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 [1]. 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 [2]

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:

- Uniform: .
- SIR-equalizing [3]: , 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,

(1)

and . For the special case of , the Kummer function simplifies giving

(2)

With an equal-SIR power allocation

(3)

and .

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 [1], there is a broad operating range (– with very conservative premises) where these effects are rather minor, and the analysis is hence applicable.

[1] G. George, A. Lozano, M. Haenggi, “Massive MIMO forward link analysis for cellular networks,” arXiv:1811.00110, 2018.

[2] T. Marzetta, E. Larsson, H. Yang, and H. Ngo, Fundamentals of Massive MIMO. Cambridge University Press, 2016.

[3] H. Yang and T. L. Marzetta, “A macro cellular wireless network with uniformly high user throughputs,” IEEE Veh. Techn. Conf. (VTC’14), Sep. 2014.

And in this video clip I talk more in general about our book, Fundamentals of Massive MIMO:

You can read more about this book in a previous blog post and also watch this new video, where I talk about the content and motivation behind the writing of the book.

]]>The derivation was based on a very simple third-order polynomial model. Questioning that model, or contesting the conclusions? Let’s run WebLab. WebLab is a web-server-based interface to a real power amplifier operating in the lab, developed and run by colleagues at Chalmers University of Technology in Sweden. Anyone can access the equipment in real time (though there might be a queue) by submitting a waveform and retrieving the amplified waveform using a special Matlab function, “weblab.m”, obtainable from their webpages. Since accurate characterization and modeling of amplifiers is a hard nonlinear identification problem, WebLab is a great tool to researchers who want to go beyond polynomial and truncated Volterra-type toy models.

A -spaced uniform linear array with 50 elements beamforms in free space line-of-sight to two terminals at (arbitrarily chosen) angles -9 respectively +34 degrees. A sinusoid with frequency is sent to the first terminal, and a sinusoid with frequency is transmitted to the other terminal. (Frequencies are in discrete time, see the Weblab documentation for details.) The actual radiation diagram is computed numerically: line-of-sight in free space is fairly uncontroversial: superposition for wave propagation applies. However, importantly, the actual amplification all signals is run on actual hardware in the lab.

The computed radiation diagram is shown below. (Some lines overlap.) There are two large peaks at -9 and +34 degrees angle, corresponding to the two signals of interest with frequencies and . There are also secondary peaks, at angles approximately -44 and -64 degrees, at frequencies different from respectively . These peaks originate from intermodulation products, and represent the out-band radiation caused by the amplifier non-linearity. (Homework: read the paper and verify that these angles are equal to those predicted by the theory.)

The Matlab code for reproduction of this experiment can be downloaded here.

]]>Contemporary multiantenna base stations for cellular communications are equipped with 2-8 antennas, which are deployed along a horizontal line. One example is a uniform linear array (ULA), as illustrated in Figure 1 below, where the antenna spacing is uniform. All the antennas in the ULA have the same physical down-tilt, with respect to the ground, and a fixed radiation pattern and directivity.

By sending the same signal from all antennas, but with different phase-shifts, we can steer beams in different angular directions and thereby make the directivity of the radiated signal different from the directivity of the individual antennas. Since the antennas are deployed on a one-dimensional horizontal line in this example, the ULA can only steer beams in the two-dimensional (2D) azimuth plane as illustrated in Figure 1. The elevation angle is the same for all beams, which is why this is called 2D beamforming. The beamwidth in the azimuth domain shrinks the more antennas are deployed. If the array is used for multiuser MIMO, then multiple beams with different azimuth angles are created simultaneously, as illustrated by the colored beams in Figure 1.

If we would rotate the ULA so that the antennas are instead deployed at different heights above the ground, then the array can instead steer beams in different elevation angles. This is illustrated in Figure 2. Note that this is still a form of 2D beamforming since every beam will have the same directivity with respect to the azimuth plane. This antenna array can be used to steer beams towards users at different floors of a building. It is also useful to serve flying objects, such as UAVs, jointly with ground users. The beamwidth in the elevation domain shrinks the more antennas are deployed.

If we instead deploy multiple ULAs on top of each other, it is possible to control both the azimuth and elevation angle of a beam. This is called 3D beamforming and is illustrated in Figure 3 using a planar array with a “massive” number of antennas. This gives the flexibility to not only steer beams towards different buildings but also towards different floors of these buildings, to provide a beamforming gain wherever the user is in the coverage area. It is not necessary to have many antennas to perform 3D beamforming – it is basically enough to have three antennas deployed in a triangle. However, as more antennas are added, the beams become narrower and easier to jointly steer in specific azimuth-elevation directions. This increases the array gain and reduces the interference between beams directed to different users, as illustrated by the colors in Figure 3.

The detailed answer to the question “3D Beamforming, is that Massive MIMO?” is as follows. Massive MIMO and 3D beamforming are two different concepts. 3D beamforming can be performed with few antennas and Massive MIMO can be deployed to only perform 2D beamforming. However, Massive MIMO and 3D beamforming is a great combination in many applications; for example, to spatially multiplex many users in a city with high-rise buildings. One should also bear in mind that, in general, only a fraction of the users are located in line-of-sight so the formation of angular beams (as shown above) might be of limited importance. The ability to control the array’s radiation pattern in 3D is nonetheless helpful to control the multipath environment such that the many signal components add constructively at the location of the intended receiver.

]]>**Can we utilize the channel hardening to estimate the channels less frequently?**

Unfortunately, the answer is no. Whenever you move approximately half a wavelength, the multi-path propagation will change each element of the channel vector. The time it takes to move such a distance is called a *coherence time*. This time is the same irrespectively of how many antennas the base station has and, therefore, you still need to estimate the channel once per coherence time. The same applies to the frequency domain, where the *coherence bandwidth* is determined by the propagation environment and not the number of antennas.

The following flow-chart shows what need to happen in every channel coherence time:

When you get a new realization (at the top of the flow-chart), you compute an estimate (e.g., based on uplink pilots), then you use the estimate to compute a new receive combining vector and transmit precoding vector. It is when you have applied these vectors to the channel that the hardening phenomena appears; that is, the randomness averages out. If you use maximum ratio (MR) processing, then the random realization * h_{1}* of the channel vector turns into an almost deterministic scalar channel ||

In conclusion, channel hardening appears after coherent combining/precoding has been applied. To maintain a hardened channel over time (and frequency), you need to estimate and update the combining/precoding as often as you would do for a single-antenna channel. If you don’t do that, you will gradually lose the array gain until the point where the channel and the combining/precoding are practically uncorrelated, so there is no array gain left. Hence, there is more to lose from estimating channels too infrequently in Massive MIMO systems than in conventional systems. This is shown in Fig. 10 in a recent measurement paper from Lund University, where you see how the array gain vanishes with time. However, the Massive MIMO system will never be worse than the corresponding single-antenna system.

]]>Since the spectral efficiency (bit/s/Hz) and many other performance metrics of interest depend on the SNR, and not the individual values of the three parameters, it is a common practice to normalize one or two of the parameters to unity. This habit makes it easier to interpret performance expressions, to select reasonable SNR ranges, and to avoid mistakes in analytical derivations.

There are, however, situations when the absolute value of the transmitted/received signal power matters, and not the relative value with respect to the noise power, as measured by the SNR. In these situations, it is easy to make mistakes if you use normalized parameters. I see this type of errors far too often, both as a reviewer and in published papers. I will give some specific examples below, but I won’t tell you who has made these mistakes, to not point the finger at anyone specifically.

**Wireless energy transfer**

Electromagnetic radiation can be used to transfer energy to wireless receivers. In such wireless energy transfer, it is the received signal energy that is harvested by the receiver, not the SNR. Since the noise power is extremely small, the SNR is (at least) a billion times larger than the received signal power. Hence, a normalization error can lead to crazy conclusions, such as being able to transfer energy at a rate of 1 W instead of 1 nW. The former is enough to keep a wireless transceiver on continuously, while the latter requires you to harvest energy for a long time period before you can turn the transceiver on for a brief moment.

**Energy efficiency**

The energy efficiency (EE) of a wireless transmission is measured in bit/Joule. The EE is computed as the ratio between the data rate (bit/s) and the power consumption (Watt=Joule/s). While the data rate depends on the SNR, the power consumption does not. The same SNR value can be achieved over a long propagation distance by using high transmit power or over a short distance by using a low transmit power. The EE will be widely different in these cases. If a “normalized transmit power” is used instead of the actual transmit power when computing the EE, one can get EEs that are one million times smaller than they should be. As a rule-of-thumb, if you compute things correctly, you will get EE numbers in the range of 10 kbit/Joule to 10 Mbit/Joule.

**Noise power depends on the bandwidth**

The noise power is proportional to the communication bandwidth. When working with a normalized noise power, it is easy to forget that a given SNR value only applies for one particular value of the bandwidth.

Some papers normalize the noise variance and channel gain, but then make the SNR equal to the unnormalized transmit power (measured in W). This may greatly overestimate the SNR, but the achievable rates might still be in the reasonable range if you operate the system in an interference-limited regime.

Some papers contain an alternative EE definition where the spectral efficiency (bit/s/Hz) is divided by the power consumption (Joule/s). This leads to the alternative EE unit bit/Joule/Hz. This definition is not formally wrong, but gives the misleading impression that one can multiply the EE value with any choice of bandwidth to get the desired number of bit/Joule. That is not the case since the SNR only holds for one particular value of the bandwidth.

**Knowing when to normalize**

In summary, even if it is convenient to normalize system parameters in wireless communications, you should only do it if you understand when normalization is possible and when it is not. Otherwise, you can make embarrassing mistakes, such as submitting a paper where the results are six orders of magnitude wrong. And, unfortunately, there are several such papers that have been published and these create a bad circle by tricking others into making the same mistakes.

]]>This price is available until the end of April when buying the book directly from the publisher through the following link:

https://www.nowpublishers.com/Order/BuyBook?isbn=978-1-68083-985-2

**Note:** The book’s authors will give a joint tutorial on April 15 at WCNC 2018. A limited number of copies of the book will be available for sale at the conference and if you attend the tutorial, you will receive even better deal on buying the book!

Christopher Mollén recently defended his doctoral thesis entitled High-End Performance with Low-End Hardware: Analysis of Massive MIMO Base Station Transceivers. In the following video, he explains the basics of how the non-linear distortion from Massive MIMO transceivers is radiated in space.

]]>The recipe is to compute the capacity bound, and depending on the code blocklength, add a dB or a few, to the required SNR. That gives the link performance prediction. The coding literature is full of empirical results, showing how far from capacity a code of a given block length is for the AWGN channel, and this gap is usually not extremely different for other channel models – although, one should always check this.

But there are three main caveats with this:

- First, the capacity bound, or the “SINR” that it often contains, must be information-theoretically correct. A great deal of papers get this wrong. Emil explained in his blog post last week some common errors. The recommended approach is to map the channel onto one of the canonical cases in Figure 2.9 in Fundamentals of Massive MIMO, verify that the technical conditions are satisfied, and use the corresponding formula.
- When computing expressions of the type E[log(1+”SINR”)], then the average should be taken over all quantities that are random within the duration of a codeword. Typically, this means averaging over the randomness incurred by the noise, channel estimation errors, and in many cases the small-scale fading. All other parameters must be kept fixed. Typically, user positions, path losses, shadow fading, scheduling and pilot assignments, are fixed, so the expectation is conditional on those. (Yet, the interference statistics may vary substantially, if other users are dropping in and out of the system.) This in turn means that many “drops” have to be generated, where these parameters are drawn at random, and then CDF curves with respect to that second level of randomness needs be computed (numerically).Think of the expectation E[log(1+”SINR”)] as a “link simulation”. Every codeword sees many independent noise realizations, and typically small-scale fading realizations, but the same realization of the user positions. Also, often, neat (and tight) closed-form bounds on E[log(1+”SINR”)] are available.
- Care is advised when working with relatively short blocks (less than a few hundred bits) and at rates close to the constrained capacity with the foreseen modulation format. In this case, many of the “standard” capacity bounds become overoptimistic.As a rule of thumb, compare the capacity of an AWGN channel with the constrained capacity of the chosen modulation at the spectral efficiency of interest, and if the gap is small, the capacity bounds will be useful. If not, then reconsider the choice of modulation format! (See also homework problem 1.4.)

How far are the bounds from the actual capacity typically? Nobody knows, but there are good reasons to believe they are extremely close. Here (Figure 1) is a nice example that compares a decoder that uses the measured channel likelihood, instead of assuming a Gaussian (which is implied by the typical bounding techniques). From correspondence with one of the authors: “The dashed and solid lines are the lower bound obtained by Gaussianizing the interference, while the circles are the rate achievable by a decoder exploiting the non-Gaussianity of the interference, painfully computed through days-long Monte-Carlo. (This is not exactly the capacity, because the transmit signals here are Gaussian, so one could deviate from Gaussian signaling and possibly do slightly better — but the difference is imperceptible in all the experiments we’ve done.)”

Concerning Massive MIMO and its capacity bounds, I have met for a long time with arguments that these capacity formulas aren’t useful estimates of actual performance. But in fact, they are: In one simulation study we were less than one dB from the capacity bound by using QPSK and a standard LDPC code (albeit with fairly long blocks). This bound accounts for noise and channel estimation errors. Such examples are in Chapter 1 of Fundamentals of Massive MIMO, and also in the ten-myth paper:

(I wrote the simulation code, and can share it, in case anyone would want to reproduce the graphs.)

So in summary, while capacity bounds are sometimes done wrong; **when done right** they give pretty good estimates of actual link performance with modern coding.

(With thanks to Angel Lozano for discussions.)

]]>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.

]]>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.

]]>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.

]]>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.

]]>- 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.
- 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. - 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.

]]>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.

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.

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:

]]>Asymptotic analysis is a popular tool within statistical signal processing (infinite SNR or number of samples), information theory (infinitely long blocks) and more recently, [massive] MIMO wireless communications (infinitely many antennas).

Some caution is strongly advisable with respect to the latter. In fact, there are compelling reasons to avoid asymptotics in the number of antennas altogether:

- First, elegant, rigorous and intuitively comprehensible capacity bound formulas are available in closed form.

The proofs of these expressions use basic random matrix theory, but no asymptotics at all. - Second, the notion of “asymptotic limit” or “asymptotic behavior” helps propagate the myth that Massive MIMO somehow relies on asymptotics or “infinite” numbers (or even exorbitantly large numbers) of antennas.
- Third, many approximate performance results for Massive MIMO (particularly “deterministic equivalents”) based on asymptotic analysis are complicated, require numerical evaluation, and offer little intuitive insight. (And, the verification of their accuracy is a formidable task.)

Finally, and perhaps most importantly, careless use of asymptotic arguments may yield erroneous conclusions. For example in the effective SINRs in multi-cell Massive MIMO, the coherent interference scales with M (number of antennas) – which yields the commonly held misconception that coherent interference is the main impairment caused by pilot contamination. But in fact, in many relevant circumstances it is not (see case studies here): the main impairment for “reasonable” values of M is the reduction in coherent beamforming gain due to reduced estimation quality, which in turn is independent of M.

In addition, the number of antennas beyond which the far-field assumption is violated is actually smaller than what one might first think (problem 3.14).

]]>IEEE SPS members can watch the videos for free but it is necessary to log in through the IEEE website.

]]>First, four carefully selected articles are offered free of charge, see the screenshot below and click here for details.

More precisely, IEEE offers free access to the published versions of these articles, while the accepted versions were already openly available: Paper 1, Paper 2, Paper 3, and Paper 4.

Second, a live webinar entitled “5G Massive MIMO: Achieving Spectrum Efficiency” is organized by IEEE ComSoc on August 24. The speaker is Professor Liesbet Van der Perre from KU Leuven. She was the scientific leader of the MAMMOET project, which is famous for demonstrating that Massive MIMO works in practice. You can expect a unique mix of theoretical concepts and practical implementation insights from this webinar.

]]>The first step towards reproducibility is to describe the simulation procedure in such detail that another researcher can repeat the simulation, but a major effort is typically needed to reimplement everything. The second step is to make the simulation code publicly available, so that any scientist can review it and easily reproduce the results. While the first step is mandatory for publishing a scientific study, there is a movement towards open science that would make also the second step a common practice.

I understand that some researchers are skeptical towards sharing their simulation code, in fear of losing their competitive advantage towards other research groups. My personal principle is to not share any code until the research study is finished and the results have been accepted for publication in a full-length journal. After that, I think that the society benefits the most if other researcher can focus on improving my and others’ research, instead of spending excessive amount of time on reimplementing known algorithms. I also believe that the primary competitive advantage in research is the know-how and technical insights, while the simulation code is of secondary importance.

On my GitHub page, I have published Matlab code packages that reproduces the simulation results in one book, one book chapter, and more than 15 peer-reviewed articles. Most of these publications are related to MIMO or Massive MIMO. I see many benefits from doing this:

1) It increases the credibility of my research group’s work;

2) I write better code when I know that other people will read it;

3) Other researchers can dedicate their time into developing new improved algorithms and compare them with my baseline implementations;

4) Young scientists may learn how to implement a basic simulation environment by reading the code.

I hope that other Massive MIMO researchers will also make their simulation code publicly available. Maybe you have already done that? In that case, please feel free to write a comment to this post with a link to your code.

]]>The article details the teaching principles and experiences that the teachers and students had from the 2015 edition of the CDIO-project. This was also described in a previous blog post. In the following video, the students describe and demonstrate the end-result of the 2016 edition of the project. The acoustic testbed is now truly massive, since 64 loudspeakers were used.

]]>**2017 Joint IEEE SPS and EURASIP Summer School on Signal Processing for 5G **

Signal processing is at the core of the emerging fifth generation (5G) cellular communication systems, which will bring revolutionary changes to the physical layer. Unlike other 5G events, the objective of this summer school is to teach the main physical-layer techniques for 5G from a signal-processing perspective. The lectures will provide a background on the 5G wireless communication concepts and their formulation from a signal processing perspective. Emphasis will be placed on showing specifically how cutting-edge signal processing techniques can and will be applied to 5G. Keynote speeches by leading researchers from Ericsson, Huawei, China Mobile, and Volvo complement the technical lectures.

The summer school covers the following specific topics:

- Massive MIMO communication in TDD and FDD
- mmWave communications and compressed sensing
- mmWave positioning
- Wireless access for massive machine-type communications

The school takes place in Gothenburg, Sweden, from May 29th to June 1st, in the week after ICC in Paris.

This event belongs to the successful series of IEEE SPS and EURASIP Seasonal Schools in Signal Processing. The 2017 edition is jointly organized by Chalmers University of Technology, Linköping University, The University of Texas at Austin, Aalborg University and the University of Vigo.

Registration is now open. A limited number of student travel grants will be available.

For more information and detailed program, see: http://www.sp-for-5g.com/

]]>**Problem set:** We have developed an extensive set of problems to go with the book. This problem set can be downloaded from the Cambridge resource page, www.cambridge.org/Marzetta, or from this direct link.

The difficulty level of the problem varies widely, rendering the material suitable for instruction at all levels. The problem set is very much a living document and may be extended or improved in the future. Many, though not all, of the problems have been tested on my students when I taught the subject last year. We appreciate, as always, comments or suggestions on the material.

A detailed solution manual is available to instructors who adopt the book.

**List of errata:** There is also a list of errata to the book – available via this direct link, or from the Cambridge resource page.

*Have no fear of perfection — you’ll never reach it.* — Salvador Dali

My colleagues Erik G. Larsson and Hien Quoc Ngo have written a book entitled “Fundamentals of Massive MIMO” together with Thomas L. Marzetta and Hong Yang at Bell Labs, Nokia. The book is published this October/November by Cambridge University Press.

I have read the book and I think it serves as an excellent introduction to the topic. The text is suitable for graduate students, practicing engineers, professors, and doctoral students who would like to learn the basic Massive MIMO concept, results and properties. It also provides a clean introduction to the theoretical tools that are suitable for analyzing the Massive MIMO performance.

I personally intend to use this book as course material for a Master level course on Multiple-antenna communications next year. I recommend other teachers to also consider this possibility!

A preview of the book can be found on Google Books:

**Update:** Since November 2017, there is another book: “Massive MIMO Networks: Spectral, Energy, and Hardware Efficiency“.