All posts by Emil Björnson

When Normalization is Dangerous

The signal-to-noise ratio (SNR) generally depends on the transmit power, channel gain, and noise power:

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.

Further Differences Between Massive MIMO for Sub-6 GHz and mmWave

One of the most read posts on this blog is Six differences between Massive MIMO for sub-6 GHz and mmWave, where we briefly outlined the key differences between how the Massive MIMO technology would be implemented and utilized in different frequency bands. Motivated by the great feedback and interest in this topic, we joined forces with Liesbet Van der Perre and Stefano Buzzi to write a full-length magazine article. It has recently been submitted to IEEE Wireless Communications and a pre-print can be found on

Massive MIMO in Sub-6 GHz and mmWave: Physical, Practical, and Use-Case Differences

30% Discount on “Massive MIMO Networks” Book

The hardback version of the massive new book Massive MIMO Networks: Spectral, Energy, and Hardware Efficiency (by Björnson, Sanguinetti, Hoydis) is currently available for the special price of $70 (including worldwide shipping). The original price is $99.

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

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!

How Distortion from Nonlinear Massive MIMO Transceivers is Radiated Spatially

While the research literature is full of papers that design wireless communication systems under constraints on the maximum transmitted power, in practice, it might be constraints on the equivalent isotropically radiated power (EIRP) or the out-of-band radiation that limit the system operation.

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.

A Basic Way to Quantify the Massive MIMO Gain

Several people have recently asked me for a simple way to quantify the spectral efficiency gains that we can expect from Massive MIMO. In theory, going from 4 to 64 antennas is just a matter of changing a parameter value. However, many practical issues need be solved to bring the technology into reality and the solutions might only be developed if we can convince ourselves that the gains are sufficiently large.

While there is no theoretical upper limit on how spectrally efficient Massive MIMO can become when adding more antennas, we need to set some reasonable first goals.  Currently, many companies are trying to implement analog beamforming in a cost-efficient manner. That will allow for narrow beamforming, but not spatial multiplexing.

By following the methodology in Section 3.3.3 in Fundamentals of Massive MIMO, a simple formula for the downlink spectral efficiency is:

(1)   \begin{equation*}K \cdot \left( 1 - \frac{K}{\tau_c} \right) \cdot \log_2 \left( 1+ \frac{ c_{ \textrm{\tiny CSI}} \cdot M \cdot \frac{\mathrm{SNR}}{K}}{\mathrm{SNR}+ 1} \right)\end{equation*}

where $M$ is the number of base-station antennas, $K$ is the number of spatially multiplexed users, $c_{ \textrm{\tiny CSI}}  \in [0,1]$ is the quality of the channel estimates, and $\tau_c$ is the number of channel uses per channel coherence block. For simplicity, I have assumed the same pathloss for all the users. The variable $\mathrm{SNR}$ is the nominal signal-to-noise ratio (SNR) of a user,  achieved when $M=K=1$. Eq. (1) is a rigorous lower bound on the sum capacity, achieved under the assumptions of maximum ratio precoding, i.i.d. Rayleigh fading channels, and equal power allocation. With better processing schemes, one can achieve substantially higher performance.

To get an even simpler formula, let us approximate (1) as

(2)   \begin{equation*}K \log_2 \left( 1+ \frac{ c_{ \textrm{\tiny CSI}} M}{K} \right)\end{equation*}

by assuming a large channel coherence and negligible noise.

What does the formula tell us?

If we increase $M$ while $K$ is fixed , we will observe a logarithmic improvement in spectral efficiency. This is what analog beamforming can achieve and, hence, I am a bit concerned that the industry will be disappointed with the gains that they will obtain from such beamforming in 5G.

If we instead increase $M$ and $K$ jointly, so that  $M/K$ stays constant, then the spectral efficiency will grow linearly with the number of users. Note that the same transmit power is divided between the $K$ users, but the power-reduction per user is compensated by increasing the array gain $M$ so that the performance per user remains the same.

The largest gains come from spatial multiplexing

To give some quantitative numbers, consider a baseline system with $M=4$ and $K=1$ that achieves 2 bit/s/Hz. If we increase the number of antennas to $M=64$, the spectral efficiency will become 5.6 bit/s/Hz. This is the gain from beamforming. If we also increase the number of users to $K=16$ users, we will get 32 bit/s/Hz. This is the gain from spatial multiplexing. Clearly, the largest gains come from spatial multiplexing and adding many antennas is a necessary way to facilitate such multiplexing.

This analysis has implicitly assumed full digital beamforming. An analog or hybrid beamforming approach may achieve most of the array gain for $K=1$, but I believe that the gains from spatial multiplexing will be substantially smaller.

Holographic Beamforming versus Massive MIMO

Last year, the startup company Pivotal Commware secured venture capital (e.g., from Bill Gates) to bring its holographic beamforming technology to commercial products. Despite the word “holographic”, this is not a technology focused on visual-light communications. Instead, the company uses passive electronically steered antennas (PESAs) that are designed for radio-frequencies (RFs) in the micro- and millimeter-wave bands. It is the impedance pattern created in the distribution network over the array that is called a “hologram” and different holograms lead to beamforming in different spatial directions. The company reportedly aims at having commercial products ready this year.

Will the futuristic-sounding holographic beamforming make Massive MIMO obsolete? Not at all, because this is a new implementation architecture, not a new beamforming scheme or spatial multiplexing method. According to the company’s own white paper, the goal is to deliver “a new dynamic beamforming technique using a Software Defined Antenna (SDA) that employs the lowest C-SWaP (Cost, Size, Weight, and Power)“. Simply speaking, it is a way to implement a phased array in a thin, conformable, and affordable way. The PESAs are constructed using high volume commercial off-the-shelf components. Each PESA has a single RF-input and a distribution network that is used to vary the directivity of the beamforming. With a single RF-input, only single-user single-stream beamforming is possible. As explained in Section 1.3 in my recent book, such single-user beamforming can improve the SINR, but the rate only grows logarithmically with the number of antennas. Nevertheless, cost-efficient single-stream beamforming from massive arrays is one of the first issues that the industry tries to solve, in preparation for a full-blown Massive MIMO deployment.

The largest gains from multiple antenna technologies come from spatial multiplexing of many users, using a Massive MIMO topology where the inter-user interference is reduced by making the beams narrower as more users are to be multiplexed. The capacity then grows linearly with the number of users, as also explained in Section 1.3 of my book.

Can holographic beamforming be used to implement Massive MIMO with spatial multiplexing of tens of users? Yes, similar to hybrid beamforming, one could deploy an array of PESAs, where each PESA is used to transmit to one user. Eric J. Black, CTO and founder of Pivotal Commware, refers to this as “sub-aperture based SDMA“. If you want the capability of serving ten users simultaneously, you will need ten PESAs.

If the C-SWaP of holographic beamforming is as low as claimed, the technology might have the key to cost-efficient deployment of Massive MIMO. The thin and conformable form factor also makes me think about the recent concept of Distributed Large Intelligent Surface, where rooms are decorated with small antenna arrays to provide seamless connectivity.

Origin of the “Massive MIMO” Name

“A dear child has many names” is a Swedish saying and it certainly applies to Massive MIMO. It is commonly claimed that the Massive MIMO concept originates from the seminal paper “Noncooperative Cellular Wireless with Unlimited Numbers of Base Station Antennas” by Thomas Marzetta, published in 2010. This is basically correct, except for the fact that the paper only talks about “Multi-user MIMO systems with very large antenna arrays“. Marzetta then published several papers using the large‐scale antenna systems (LSAS) terminology, before switching to calling it Massive MIMO in more recent years. Over the years, various papers have also called it “very large multiuser MIMO” and “large-scale MIMO“. Nowadays, Massive MIMO is used by almost everyone in the research community, and even by marketing people.

If you search at IEEEXplore, the origin of the name remains puzzling. The earliest papers are “Massive MIMO: How many antennas do we need?” by Hoydis/ten Brink/Debbah and “Achieving Large Spectral Efficiency with TDD and Not-so-Many Base-Station Antennas” by Huh/Giuseppe Caire/Papadopoulos/Ramprashad, both from 2011. However, these papers are referring to Marzetta’s seminal paper, which doesn’t call it “Massive MIMO”.

If you instead read the news reports by ZDNet and Silicon from the 2010 Bell Labs Open Days in Paris, the origin of “Massive MIMO” becomes clearer. Marzetta presented his concept and reportedly said that “We haven’t been able to come up with a catchy name”, but told ZDNet that “massive MIMO” and “large-scale MIMO” were two candidates. To the Massive MIMO blog, Marzetta now explains why he initially abandoned these potential names, in favor for LSAS:

When I explained the concept to the Bell Labs Director of Research, he commented that it didn’t sound at all like MIMO to him. He recommended strongly that I think of a name that didn’t contain the acronym “MIMO”, hence, LSAS. Eventually (after everyone else called it Massive MIMO) I abandoned “LSAS” and started to call it “Massive MIMO”.

In conclusion, the Massive MIMO name came originally from Marzetta, who used it when first describing the concept to the public, but the name was popularized by other researchers.