Category Archives: Technical insights

Is It Time to Forget About Antenna Selection?

Channel fading has always been a limiting factor in wireless communications, which is why various diversity schemes have been developed to combat fading (and other channel impairments). The basic idea is to obtain many “independent” observations of the channel and exploit that it is unlikely that all of these observations are subject to deep fade in parallel. These observations can be obtained over time, frequency, space, polarization, etc.

Only one antenna is used at a time when using antenna selection.

Antenna selection is the basic form of space diversity. Suppose a base station (BS) equipped with multiple antennas applies antenna selection. In the uplink, the BS only uses the antenna that currently gives the highest signal-to-interference-and-noise ratio (SINR). In the downlink, the BS only transmits from the antenna that currently has the highest SINR. As the user moves around, the fading changes and we, therefore, need to reselect which antenna to use.

The term antenna selection diversity can be traced back to the 1980s, but this diversity scheme was analyzed already in the 1950s. One well-cited paper from that time is Linear Diversity Combining Techniques by D. G. Brennan. This paper demonstrates mathematically and numerically that selection diversity is suboptimal, while the scheme called maximum-ratio combining (MRC) always provides higher SINR. Hence, instead of only selecting one antenna, it is preferable for the BS to coherently combine the signals from/to all the antennas to maximize the SINR. When the MRC scheme is applied in Massive MIMO with a very large number of antennas, we often talk about channel hardening but this is nothing but an extreme form of space diversity that almost entirely removes the fading effect.

Even if the suboptimality of selection diversity has been known for 60 years, the antenna selection concept has continued to be analyzed in the MIMO literature and recently also in the context of Massive MIMO. Many recent papers are considering a generalization of the scheme that is known as antenna subset selection, where a subset of the antennas is selected and then MRC is applied using only these ones.

Why use antenna selection?

A common motivation for using antenna selection is that it would be too expensive to equip every antenna with a dedicated transceiver chain in Massive MIMO, therefore we need to sacrifice some of the performance to achieve a feasible implementation. This is a misleading motivation since Massive MIMO capable base stations have already been developed and commercially deployed. I think a better motivation would be that we can save power by only using a subset of the antennas at a time, particularly, when the traffic load is below the maximum system capacity so we don’t need to compromise with the users’ throughput.

The recent papers [1], [2], [3] on the topic consider narrowband MIMO channels. In contrast, Massive MIMO will in practice be used in wideband systems where the channel fading is different between subcarriers. That means that one antenna will give the highest SINR on one subcarrier, while another antenna will give the highest SINR on another subcarrier. If we apply the antenna selection principle on a per-subcarrier basis in a wideband OFDM system with thousands of subcarriers, we will probably use all the antennas on at least one of the subcarrier. Consequently, we cannot turn off any of the antennas and the power saving benefits are lost.

We can instead apply the antenna selection scheme based on the average received power over all the subcarriers, but most channel models assume that this average power is the same for every base station antenna (this applies to both i.i.d. fading and correlated fading models, such as the one-ring model). That means that if we want to turn off antennas, we can select them randomly since all random selections will be (almost) equally good, and there are no selection diversity gains to be harvested.

This is why we can forget about antenna selection diversity in Massive MIMO!

It is only when the average channel gain is different among the antennas that antenna subset selection diversity might have a role to play. In that case, the antenna selection is governed by variations in the large-scale fading instead of variations in the small-scale fading, as conventionally assumed. This paper takes a step in that direction. I think this is the only case of antenna (subset) selection that might deserve further attention, while in general, it is a concept that can be forgotten.

Five Promising Research Directions for Antenna Arrays

Ever since I finished the writing of the book Massive MIMO Networks: Spectral, Energy, and Hardware Efficiency, I have felt that I’m somewhat done with my research on conventional Massive MIMO. The spectral efficiency, energy efficiency, resource allocation, and pilot contamination phenomenon are well understood by now. This is not a bad thing—as researchers, we are supposed to solve the problems we are analyzing. But it means that this is a good time to look for new research directions. It should preferably be something where we can utilize our skills as Massive MIMO researchers to do something new and exciting!

With this in mind, I gathered a team consisting of myself, Luca Sanguinetti, Henk Wymeersch, Jakob Hoydis, and Thomas L. Marzetta. Each one of us has written about one promising new direction of research related to antenna arrays and MIMO, including the background of the topic, our long-term vision, and pertinent open problem. This resulted in the paper:

Massive MIMO is a Reality – What is Next? Five Promising Research Directions for Antenna Arrays

You can find the preprint on or by clicking on the name of the paper. I hope that you will find it as interesting to read as it was for us to write!

Efficient DSP and Circuit Architectures for Massive MIMO: State-of-the-Art and Future Directions

Come listen to Liesbet Van der Perre, Professor at KU Leuven (Belgium) on Monday February 18 at 2.00 pm EST.

She gives a webinar on state-of-the-art circuit implementations of Massive MIMO, and outlines future research challenges. The webinar is based on, among others, this paper.

In more detail the webinar will summarize the fundamental technical contributions to efficient digital signal processing for Massive MIMO. The opportunities and constraints on operating on low-complexity RF and analog hardware chains are clarified. It will explain how terminals can benefit from improved energy efficiency. The status of technology and real-life prototypes will be discussed. Open challenges and directions for future research are suggested.

Listen to the webinar by following this link.

Could chip-scale atomic clocks revolutionize wireless access?

This chip-scale atomic clock (CSAC) device, developed by Microsemi, brings atomic clock timing accuracy (see the specs available in the link) in a volume comparable to a matchbox, and 120 mW power consumption.  This is way too much for a handheld gadget, but undoubtedly negligible for any fixed installation powered from the grid.  An alternative to synchronization through GNSS that works anywhere, including indoor in GNSS-denied environments.

I haven’t seen a list price, and I don’t know how much exotic metals and what licensing costs that its manufacture requires, but let’s ponder the possibility that a CSAC could be manufactured for the mass-market for a few dollars each. What new applications would then become viable in wireless?

The answer is mostly (or entirely) speculation. One potential application that might become more practical is positioning using distributed arrays.  Another is distributed multipair relaying. Here and here are some specific ideas that are communication-theoretically beautiful, and probably powerful, but that seem to be perceived as unrealistic because of synchronization requirements. Perhaps CoMP and distributed MIMO, a.k.a. “cell-free Massive MIMO”, applications could also benefit.

Other applications might be applications for example in IoT, where a device only sporadically transmits information and wants to stay synchronized (perhaps there is no downlink, hence no way of reliably obtaining synchronization information).  If a timing offset (or frequency offset for that matter) is unknown but constant over a very long time, it may be treated as a deterministic unknown and estimated. The difficulty with unknown time and frequency offsets is not their existence per se, but the fact that they change quickly over time.

It’s often said (and true) that the “low” speed of light is the main limiting factor in wireless.  (Because channel state information is the main limiting factor of wireless communications.  If light were faster, then channel coherence would be longer, so acquiring channel state information would be easier.) But maybe the unavailability of a ubiquitous, reliable time reference is another, almost as important, limiting factor. Can CSAC technology change that?  I don’t know, but perhaps we ought to take a closer look.

Beamforming From Distributed Arrays

When an antenna array is used to focus a transmitted signal on a receiver, we call this beamforming (or precoding) and we usually illustrate it as shown to the right. This cartoonish illustration is only applicable when the antennas are gathered in a compact array and there is a line-of-sight channel to the receiver.

If we want to deploy very many antennas, as in Massive MIMO, it might be preferable to distribute the antennas over a larger area. One such deployment concept is called Cell-free Massive MIMO. The basic idea is to have many distributed antennas that are transmitting phase-coherently to the receiving user. In other words, the antennas’ signal components add constructively at the location of the user, just as when using a compact array for beamforming. It is therefore convenient to call it beamforming in both cases—algorithmically it is the same thing!

The question is: How can we illustrate the beamforming effect when using a distributed array?

The figure below shows how to do it. I consider a toy example with 80 star-marked antennas deployed along the sides of a square and these antennas are transmitting sinusoids with equal power, but different phases. The phases are selected to make the 80 sine-components phase-aligned at one particular point in space (where the receiving user is supposed to be):

Clearly, the “beamforming” from a distributed array does not give rise to a concentrated signal beam, but the signal amplification is confined to a small spatial region (where the color is blue and the values on the vertical axis are close to one). This is where the signal components from all the antennas are coherently combined. There are minor fluctuations in channel gain at other places, but the general trend is that the components are non-coherently combined everywhere except at the receiving user. (Roughly the same will happen in a rich multipath channel, even if a compact array is used for transmission.)

By looking at a two-dimensional version of the figure (see below), we can see that the coherent combination occurs in a circular region that is roughly half a wavelength in diameter. At the carrier frequencies used for cellular networks, this region will only be a few centimeters or millimeters wide. It is almost magical how this distributed array can amplify the signal at such a tiny spatial region! This spatial region is probably what the company Artemis is calling a personal cell (pCell) when marketing their distributed MIMO solution.

If you are into the details, you might wonder why I simulated a square region that is only a few wavelengths wide, and why the antenna spacing is only a quarter of a wavelength. This assumption was only made for illustrative purposes. If the physical antenna locations are fixed but we would reduce the wavelength, the size of the circular region will reduce and the ripples will be more frequent. Hence, we would need to compute the channel gain at many more spatial sample points to produce a smooth plot.

Reproduce the results: The code that was used to produce the plots can be downloaded from my GitHub.

Adaptive Beamforming and Antenna Arrays

Adaptive beamforming for wireless communications has a long history, with the modern research dating back to the 70s and 80s. There is even a paper from 1919 that describes the development of directive transatlantic communication practices that were developed during the First World War. Many of the beamforming methods that are considered today can be found already in the magazine paper Beamforming: A Versatile Approach to Spatial Filtering from 1988. Plenty of further work was carried out in the 90s and 00s, before the Massive MIMO paradigm.

I think it is fair to say that no fundamentally new beamforming methods have been developed in the Massive MIMO literature, but we have rather taken known methods and generalized them to take imperfect channel state information and other practical aspects into account. And then we have developed rigorous ways to quantify the achievable rates that these beamforming methods achieve and studied the asymptotic behaviors when having many antennas. Closed-form expressions are available in some special cases, while Monte Carlo simulations can be used to compute these expressions in other cases.

As beamforming has evolved from an analog phased-array concept, where angular beams are studied, to a digital concept where the beamforming is represented in multi-dimensional vector spaces, it easy to forget the basic properties of array processing. That is why we dedicated Section 7.4 in Massive MIMO Networks to describe how the physical beam width and spatial resolution depend on the array geometry.

In particular, I’ve observed a lot of confusion about the dimensionality of MIMO arrays, which are probably rooted in the confusion around the difference between an antenna (which is something connected to an RF chain) and a radiating element. I explained this in detail in a previous blog post and then exemplified it based on a recent press release. I have also recorded the following video to visually explain these basic properties:

A recent white paper from Ericsson is also providing a good description of these concepts, particularly focused on how an array with a given geometry can be implemented with different numbers of RF chains (i.e., different numbers of antennas) depending on the deployment scenario. While having as many antennas as radiating element is preferable from a performance perspective, but the Ericsson researchers are arguing that one can get away with fewer antennas in the vertical direction in deployments where it is anyway very hard to resolve users in the elevation dimension.

Downlink Massive MIMO Analysis

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 N-antenna base station (BS) serves K single-antenna users. The large-scale channel gains include pathloss with exponent $\eta$ and shadowing having log-scale standard deviation \sigma_{\scriptscriptstyle \rm dB}, with the gain between the \ellth BS and the kth user served by a BS of interest denoted by G_{\ell;k}.  With conjugate beamforming and receivers reliant on channel hardening, the signal-to-interference ratio (SIR) at such user is [2]

    $$\mathsf{SIR}_k = \frac{N p_k\,G_{k}}{\sum_{\ell} G_{\ell:k} } . $$

where G_{k} is the gain from the serving BS and p_k is the share of that BS’s power allocated to user k. Two power allocations can be analyzed:

  1. Uniform: p_k = 1/K.
  2. SIR-equalizing [3]: p_{k} \propto \frac{\sum_{\ell} G_{\ell;k}}{G_{k}}, with the proportionality constant ensuring that \sum_k p_k = 1. This makes \mathsf{SIR}_k = \mathsf{SIR} \, \forall k. Moreover, as N and K grow large, \mathsf{SIR} \rightarrow \frac{N}{K} \, (1- 2 / \eta) .

The analysis is conducted for \sigma_{\scriptscriptstyle \rm dB} \to \infty, which makes it valid for arbitrary BS locations.


For notational compactness, let \delta = 2/\eta. Define s<0 as the solution to  {s}^\delta \,\gamma(-\delta,s)=0, where \gamma(\cdot) is the lower incomplete gamma function. For \eta=4, in particular, s = -0.85. Under a uniform power allocation, the CDF of \mathsf{SIR}_k is available in an explicit form involving the Gauss hypergeometric function {}_2 F_1 (available in MATLAB and Mathematica):

$\!\!\!\!\!\!\begin{cases} F_{\mathsf{SIR}_k}(\theta) \simeq e^{s \left(\frac{N}{\theta \,K}-1\right)}  & 0 \leq \theta < \frac{N/K}{3 + \epsilon} \\ F_{\mathsf{SIR}_k}(\theta) = 1 - \left(\frac{N}{\theta \,K}-1\right)^{\delta} \mathrm{sinc} \, \delta + B \! \left(\frac{\theta \,K}{N-2\,\theta \,K}\right) & \frac{N/K}{3} \leq \theta < \frac{N / K}{2 } \\ F_{\mathsf{SIR}_k}(\theta) = 1 - \left(\frac{N}{\theta \,K}-1\right)^{\delta} \mathrm{sinc} \, \delta \quad\qquad & \frac{N / K}{2} \leq \theta<\frac{N}{K}\end{cases}$

where “\simeq” indicates asymptotic (\theta \to 0) equality, \epsilon is such that the CDF is continuous, and

    $$B(x) = \frac{ {}_2 F_1 \big(1, \delta+1; 2 \, \delta + 2; -1/x \big) \, \delta }{x^{1+2\,\delta}\;\Gamma (2\,\delta + 2)\,{\Gamma^2 (1-\delta)}} .$$

Alternatively, the CDF can be obtained by solving (e.g., with Mathematica) a single integral involving the Kummer function {}_1 F_1:

$\!\!\!\!\!\! F_{\mathsf{SIR}_k}(\theta)=\frac{1}{2}-\frac{1}{\pi}\int_{0}^{\infty}\Im\!\left\{\frac{e^{\frac{i\omega}{1-\theta K/N}}}{{}_1 F_1\left(1,1-\delta,\frac{i\theta\omega}{N/K-\theta}\right)}\right\}\frac{d\omega}{\omega}\,\,\,0<\theta<\frac{N}{K}.$

This latter solution can be modified for the SIR-equalizing power allocation as

$\!\!\!\!\!\!\!\!F_{\mathsf{SIR}}(\theta) = \frac{1}{2} - \frac{1}{\pi} \int_{0}^{\infty} \Im \! \left\{\frac{e^{i\,\omega}}{\left\{{}_1 F_1\!\left(1,1-\delta,i \,\theta\,\omega/N\right)\right\}^K}\right\} \frac{d\omega}{\omega} \,\,\, 0<\theta<\frac{N}{K}.$

Spectral Efficiency

The spectral efficiency of user k is C_k=\log_2(1+\mathsf{SIR}_k), with CDF F_{C_k}(\zeta) = F_{\mathsf{SIR}_k}(2^\zeta-1) readily characterizable from the expressions given earlier. From C_k, the sum spectral efficiency at the BS of interest can be found as C_{\Sigma} = \sum_{k} C_k . Expressions for the averages \bar{C} = \mathbb{E} \big[ C_k \big] and \bar{C}_{\scriptscriptstyle \Sigma} = \mathbb{E} \! \left[ C_{\scriptscriptstyle \Sigma} \right] are further available in the form of single integrals.

With a uniform power allocation,

(1)   \begin{equation*}\bar{C} =  \log_2(e) \,\int_{0}^{\infty} \frac{ 1-e^{-z N/K}}{ {}_1 F_1 \big( 1,1-\delta,z \big)} \, \frac{{d}z}{z}\end{equation*}

and \bar{C}_{\scriptscriptstyle \Sigma} = K \bar{C}. For the special case of \eta=4, the Kummer function simplifies giving

(2)   \begin{equation*}\bar{C}=\log_2(e) \,\int_{0}^{\infty} \frac{ 1-e^{-z N/K}}{1 + e^z \sqrt{\pi z} \, \erf\sqrt{z}} \, \frac{{d}z}{z} .\end{equation*}

With an equal-SIR power allocation

(3)   \begin{equation*}\bar{C}=\log_2(e)\,\int_{0}^{\infty} \frac{ 1-e^{-z}}{{}_1 F_1\left(1,1-\delta,z/N \right)^K} \, \frac{{d}z}{z}\end{equation*}

and \bar{C}_{\scriptscriptstyle \Sigma} = K \bar{C}.

Application to Relevant Networks

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 N=100, K=10 and \eta=4. Figs. 1a-1b compare the simulations for \sigma_{\scriptscriptstyle \rm dB}= 1014 dB with the analysis. The behaviors with these typical outdoor values of \sigma_{\scriptscriptstyle \rm dB} 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.

Figure 1: Analysis vs hexagonal network simulations with lognormal shadowing

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 K of them are served by the central BS. As before, N=100, K = 10 and \eta =4. Figs. 2a-2b compare the simulations for \sigma_{\scriptscriptstyle \rm dB} = 10 dB with the analysis, and the agreement is now complete. The simulated average spectral efficiency with a uniform power allocation is \bar{C}=2.77 b/s/Hz/user while (2) gives \bar{C}=2.76 b/s/Hz/user.

Figure 2: Analysis vs Poisson network simulations with lognornmal shadowing.

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 (N \lesssim 150200 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.