The deployment cost is a key factor that will limit the first generations of Massive MIMO networks. Despite all the theoretic research that has demonstrated that each antenna branch can be built using low-resolution hardware, when there are many antennas, one should not forget the higher out-of-band radiation that it can lead to. We need to comply with the spectral emission masks – spectrum is incredibly expensive so a licensee cannot accept interference from adjacent bands. For this reason, several panelists from the industry expressed the view that we need to use similar hardware components in Massive MIMO as in contemporary base stations and, therefore, the hardware cost grows linearly with the number of antennas. On the other hand, Larsson pointed out that the futuristic devices that you could see in James Bond movies 10 years ago can now be bought for \$100 in any electronic store; hence, when the technology evolves and the economy of scale kicks in, the cost per antenna should not be more than in a smartphone.

A related debate is the one between analog and digital beamforming. Several panelists said that analog and hybrid approaches will be used to cut cost in the first deployments. To rely on analog technology is somewhat weird in an age when everything is becoming digital, but Yazdan pointed out that it is only a temporary solution. The long-term vision is to do fully digital beamforming, even in mmWave bands.

Another implementation challenge that was discussed is the acquisition of CSI for mobile users. This is often brought up as a showstopper since hybrid beamforming methods have such difficulties – it is like looking at a running person in a binocular and trying to follow the movement. This is a challenging issue for any radio technology, but if you rely on uplink pilots for CSI acquisition, it will not be harder than in a system of today. This has also been demonstrated by measurements.

Open problems

The panel was asked to describe the most important open problems in the Massive MIMO area, from a deployment perspective. One obvious issue, which we called the “grand question” in a previous paper, is to provide better support for Massive MIMO in FDD.

The control plane and MAC layer deserve more attention, according to Larsson. Basic functionalities such as ACK/NACK feedback is often ignored by academia, but incredibly important in practice.

The design of “cell-free” densely distributed Massive MIMO systems also deserve further attention. Connecting all existing antennas together to perform joint transmission seems to be the ultimate approach to wireless networks. Although there is no practical implementation yet, Yazdan stressed that deploying such networks might actually be more practical than it seems, given the growing interest in C-RAN technology.

10 years from now

I asked the panel what will be the status of Massive MIMO in 10 years from now. Rao predicted that we will have Massive MIMO everywhere, just as all access point supports small-scale MIMO today. Yazdan believed that the different radio technology (e.g., WiFi, LTE, NR) will converge into one interconnected system, which also allows operators to share hardware. Larsson thinks that over the next decade many more people will have understood the fundamental benefits of utilizing TDD and channel reciprocity, which will have a profound impact on the regulations and spectrum allocation.

# New Massive MIMO Book

For the past two years, I’ve been writing on a book about Massive MIMO networks, together with my co-authors Jakob Hoydis and Luca Sanguinetti. It has been a lot of hard work, but also a wonderful experience since we’ve learned a lot in the writing process. We try to connect all dots and provide answers to many basic questions that were previously unanswered.

The book has now been published:

Emil Björnson, Jakob Hoydis and Luca Sanguinetti (2017), “Massive MIMO Networks: Spectral, Energy, and Hardware Efficiency”, Foundations and Trends® in Signal Processing: Vol. 11, No. 3-4, pp 154–655. DOI: 10.1561/2000000093.

What is new with this book?

Marzetta et al. published Fundamentals of Massive MIMO last year. It provides an excellent, accessible introduction to the topic. By considering spatially uncorrelated channels and two particular processing schemes (MR and ZF), the authors derive closed-form capacity bounds, which convey many practical insights and also allow for closed-form power control.

In the new book, we consider spatially correlated channels and demonstrate how such correlation (which always appears in practice) affects Massive MIMO networks. This modeling uncovers new fundamental behaviors that are important for practical system design. We go deep into the signal processing aspects by covering several types of channel estimators and deriving advanced receive combining and transmit precoding schemes.

In later chapters of the book, we cover the basics of energy efficiency, transceiver hardware impairments, and various practical aspects; for example, spatial resource allocation, channel modeling, and antenna array deployment.

The book is self-contained and written for graduate students, PhD students, and senior researchers that would like to learn Massive MIMO, either in depth or at an overview level. All the analytical proofs, and the basic results on which they build, are provided in the appendices.

On the website massivemimobook.com, you will find Matlab code that reproduces all the simulation figures in the book. You can also download exercises and other supplementary material.

Update: Get a free copy of the book

From August 2018, you can download a free PDF of the authors’ version of the manuscript. This version is similar to the official printed books, but has a different front-page and is also regularly updated to correct typos that have been identified.

# Ten Questions and Answers About Massive MIMO

After the IEEE ComSoc Webinar that I gave this month, there was a 15 minute online Q/A session.

Unfortunately, there was not enough time for me to answer all the questions that I received, so I had to answer many of them afterwards. I have gathered ten questions and my answers below. I can also announce that I will give another Massive MIMO webinar in January 2018 and it will also be followed by a Q/A session.

1. What are the differences between 4G and 5G that will affect how Massive MIMO can be implemented?

The channel estimation must be implemented in the right way (i.e., exploiting uplink pilots and channel reciprocity) to obtain sufficiently accurate channel state information (CSI) to perform spatial multiplexing of many users, otherwise the inter-user interference will eliminate most of the gains. Accurate CSI  is hard to achieve within the 4G standard, although there are several Massive MIMO field trials for TDD LTE that show promising results. However, if 5G is designed properly, it will support Massive MIMO from scratch, while in 4G it will always be an add-on that must to adhere to the existing air interface.

2. How easy it is to deploy MIMO antennas on the current infrastructure?

Generally speaking, we can reuse the current infrastructure when deploying Massive MIMO, which is why operators show much interest in the technology. You upgrade the radio base stations but keep the same backhaul infrastructure and core network. However, since Massive MIMO supports much higher data rates, some of the backhaul connections might also need to be upgraded to deliver these rates.

3. What are the most suitable channel models for Massive MIMO?

I recommend the channel model that was developed in the MAMMOET project. It is a refinement of the COST 2100 model that takes particular phenomena of having large antenna arrays into account. Check out Deliverable D1.2 from that project.

4. For planar arrays, what is the height to width ratio that gives the highest performance?

You typically need more antennas in the horizontal direction (width) than in the vertical direction (height), because the angular variations between users is larger in the horizontal domain. For example, the array might cover a horizontal sector of 120-180 degrees, while the users’ elevation angles might only differ by a few tens of degrees. This is the reason that 8-antenna LTE base stations use linear arrays in the horizontal direction.

There is no optimal answer to the question. It depends on the deployment scenario. If you have high-rise buildings, users at different floors can have rather different elevation angles (it can differ up to 90 degrees) and you can benefit more from having many antennas in the vertical direction. If all users have almost the same elevation angle, it is preferable to have many antennas in the horizontal direction. These things are further discussed in Sections 7.3 and 7.4 in my new book.

5. What are the difficulties we face in deploying Massive MIMO in FDD systems?

The difficulty is to acquire channel state information at the base station for the frequency band used in the downlink, since it is very resource-demanding to send downlink pilots from a large array; particularly, if you want to spatially multiplex many users. This is an important but challenging problem that researchers have been working on since the 1990s. You can read more about it in Myth 3 and the grand question in the paper Massive MIMO: ten myths and one grand question.

6. Do you believe that there is a value in coordinated resource allocation schemes for Massive MIMO?

Yes, but the resource allocation in Massive MIMO is different from conventional systems. Scheduling might not be so important, since you can multiplex many users spatially, but pilot assignment and power allocation are important aspects that must be addressed. I refer to these things as spatial resource allocation. You can read more about this in Sections 7.1 and 7.2 in my new book, but as you can see from those sections, there are many open problems to be solved.

7. What is channel hardening and what implications does it have on the frequency allocation (in OFDMA networks, for example)?

Channel hardening means that the effective channel after beamforming is almost constant so that the communication link behaves as if there is no small-scale fading. A consequence is that all frequency subcarriers provide almost the same channel quality to a user. Regarding channel assignment, since you can multiplex many tens of users spatially in Massive MIMO, you can assign the entire bandwidth (all subcarriers) to every user; there is no need to use OFDMA to allocate orthogonal frequency resources to the users.

8. Is it practical to estimate the channel for each subcarrier in an OFDM system?

To limit the pilot overhead, you typically place pilots only on a small subset of the subcarriers. The distance between the pilots in the frequency domain can be selected based on how frequency-selective the channels are; if a user has L strong channel taps, it is sufficient to send pilots on L subcarriers, even if you many more subcarriers than that. Based on the received pilot signals, one can either estimate the channels on every subcarrier or estimate the channels on some of them and interpolate to get estimates on the remaining subcarriers.

9. How sensitive are the Massive MIMO spectral efficiency gains to TDD frame synchronization?

If you consider an OFDM system, then timing synchronization mismatches that are smaller than the cyclic prefix can basically be ignored. This is the case in TDD LTE systems and will not change when considering Massive MIMO systems that are implemented using OFDM. However, the synchronization across cells will not be perfect. The implications are investigated in a recent paper.

10. How does the higher computational complexity and delay in Massive MIMO processing affect the system performance?

I used to think that the computational complexity would be a bottleneck, but it turns out that it is not a big deal since all of the operations are standard (i.e., matrix multiplications and matrix inversions). For example, the circuit that was developed at Lund University shows that MIMO detection and precoding for a 20 MHz channel can be implemented very efficiently and only consumes a few mW.

# Massive MIMO for 5G: How Big Can it Get?

I gave an IEEE ComSoc Webinar entitled Massive MIMO for 5G: How Big Can it Get earlier this month. If you missed it, you can view IEEE’s recording of the webinar. Alternatively, you check out the following video where I give the same talk:

Other Massive MIMO videos can be found on our Youtube channel.