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

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.

]]>- J. H. Thompson from Qualcomm gave a keynote on 5G, relaying several important insights. He stressed the fundamental role of Massive MIMO, utilizing reciprocity (which in turn, of course, implies TDD). This is a message we have been preaching for years now, and it is reassuring to hear a main industry leader echo it at such an important event. He pointed to distributed Massive MIMO (that we know of as “cell-free massive MIMO“) as a forthcoming technology, not only because of the macro-diversity but also because of the improved channel rank it offers to multiple-antenna terminals. This new technology may enable AR/VR/XR, wireless connectivity in factories and much more… where conventional massive MIMO might not be sufficient.
- In the exhibition hall Nokia showcased a 64×2=128 Massive MIMO array, with fully digital transceiver chains, small dual-polarized path antennas, operating at 2.5 GHz and utilizing reciprocity – though it wasn’t clear exactly what algorithmic technology that went inside. (See photographs below.) Sprint already has deployed this product commercially, if I understood well, with an LTE TDD protocol. Ericsson had a similar product, but it was not opened, so difficult to tell exactly what the actual array looked like. The Nokia base station was only slightly larger, physically, than the flat-screen-base-station vision I have been talking about for many years now, and along the lines that T. Marzetta from Bell Labs had already back in 2006. Now that cellular Massive MIMO is a commercial reality… what should the research community do? Granted there are still lots of algorithmic innovation possible (and needed), but …. Cell-free massive MIMO with RF over fiber is the probably the obvious next step.
- T. Marzetta from NYU gave an industry distinguished talk, speculating about the future of wireless beyond Massive MIMO. What, if anything at all, could give us another 10x or 100x gain? A key point of the talk was that we have to go back to (wave propagation) physics and electromagnetics, a message that I very much subscribe to: the “y=Hx+w” models we typically use in information and communication theory are in many situations rather oversimplified. Speculations included the use of super-directivity, antenna coupling and more… It will be interesting to see where this leads, but at any rate, it is interesting fundamental physics.

There were also lots of other (non-Massive MIMO) interesting things: UAV connectivity, sparsity… and a great deal of questions and discussion on how machine learning could be leveraged, more about that at a later point in time.

It looks to me now that two of these speculations were wrong:

- First, “Massive MIMO increases the robustness against both unintended man-made interference and intentional jamming.” This is only true with some qualifiers, or possibly not true at all. (Actually I don’t really know, and I don’t think it is known for sure. It seems that this question remains a rather pertinent research direction for anyone interested in physical layer security and MIMO.) Subsequent research by others showed that Massive MIMO can be extraordinarily susceptible to attacks on the pilot channels, revealing an important, fundamental vulnerability at least if standard pilot-based channel estimation is used and no excess dimensions are “wasted” on interference suppression or detection. Basically this pilot channel attack exploits the so-called pilot contamination phenomenon, “hijacking” the reciprocity-based beamforming mechanism.
- Second, “In a way, massive MIMO relies on the law of large numbers to make sure that noise, fading, and hardware imperfections average out when signals from a large number of antennas are combined in the air.” This is not generally true, except for in-band distortion and with many simultaneously multiplexed users and frequency selective Rayleigh fading. In general the distortion that results from hardware imperfections is correlated among the antennas. In the special case of line-of-sight with a single terminal, an important basic reference case, the distortion is identical (up to a phase shift) at all antennas, hence resulting in a rank-one transmission: the distortion is beamformed in the same direction as the signal of interest and hardware imperfections do not “average out” at all.

This is particularly serious for out-band effects. Readers interested in a thorough mathematical treatment may consult my student’s recent Ph.D. dissertation.

Have you found any more? Let me know. The knowledge in the field continues to evolve.

]]>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.)

]]>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.

]]>Until recently, a more rigorous analysis was unavailable. Some weeks ago the authors of this paper argued, that all things considered, the use of superimposed pilots does not offer any appreciable gains for practically interesting use cases. The analysis was based on a capacity-bounding approach for finite numbers of antennas and finite channel coherence, but it assumed the most basic form of signal processing for detection and decoding.

There still remains some hope of seeing improvements, by implementing more advanced signal processing, like zero-forcing, multicell MMSE decoding, or iterative decoding algorithms, perhaps involving “turbo” information exchange between the demodulator, channel estimation, and detector. It will be interesting to follow future work by these two groups of authors to understand how large improvements (if any) superimposed pilots eventually can give.

There are, at least, two general lessons to learn here. First, that performance predictions based on asymptotics can be misleading in practically relevant cases. (I have discussed this issue before.) The best way to perform analysis is to use rigorous capacity lower bounds, or possibly, in isolated cases of interest, link-level simulations with channel coding (for which, as it turns out, capacity bounds are a very good proxy). Second, more concretely, that while it may be tempting, to superimpose-squeeze multiple symbols into the same time-frequency-space resource, once all sources of impairments (channel estimation errors, interference) are accurately accounted for, the gains tend to evaporate. (It is for the same reason that NOMA offers no substantial gains in MIMO systems – a topic that I may return to at a later time.)

]]>- 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.

]]>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.

]]>The important fact is that ergodic capacity can be lower-bounded by a formula of the form log2(1+SINR), where SINR is an “effective SINR” (that includes, among others, the effects of the terminal’s lack of channel knowledge).

This effective SINR scales proportionally to M (number of antennas), for fixed total radiated power. Compared to a single-antenna system, reciprocity always offers M times better “beamforming gain” regardless of the system’s operating point. (In fact one of the paradoxes of Massive MIMO is that performance always increases with M, despite the fact that there are “more unknowns to estimate”!) And yes, at very low SNR, the effective SINR is proportional to SNR^2 so reciprocity-based beamforming does “break down”, however, it is still M times better than a single-antenna link (with the same total radiated power). One will also, eventually, reach a point where the capacity bound for omnidirectional transmission (e.g. using a space-time code with appropriate dimension reduction in order to host the required downlink pilots) exceeds that of reciprocity-based beamforming, however, importantly, in this regime the bounds may be loose.

These matters, along with numerous case studies involving actual link budget calculations, are of course rigorously explained in our recent textbook.

]]>I also touched the (for sub-5 GHz bands somewhat controversial) topic of hybrid beamforming, and whether that would reduce the required amount of hardware.

A question from the audience was whether the use of antennas with larger physical aperture (i.e., intrinsic directivity) would change the conclusions. The answer is no: the use of directional antennas is more or less equivalent to sectorization. The problem is that to exploit the intrinsic gain, the antennas must a priori point “in the right direction”. Hence, in the array, only a subset of the antennas will be useful when serving a particular terminal. This impacts both the channel gain (reduced effective aperture) and orthogonality (see, e.g, Figure 7.5 in this book).

There was also a stimulating panel discussion afterwards. One question discussed in the panel concerned the necessity, or desirability, of using multiple terminal antennas at mmWave. Looking only at the link budget, base station antennas could be traded against terminal antennas – however, that argument neglects the inevitably lost orthogonality, and furthermore it is not obvious how beam-finding/tracking algorithms will perform (millisecond coherence time at pedestrian speeds!). Also, obviously, the comparison I presented is extremely simplistic – to begin with, the line-of-sight scenario is extremely favorable for mmWaves (blocking problems), but also, I entirely neglected polarization losses. Solely any attempts to compensate for these problems are likely to require multiple terminal antennas.

Other topics touched in the panel were the viability of Massive MIMO implementations. Perhaps the most important comment in this context made was by Ian Wong of National Instruments: “In the past year, we’ve actually shown that [massive MIMO] works in reality … To me, the biggest development is that the skeptics are being quiet.” (Read more about that here.)

]]>Impressive, and important.

Granted, this number does not include the complexity of FFTs, sampling rate conversions, and several other (non-insignificant) tasks; however, it does include the bulk of the “Massive-MIMO”-specific digital processing. The design exploits a number of tricks and Massive-MIMO specific properties: diagonal dominance of the channel Gramian, in particular, in sufficiently favorable propagation.

When I started work on Massive MIMO in 2009, the common view held was that the technology would be infeasible because of computational complexity. Particularly, the sheer idea of performing zero-forcing processing in real time was met with, if not ridicule, extreme skepticism. We quickly realized, however, that a reasonable DSP implementation would require no more than some ten Watt. While that is a small number in itself, it turned out to be an overestimate by orders of magnitude!

I spoke with some of the lead inventors of the chip, to learn more about its design. First, the architectures for decoding and for precoding differ a bit. While there is no fundamental reason for why this has to be so, one motivation is the possible use of nonlinear detectors on uplink. (The need for such detectors, for most “typical” cellular Massive MIMO deployments, is not clear – but that is another story.)

Second, and more importantly, the scalability of the design is not clear. While the complexity of the matrix operations themselves scale fast with the dimension, the precision in the arithmetics may have to be increased as well – resulting in a much-faster-than-cubically overall complexity scaling. Since Massive MIMO operates at its best when multiplexing to many tens of terminals (or even thousands, in some applications), significant challenges remain for the future. That is good news for circuit engineers, algorithm designers, and communications theoreticians alike. The next ten years will be exciting.

]]>While it is known that grid-of-beams solutions perform poorly in isotropic scattering, no prior experimental results are known. This new paper:

Massive MIMO Performance—TDD Versus FDD: What Do Measurements Say?

answers this performance question through the analysis of real Massive MIMO channel measurement data obtained at the 2.6 GHz band. Except for in certain line-of-sight (LOS) environments, the original reciprocity-based TDD Massive MIMO represents the only effective implementation of Massive MIMO at the frequency bands under consideration.

]]>The basic presumption of TDD/reciprocity-based Massive MIMO is that all activity, comprising the transmission of uplink pilots, uplink data and downlink data, takes place inside of a coherence interval:

At fixed mobility, in meter/second, the dimensionality of the coherence interval is proportional to the wavelength, because the Doppler spread is proportional to the carrier frequency.

In a single cell, with max-min fairness power control (for uniform quality-of-service provision), the sum-throughput of Massive MIMO can be computed analytically and is given by the following formula:

In this formula,

- = bandwidth in Hertz (split equally between uplink and downlink)
- = number of base station antennas
- = number of multiplexed terminals
- = coherence bandwidth in Hertz (independent of carrier frequency)
- = coherence time in seconds (inversely proportional to carrier frequency)
- SNR = signal-to-noise ratio (“normalized transmit power”)
- = path loss for the k:th terminal
- = constant, close to with sufficient pilot power

This formula assumes independent Rayleigh fading, but the general conclusions remain under other models.

The factor that pre-multiplies the logarithm depends on .

The pre-log factor is maximized when . The maximal value is , which is proportional to , and therefore proportional to the wavelength. Due to the multiplication $B T_c$, one can get same pre-log factor using a smaller bandwidth by instead increasing the wavelength, i.e., reducing the carrier frequency. At the same time, assuming appropriate scaling of the number of antennas, , with the number of terminals, , the quantity inside of the logarithm is a constant.

Concluding, the sum spectral efficiency (in b/s/Hz) easily can double for every doubling of the wavelength: a megahertz of bandwidth at 100 MHz carrier is ten times more worth than a megahertz of bandwidth at a 1 GHz carrier. So while there is more bandwidth available at higher carriers, the potential multiplexing gains are correspondingly smaller.

In this example,

all three setups give the same sum-throughput, however, the throughput per terminal is vastly different.

**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/

]]>- At lower frequencies, larger areas are covered, thus most connections are likely to experience non-line-of-sight propagation. Since channel coherence is large (scales inverse proportionally to the Doppler), there is room for many terminals to transmit uplink pilots from which the base station consequently can obtain CSI. Reciprocity-based beamforming in TDD operation is scalable with respect to the number of base station antennas and delivers great value.
- As the carrier frequency is increased, the coverage area shrinks; connections are more and more likely to experience line-of-sight propagation. At mmWave frequencies, all connections are either line-of-sight, or consist of a small number of reflected components. Then the channel can be parameterized with only few angular parameters; FDD operation with appropriate flavors of beam tracking may work satisfactorily. Reciprocity certainly would be desirable in this case too, but may not be necessary for the system to function.

Physics has given us the reciprocity principle. It should be exploited in wireless system design.

]]>A good system design definitely must not ignore pilot interference. While it is easily removed “on the average” through greater-than-one reuse, the randomness present in wireless communications – especially the shadow fading – will occasionally cause a few terminals to be severely hit by pilot contamination and bring down their performance. This is problematic whenever we are concerned about the provision of uniformly great service in the cell – and that is one of the principal selling arguments for Massive MIMO. Notwithstanding, the impact of pilot contamination can be reduced significantly in practice by appropriate pilot reuse and judicious power control. (Chapters 5-6 in Fundamentals of Massive MIMO gives many details.)

A more fundamental question is whether pilot contamination could be entirely overcome: Does there exist an upper bound on capacity that saturates as the number of antennas, *M*, is increased indefinitely? Some have speculated that it cannot; much in line with known capacity upper bounds for cellular base station cooperation. While this question may be of more academic than practical interest, it has long been open except for in some trivial special cases: If the channels of two terminals lie in non-overlapping subspaces and Bayesian channel estimation is used, the channel estimates will not be contaminated; capacity grows as log(*M*) when *M* increases without bound.

A much deeper result is established in this recent paper: the subspaces of the channel covariances may overlap, yet capacity grows as log(*M*). Technically, a Rayleigh fading with spatial correlation is assumed, and the correlation matrices for the contaminating terminals must only be linearly independent as *M* goes to infinity (exact conditions in the paper). In retrospect, this is not unreasonable given the substantial a priori knowledge exploited by the Bayesian channel estimator, but I found it amazing how weak the required conditions on the correlation matrices are. It remains unclear whether the result generalizes to the case of a growing number of interferers: letting the number of antennas go to infinity and then growing the network is not the same thing as taking an “infinite” (scalable) network and increasing the number of antennas. But this paper elegantly and rigorously answers a long-standing question that has been the subject of much debate in the community – and is a recommended read for anyone interested in the fundamental limits of Massive MIMO.

**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

I have argued before that in a mobile access environment, no more than a few hundred of antennas per base station will be useful. In an environment without significant mobility, however, the answer is different. In [1, Sec. 6.1], one case study establishes the feasibility of providing (fixed) wireless broadband service to 3000 homes, using a single isolated base station with 3200 antennas (zero-forcing processing and max-min power control). The power consumption of the associated digital signal processing is estimated in [1, homework #6.6] to less than 500 Watt. The service of this many terminals is enabled by the long channel coherence (50 ms in the example).

Is this as massive as MIMO could ever get? Perhaps not. Conceivably, there will be environments with even larger channel coherence. Consider, for example, an outdoor city square with no cars or other traffic – hence no significant mobility. Eventually only measurements can determine the channel coherence, but assuming for the sake of argument 200 ms by 400 kHz, gives room for training of 40,000 terminals (assuming no more than 50% of resources are spent on training). Multiplexing these terminals would require at least 40,000 antennas, which would, at 3 GHz and half wavelength-spacing, occupy an area of 10 x 10 meters (say with a rectangular array for the sake of argument) – easily integrated onto the face of a skyscraper.

- What gross rate would the base station offer? Assuming, conservatively, 1 bit/s/Hz spectral efficiency (with the usual uniform-service-for-all design), the gross rate in a 25 MHz bandwidth would amount to 1 Tbit/s.
- How much power would the digital processing require? A back-of-the envelope calculation along the lines of the homework cited above suggests some 15 kW – the equivalent of a few domestic space heaters (I will return to the “energy efficiency” hype later on this blog).
- How much transmit power is required? The exact value will depend on the coverage area, but to appreciate the order of magnitude, observe that when doubling the number of antennas, the array gain is doubled. If, simultaneously, the number of terminals is doubled, then the total radiated power will be independent of the array size. Hence, transmit power is small compared to the power required for processing.

Is this science fiction or will we be seeing this application in the future? The application is fully feasible, with today’s circuit technology, and does not violate known physical or information theoretic constraints. Machine-to-machine, IoT, or perhaps virtual-reality-type applications may eventually create the desirability, or need, to build extreme Massive MIMO.

]]>Merouane Debbah, Vice-President of the Huawei France R&D center, confirms to the Massive MIMO blog that this spectral efficiency was achieved in the downlink, using TDD and exploiting channel reciprocity. This comes as no surprise, as it is not plausible that this performance could be sustained with FDD-style CSI feedback.

Another piece of evidence, that the theoretical predictions of Massive MIMO performance are for real.

]]>In that application, I don’t think so. Here is why.

What ultimately limits Massive MIMO is mobility: no more than half of the coherence time-bandwidth product should be occupied by pilot transmission activities. (This is the “half and half rule”.) In macro-cellular at 3 GHz, with highway mobility we may have on the order of 200 kHz x 1 millisecond coherence; that is 200 samples. With pilot reuse of 3 (that practically does away with pilot contamination), we could, then ultimately learn the channel to some 30 simultaneously served terminals – assuming mutually orthogonal pilots. Once the number of base station antennas M reaches beyond twice this number, with some margin – say M=100, the spectral efficiency grows logarithmically with M. That means, even doubling M yields only a 3dB effective SINR increase, that is a single extra bit per second/Hz per terminal. Beyond M=100 or M=200, it may not be worth it. Multiple antennas are only truly useful if they are used to multiplex, and mobility limits the amount of multiplexing we can perform.

So why not quadruple the number of antennas for additional coverage? May not be worth it either. Going from M=200 to M=2000 gives 10 dB – that pays for a 75% range extension, or, alternatively, a tenth of the losses incurred by an energy-saving coated window glass.

In stationary environments, the story is different – a topic that we will be returning to.

]]>- Channel Estimation and Performance Analysis of One-Bit Massive MIMO Systems
- One-bit massive MIMO: Channel estimation and high-order modulations,
- Performance of the Wideband Massive Uplink MIMO with One-Bit ADCs,

suggest the use of 1-bit ADCs in Massive MIMO base station receivers. Important studies of a concept, that offers great potential for cost saving and simplification of transceiver hardware.

Granted, much lower resolution will be sufficient in Massive MIMO than in conventional MIMO, but will one bit be sufficient? These papers indicate that the price to pay is not insignificant: the number of antennas may have to be doubled in some cases. Also, while the use of symbol-sampled models as in these studies may give correct order-of-magnitude estimates of capacity, much future work appears to remain to understand the effects of digital channelization/prefiltering and sampling rate conversion if 1-bit frontends are going to be used.

]]>The interesting part starts at 2:48, with the terminals onboard cars. While it has been contested whether Massive MIMO can work in mobility (because of channel aging) this experiment confirms that it does — as predicted by theory for a long time. In fact, at 3.7 GHz carrier and with a slot length of 0.5 ms, the maximum permitted mobility (assuming a two-ray model with Nyquist sampling, and a factor-of-two design margin) is over 140 km/h. So the experiment is probably still not close to the physical limits.

]]>