Category Archives: Education

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 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 uplink bound 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 uplink bound 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.

I Never Thought It Would Happen So Fast

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.

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.

Limited-time offer: Get a free copy of the book

Next week, we are giving a tutorial at the Globecom conference. In support of this, the publisher is currently providing free digital copies of the book on their website. This offer is available until December 7.

If you like the book, you can also buy a printed copy from the publisher’s website for the special price of $40! Use the discount code 552568, which is valid until December 31, 2017.

Upcoming Massive MIMO Webinars

IEEE ComSoc is continuing to deliver webinars on 5G topics and Massive MIMO is a key part of several of them. The format is a 40 minute presentation followed by a 20 minuter Q/A session. Hence, if you attend the webinars “live”, you have the opportunity to ask questions to the presenters. Otherwise, you can also watch each webinar afterwards. For example, 5G Massive MIMO: Achieving Spectrum Efficiency, which was given in August by Liesbet Van der Perre (KU Leuven), can still be watched.

In November, the upcoming Massive MIMO webinars are:

Massive MIMO for 5G: How Big Can it Get? by Emil Björnson (Linköping University), Thursday, 9 November 2017, 3:00 PM EST, 12:00 PM PST, 20:00 GMT.

Real-time Prototyping of Massive MIMO: From Theory to Reality by Douglas Kim (NI) and Fredrik Tufvesson (Lund University), Wednesday, 15 November 2017, 12:00 PM EST, 9:00 AM PST, 17:00 GMT.

Some Impactful Rejected Papers

Yes, my group had its share of rejected papers as well. Here are some that I specially remember:

  1. Massive MIMO: 10 myths and one critical question. The first version was rejected by the IEEE Signal Processing Magazine. The main comment was that nobody would think that the points that we had phrased as myths were true. But in reality, each one of the myths was based on an actual misconception heard in public discussions! The paper was eventually published in the IEEE Communications Magazine instead in 2016, and has been cited more than 180 times.
  2. Massive MIMO with 1-bit ADCs. This paper was rejected by the IEEE Transactions on Wireless Communications. By no means a perfect paper… but the review comments were mostly nonsensical. The editor stated: “The concept as such is straightforward and the conceptual novelty of the manuscript is in that sense limited.” The other authors left my group shortly after the paper was written. I did not predict the hype on 1-bit ADCs for MIMO that would ensue (and this happened despite the fact that yes, the concept as such is straightforward and its conceptual novelty is rather limited!). Hence I didn’t prioritize a rewrite and resubmission. The paper was never published, but we put the rejected manuscript on arXiv in 2014, and it has been cited 80 times.
  3. Finally, a paper that was almost rejected upon its initial submission: Energy and Spectral Efficiency of Very Large Multiuser MIMO Systems, eventually published in the IEEE Transactions on Communications in 2013. The review comments included obvious nonsense, such as “Overall, there is not much difference in theory compared to what was studied in the area of MIMO for the last ten years.” The paper subsequently won the IEEE ComSoc Stephen O. Rice Prize, and has more than 1300 citations.

There are several lessons to learn here. First, that peer review may be the best system we know, but it isn’t perfect: disturbingly, it is often affected by incompetence and bias. Second, notwithstanding the first, that many paper rejections are probably also grounded in genuine misunderstandings: writing well takes a lot of experience, and a lot of hard, dedicated work. Finally, and perhaps most significantly, that persistence is really an essential component of success.

What is the Difference Between Beamforming and Precoding?

I’ve got an email with this question last week. There is not one but many possible answers to this question, so I figured that I write a blog post about it.

One answer is that beamforming and precoding are two words for exactly the same thing, namely to use an antenna array to transmit one or multiple spatially directive signals.

Another answer is that beamforming can be divided into two categories: analog and digital beamforming. In the former category, the same signal is fed to each antenna and then analog phase-shifters are used to steer the signal emitted by the array. This is what a phased array would do. In the latter category, different signals are designed for each antenna in the digital domain. This allows for greater flexibility since one can assign different powers and phases to different antennas and also to different parts of the frequency bands (e.g., subcarriers). This makes digital beamforming particularly desirable for spatial multiplexing, where we want to transmit a superposition of signals, each with a separate directivity. It is also beneficial when having a wide bandwidth because with fixed phases the signal will get a different directivity in different parts of the band. The second answer to the question is that precoding is equivalent to digital beamforming. Some people only mean analog beamforming when they say beamforming, while others use the terminology for both categories.

Analog beamforming uses phase-shifters to send the same signal from multiple antennas but with different phases. Digital beamforming designs different signals for each antennas in the digital baseband. Precoding is sometimes said to be equivalent to digital beamforming.

A third answer is that beamforming refers to a single-user transmission with one data stream, such that the transmitted signal consists of one main-lobe and some undesired side-lobes. In contrast, precoding refers to the superposition of multiple beams for spatial multiplexing of several data streams.

A fourth answer is that beamforming refers to the formation of a beam in a particular angular direction, while precoding refers to any type of transmission from an antenna array. This definition essentially limits the use of beamforming to line-of-sight (LoS) communications, because when transmitting to a non-line-of-sight (NLoS) user, the transmitted signal might not have a clear angular directivity. The emitted signal is instead matched to the multipath propagation so that the multipath components that reach the user add constructively.

A fifth answer is that precoding consists of two parts: choosing the directivity (beamforming) and choosing the transmit power (power allocation).

I used to use the word beamforming in its widest meaning (i.e., the first answer), as can be seen in my first book on the topic. However, I have since noticed that some people have a more narrow or specific interpretation of beamforming. Therefore, I nowadays prefer only talking about precoding. In Massive MIMO, I think that precoding is the right word to use since what I advocate is a fully digital implementation, where the phases and powers can be jointly designed to achieve high capacity through spatial multiplexing of many users, in both NLoS and LOS scenarios.

5.5 Hours of Massive MIMO Tutorials

Video recordings from the 2017 Joint IEEE SPS and EURASIP Summer School on Signal Processing for 5G Wireless Access are available for IEEE members, as we wrote about in a previous post. Now two of the Massive MIMO tutorial talks are openly available on Youtube.

Prof. Erik. G. Larsson gave a 2.5 hour tutorial on the fundamentals of Massive MIMO, which is highly recommended for anyone learning this topic. You can then follow up by reading his book with the same topic.

When you have viewed Erik’s introduction, you can learn more about the state-of-the-art signal processing schemes for Massive MIMO from another talk at the summer school. Dr. Emil Björnson gave a 3 hour tutorial on this topic: