All posts by Emil Björnson

5G, Commentary, Technical insights

Relax and Conquer

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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.

Commentary, Education, Technical insights

The Common SINR Mistake

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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 K single-antenna users and M antennas at the base station. The received M-dimensional signal is

    $$\mathbf{y} = \sum_{i=1}^{K} \mathbf{h}_{i} x_{i} + \mathbf{n}$$

where $x_{i}$ is the unit-power information signal from user $i$, $\mathbf{h}_{i} \in \mathbb{C}^{M}$ is the fading channel from this user, and $\mathbf{n}\in \mathbb{C}^{M}$ is unit-power additive Gaussian noise. In general, the base station will only have access to an imperfect estimate $\hat{\mathbf{h}}_{i} \in \mathbb{C}^{M}$ of $\mathbf{h}_{i}$, for $i=1,\ldots,K.$

Suppose the base station uses  $\hat{\mathbf{h}}_{1},\ldots,\hat{\mathbf{h}}_{K}$ to select a receive combining vector $\mathbf{v}_k\in \mathbb{C}^{M}$ for user $k$. The base station then multiplies it with $\mathbf{y}$ to form a scalar that is supposed to resemble the information signal $x_{k}$:

    $$\mathbf{v}_k^H \mathbf{y} = \underbrace{\mathbf{v}_k^H \mathbf{h}_{k} x_{k}}_\textrm{Desired signal} + \underbrace{\sum_{i=1, i \neq k}^{K} \mathbf{v}_k^H\mathbf{h}_{i} x_{i}}_\textrm{Interference} + \underbrace{\mathbf{v}_k^H \mathbf{w}}_\textrm{Noise}.$$

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

    $$\mathrm{SINR}_k^\textrm{wrong} = \frac{| \mathbf{v}_k^H \mathbf{h}_{k}|^2}{ \sum_{i=1, i \neq k}^{K} | \mathbf{v}_k^H \mathbf{h}_{i}|^2 + \| \mathbf{v}_k \|^2},$$

which is obtained by computing the power of each of the terms (averaged over the signal and noise), and then claim that $\mathbb{E}\{\log_2(1+\mathrm{SINR}_k^\textrm{wrong} )\}$ 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, $\mathrm{SINR}_k^\textrm{wrong} $ 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 $\mathrm{SINR}_k^\textrm{wrong} $ 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 $\mathrm{SINR}_k^\textrm{wrong}$ fails this basic test since you need to know the exact channel realizations \mathbf{h}_{1},\ldots,\mathbf{h}_{K} 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 $\hat{\mathbf{h}}_{1}, \ldots, \hat{\mathbf{h}}_{K}$ are the MMSE estimates with the corresponding estimation error covariance matrices $\mathbf{C}_{1},\ldots,\mathbf{C}_{K}$. The ergodic capacity of user $k$ is then lower bounded by

$$R_k^{(1)} = \mathbb{E} \left\{ \log_2 \left( 1 + \frac{| \mathbf{v}_k^H \hat{\mathbf{h}}_{k}|^2}{ \sum_{i=1, i \neq k}^{K} | \mathbf{v}_k^H \hat{\mathbf{h}}_{i}|^2 + \sum_{i=1}^{K} \mathbf{v}_k^H \mathbf{C}_{i} \mathbf{v}_k + \| \mathbf{v}_k \|^2} \right) \right\}.

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 $\mathrm{SINR}_k^\textrm{wrong}$ 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 $\mathbf{v}_k^H \mathbf{C}_{k} \mathbf{v}_k$ 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

$$R_k^{(2)} = \log_2 \left( 1 + \frac{ | \mathbb{E}\{ \mathbf{v}_k^H \mathbf{h}_{k} \} |^2}{ \sum_{i=1}^{K} \mathbb{E} \{ | \mathbf{v}_k^H \mathbf{h}_{i}|^2 \} - | \mathbb{E}\{ \mathbf{v}_k^H \mathbf{h}_{k} \} |^2+ \mathbb{E}\{\| \mathbf{v}_k \|^2\} } \right),

which can be applied for any channel fading distribution. This bound provides a value close to $R_k^{(1)}$ when there is substantial channel hardening in the system, while $R_k^{(2)}$ will greatly underestimate the capacity when $\mathbf{v}_k^H \mathbf{h}_{k}$ 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 \mathbb{E}\{ \mathbf{v}_k^H \mathbf{h}_{k} \} (which is deterministic and thus known in theory, and easy to measure in practice), but there are no approximations involved so $R_k^{(2)}$ is always a valid bound.

Since all the terms in $R_k^{(2)} $ are deterministic, the receiver can clearly compute it using its available information. The main merit of $R_k^{(2)}$ 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.

5G, Commentary, Technical insights

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

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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?

 

5G, Beyond 5G, News

Wireless Communications with UAVs: Theory and Practice

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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:

5G, Beyond 5G, Commentary, Technical insights

Challenges on the Path to Deployment

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Marina Bay Sands Expo and Convention Centre

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.

Education, News

New Massive MIMO Book

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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.