“Computers Weren’t Powerful Enough to Operate It”

Image from Wikipedia

I found an interesting news article by  where he is interviewing Martin Cooper, who is considered the father of the handheld cell phone. Cooper is talking about spatial division multiple access, the early name of the multi-user MIMO technology, and how “computers weren’t powerful enough to operate it” at the time it was invented at his startup-company ArrayComm.

Cooper points out that “Today we spray energy in all directions. Why not aim it directly?” He then provides three examples of why multi-user MIMO solves important practical problems:

First, deploying cellular and derivative technologies is costly. Second, the quality of wireless communicating must be comparable or better than that of wireline in order to compete with wireline, said Cooper. And as spectrum is finite, technology must work toward greater efficiency.

Furthermore, he says that ArrayComm’s multi-user MIMO solution “requires fewer base stations, which cuts costs. The technology also is adaptive, which simplifies network design and reduces site acquisition and installation costs.”

By the way, did I forgot to say that this interview is from 1996…?

I explained in a previous blog post why the efforts to commercialize multi-user MIMO in the nineties were not as successful as Cooper and others might have hoped for. Now, more than 20 years later, we are about to witness a large-scale deployment of 5G technology, in which MIMO is a key component. The industry has hopefully learned from the negative past experiences when Massive MIMO is now being deployed in commercial networks. One thing we know for sure is that computational complexity is not a problem anymore.

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.

Open Science and Massive MIMO

Open science is just science done right” is a quote from Prof. Jon Tennant in a recent podcast. He is referring to the movement away from the conventionally closed science community where you need to pay to gain access to research results and everyone treats data and simulation code as confidential. Since many funding agencies are requiring open access publishing and open data nowadays, we are definitely moving in the open science direction. But different research fields are at different positions on the scale between fully open and entirely closed science. The machine learning community has embraced open science to a large extent, maybe because the research requires common data sets. When the Nature Machine Intelligence journal was founded, more 3000 researchers signed a petition against its closed access and author fees and promised to not publish in that journal. However, research fields that for decades have been dominated by a few high-impact journals (such as Nature) have not reached as far.

IEEE is the main publisher of Massive MIMO research and has, fortunately, been quite liberal in terms of allowing for parallel publishing. At the time of writing this blog post, the IEEE policy is that an author is allowed to upload the accepted version of their paper on the personal website, the author’s employer’s website, and on arXiv.org. It is more questionable if it is allowed to upload papers in other popular repositories such as ResearchGate – can the ResearchGate profile pages count as personal websites?

It is we as researchers that need to take the steps towards open science. The publishers will only help us under the constraint that they can sustain their profits. For example, IEEE Access was created to have an open access alternative to the traditional IEEE journals, but its quality is no better than non-IEEE journals that have offered open access for a long time. I have published several papers in IEEE Access and although I’m sure that these papers are of good quality, I’ve been quite embarrassed by the poor review processes.

Personally, I try to make all my papers available on arXiv.org and also publish simulation code and data on my GitHub whenever I can, in an effort to support research reproducibility. My reasons for doing this are explained in the following video:

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.

SIR

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.

Outdoor Massive MIMO Demonstrations in Bristol

The University of Bristol continues to be one of the driving forces in demonstrating reciprocity-based Massive MIMO in time-division duplex. The two videos below are from an outdoor demo that was carried out in Bristol in March 2018.  A 128-antenna testbed with a rectangular array of 4 rows and 32 single-polarized antennas per row were used. The demo was carried out with a carrier frequency of 3.5 GHz and featured spatial multiplexing of video streaming to 12 users.

Prof. Mark Beach, who is leading the effort, believes that Massive MIMO in sub-6 GHz bands will be the key technology for serving the users in hotspots and sport arenas. Interestingly, Prof. Beach is also an author of one of the first paper on multiuser MIMO from 1990: “The performance enhancement of multibeam adaptive base-station antennas for cellular land mobile radio systems“.

When Are Downlink Pilots Needed?

Pilots are predefined reference signals that are transmitted to let the receiver estimate the channel. While many communication systems have pilot transmissions in both uplink and downlink, the canonical communication protocol in Massive MIMO only contains uplink pilots. In this blog post, I will explain when downlink pilots are needed and why we can omit them in Massive MIMO.

Consider the communication link between a single-antenna user and an M-antenna base station (BS). The channel vector $\mathbf{h} \in \mathbb{C}^M$ varies over time and frequency in a way that is often modeled as random fading. In each channel coherence blocks, the BS selects a precoding vector $\mathbf{w} \in \mathbb{C}^M$ and uses it for downlink transmission. The precoding reduces the multiantenna vector channel to an effective single-antenna scalar channel

    $$g = \mathbf{h}^{T} \mathbf{w}.$$

The receiving user does not need to know the full M-dimensional vectors $\mathbf{h}$ and $\mathbf{w}$. However, to decode the downlink data in a successful way, it needs to learn the complex scalar channel $g$. The difficulty in learning $g$ depends strongly on the mechanism of precoding selection. Two examples are considered below.

Codebook-based precoding

In this case, the BS tries out a set of different precoding vectors from a codebook (e.g., a grid of beams, as shown to the right) by sending one downlink pilot signal through each one of them. The user measures $g$ for each one of them and feeds back the index of the one that maximizes the channel gain |g|. The BS will then transmit data using that precoding vector. During the data transmission, $g \in \mathbb{C}$ can have any phase, but the user already knows the phase and can compensate for it in the decoding algorithm.

If multiple users are spatially multiplexed in the downlink, the BS might use another precoding vector than the one selected by the user. For example, regularized zero-forcing might be used to suppress interference. In that case, the magnitude |g| of the channel changes, but the phase remains the same. If phase-shift keying (PSK) is used for communication, such that no information is encoded in the signal amplitude, no estimation of |g| is needed for decoding (but it can help to reduce the error probability). If quadrature amplitude modulation (QAM) is used instead, the user needs to learn also |g| to decode the data. The unknown magnitude can be estimated blindly based on the received signals. Hence, no further pilot transmission is needed.

Reciprocity-based precoding

In this case, the user transmits a pilot signal in the uplink, which enables the BS to directly estimate the entire channel vector $\mathbf{h}$. For the sake of argument, suppose this estimation is perfect and that maximum ratio transmission with $\mathbf{w}=\mathbf{h}^*/\| \mathbf{h} \|$ is used for downlink data transmission. The effective channel gain will then be

    $$g = \mathbf{h}^{T} \frac{\mathbf{h}^*}{\| \mathbf{h} \|} = \| \mathbf{h} \|,$$

which is a positive scalar. Hence, the user only needs to learn the magnitude of  $g$ because the phase is always zero. Estimation of  |g| can be implemented without downlink pilots, either by relying on channel hardening or by blind estimation based on the received signals. The former only works well in Massive MIMO with very many antennas, while the latter can be done in any system (including codebook-based precoding).

Conclusion

We generally need to compensate for the channel’s phase-shift at some place in a wireless system. In codebook-based precoding, the compensation is done at the user-side, based on the received signals from the downlink pilots. This is the main approach in 4G systems, which is why downlink pilots are so commonly used. In contrast, when using reciprocity-based precoding, the phase-shifts are compensated for at the BS-side using the uplink pilots. In either case, explicit pilot signals are only needed in one direction: uplink or downlink. If the estimation is imperfect, there will be some remaining phase ambiguity, which can be estimated blindly since we know that it is small (i.e., of all possible phase-rotations that could have resulted in the received signal, the smallest one is most likely).

When we have access to TDD spectrum, we can choose between the two precoding methods mentioned above. The reciprocity-based approach is preferable in terms of less overhead signaling; one pilot per user instead of one per index in the codebook (the codebook size needs to grow with the number of antennas), and no feedback is needed. That is why this approach is taken in the canonical form of Massive MIMO.

Joint Massive MIMO Deployment for LTE and 5G

The American telecom operator Sprint is keen on mentioning Massive MIMO in the marketing of its 5G network deployments, as I wrote about a year ago. You can find their new video below and it gives new insights into the deployment strategy of their new 64-antenna BSs. Initially, the base station will be divided between LTE and 5G operation. According to CTO Dr. John Saw, the left half of the array will be used for LTE and the right half for 5G. This will lead to a 3 dB loss in SNR and also a reduced multiplexing capability, but I suppose that Sprint is only doing this temporarily until the number of 5G users is sufficiently large to motivate a 5G-only base station. Another thing that one can infer from the video is that the LTE/5G splitting is software-defined so physical changes to the base station hardware are not needed to change it.

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