Category Archives: 5G

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.

5G Dilemma: Higher throughput but also higher energy consumption

I was recently interviewed by IEEE Spectrum for the article: The 5G Dilemma: More Base Stations, More Antennas—Less Energy?

Since 5G is being built in addition to the existing cellular networks, the energy consumption of the cellular network infrastructure as a whole will certainly increase when 5G is introduced. It is too early to say how much the energy consumption will grow, but even if the implementation would be vastly more energy efficient than before, we need to spend more energy to gain more in network capacity.

It is important to keep in mind that having a high energy consumption is not necessarily a problem. The real issue is that the power plants that power the cellular networks are mainly extracting energy from non-renewable sources that have a negative impact on the environment. It is the same issue that electric cars have – these are only environmentally friendly if they are charged with energy from environmentally friendly power plants. Hence, we need to keep the energy consumption of cellular networks down until cleaner power plants are widely used.

If you want to learn more about energy efficiency after reading the article in IEEE Spectrum, I recommend the following overview video (you find all the technical details in Section 5 in my book Massive MIMO networks):

Are Pilots and Data Transmitted With the Same Power?

The user terminals in reciprocity-based Massive MIMO transmit two types of uplink signals: data and pilots (a.k.a. reference signals). A terminal can potentially transmit these signals using different power levels. In the book Fundamentals of Massive MIMO, the pilots are always sent with maximum power, while the data is sent with a user-specific power level that is optimized to deliver a certain per-user performance. In the book Massive MIMO networks, the uplink power levels are also optimized, but under another assumption: each user must assign the same power to pilots and data.

Moreover, there is a series of research papers (e.g., Ref1, Ref2, Ref3) that treat the pilot and data powers as two separate optimization variables that can be optimized with respect to some performance metric, under a constraint on the total energy budget per transmission/coherence block. This gives the flexibility to “move” power from data to pilots for users at the cell edge, to improve the channel state information that the base station acquires and thereby the array gain that it obtains when decoding the data signals received over the antennas.

In some cases, it is theoretically preferable to assign, for example, 20 dB higher power to pilots than to data. But does that make practical sense, bearing in mind that non-linear amplifiers are used and the peak-to-average-power ratio (PAPR) is then a concern? The answer depends on how the pilots and data are allocated over the time-frequency grid. In OFDM systems, which have an inherently high PAPR, it is discouraged to have large power differences between OFDM symbols (i.e., consecutive symbols in the time domain) since this will further increase the PAPR. However, it is perfectly fine to assign the power in an unequal manner over the subcarriers.

In the OFDM literature, there are two elementary ways to allocate pilots: block and comb type arrangements. These are illustrated in the figure below and some early references on the topic are Ref4, Ref5, Ref6.

(a): In the block type arrangement, at a given OFDM symbol time, all subcarriers either contain pilots or data. It is then preferable for a user terminal to use the same transmit power for pilots and data, to not get a prohibitively high PAPR. This is consistent with the assumptions made in the book Massive MIMO networks.

(b): In the comb type arrangement, some subcarriers always contain pilots and other subcarriers always contain data. It is then possible to assign different power to pilots and data at a user terminal. The power can be moved from pilot subcarriers to data subcarriers or vice versa, without a major impact on the PAPR. This approach enables the type of unequal pilot and data power allocations considered in Fundamentals of Massive MIMO or research papers that optimize the pilot and data powers under a total energy budget per coherence block.

The downlink in LTE uses a variation of the two elementary pilot arrangements, as illustrated in (c). It is easiest described as a comb type arrangement where some pilots are omitted and replaced with data. The number of omitted pilots depend on how many antenna ports are used; the more antennas, the more similar the pilot arrangement becomes to the comb type. Hence, unequal pilot and power allocation is possible in LTE but maybe not as easy to implement as described above. 5G has a more flexible frame structure but supports the same arrangements as LTE.

In summary, uplink pilots and data can be transmitted at different power levels, and this flexibility can be utilized to improve the performance in Massive MIMO. It does, however, require that the pilots are arranged in practically suitable ways, such as the comb type arrangement.

UAVs Prepare for Take-off With Massive MIMO

Drones could shape the future of technology, especially if provided with reliable command and control (C&C) channels for safe and autonomous flying, and high-throughput links for multi-purpose live video streaming. Some months ago, Prabhu Chandhar’s guest post discussed the advantages of using massive MIMO to serve drone – or unmanned aerial vehicle (UAV) – users. More recently, our Paper 1 and Paper 2 have quantified such advantages under the realistic network conditions specified by the 3GPP. While demonstrating that massive MIMO is instrumental in enabling support for UAV users, our works also show that merely upgrading existing base stations (BS) with massive MIMO might not be enough to provide a reliable service at all UAV flying heights. Indeed, hardware solutions need to be complemented with signal processing enhancements through all communications phases, namely, 1) UAV cell selection and association, 2) downlink BS-to-UAV transmissions, and 3) uplink UAV-to-BS transmissions. These are outlined below.

1. UAV cell selection and association

As depicted in Figure 1(a), most existing cellular BSs create a fixed beampattern towards the ground. Thanks to this, ground users tend to perceive a strong signal strength from nearby BSs, which they use for connecting to the network. Instead, aerial users such as the red drone in Figure 1(a) only receive weak sidelobe-generated signals from a nearby BS when flying above it. This results in a deployment planning issue as illustrated in Figure 1(b), where due to the radiation of a strong sidelobe, the tri-sector BSs located in the origin can be the preferred server for far-flung UAVs (red spots). Consequently, these UAVs might experience strong interference, since they perceive signals from a multiplicity of BSs with similar power.

Figure 1. (a) Illustration of a downtilted cellular BS and its beampattern: low (blue) UAV receives strong main lobe signals, whereas high (red) drone only receives weak sidelobe-generated signals. (b) 150-meter-high UAVs (red dots) associated with a three-cell BS site located at the origin and pointing at 30°, 150°, and 270°. The three BSs of each cellular site (orange squares) generate ground cells represented by hexagons.

On the other hand, thanks to their capability of beamforming the synchronization signals used for user association, massive MIMO systems ensure that aerial users generally connect to a nearby BS. This optimized association enhances the robustness of the mobility procedures, as well as the downlink and uplink data phases.

2. Downlink BS-to-UAV transmissions

During the downlink data phase, UAV users are very sensitive to the strong inter-cell interference generated from a plurality of BSs, which are likely to be in line-of-sight. This may result in performance degradation, preventing UAVs from receiving critical C&C information, which has an approximate rate requirement of 60-100 kbps. Indeed, Figure 2 shows how conventional cellular networks (‘SU’) can only guarantee 100 kbps to a mere 6% of the UAVs flying at 150 meters. A conventional massive MIMO system (‘mMIMO’) enhances the data rates, albeit only 33% of the UAVs reach 100 kbps when they fly at 300 meters. This is due to a well-known effect: pilot contamination. Such an effect is particularly severe in scenarios with UAV users, since they can create strong uplink interference to many line-of-sight BSs simultaneously. In contrast, the pilot contamination decays much faster with distance for ground UEs.

In a nutshell, Figure 2 tells us that complementing conventional massive MIMO with explicit inter-cell interference suppression (‘mMIMO w/ nulls’) is essential when supporting high UAVs. In a ‘mMIMO w/ nulls’ system, BSs incorporate additional signal processing features that enable them to perform a twofold task. First, leveraging channel directionality, BSs can spatially separate non-orthogonal pilots transmitted by different UAVs. Second, by dedicating a certain number of spatial degrees of freedom to place radiation nulls, BSs can mitigate interference on the directions corresponding to users in other cells that are most vulnerable to the BS’s interference. Indeed, these additional capabilities dramatically increase the percentage of UAVs that meet the 100 kbps requirement when these are flying at 300 m, from 33% (‘mMIMO’) to a whopping 98% (‘mMIMO w/ nulls’).

Figure 2. Percentage of UAVs with a downlink C&C rate above 100 kbps for three representative UAV heights. ‘SU’ denotes a conventional cellular network with a single antenna port, ‘mMIMO’ represents a system with 8×8 dual-polarized antenna arrays and 128 antenna ports, and ‘mMIMO w/ nulls’ complements the latter with additional space-domain inter-cell interference suppression techniques.

3. Uplink UAV-to-BS transmissions

Unlike the downlink, where UAVs should be protected to prevent a significant performance degradation, it is the ground users who we should care about in the uplink. This is because line-of-sight UAVs can generate strong interference towards many BSs, therefore overwhelming the weaker signals transmitted by non-line-of-sight ground users. The consequences of such a phenomenon are illustrated in Figure 3, where the uplink rates of ground users plummet as the number of UAVs increases.

Again, ‘mMIMO w/nulls’ – incorporating additional space-domain inter-cell interference suppression capabilities – can solve the above issue and guarantee a better performance for legacy ground users.

Figure 3. Average uplink rates of ground users when the number of UAVs per cell grows. ‘SU’ denotes a conventional cellular network with a single antenna port, ‘mMIMO’ represents a system with 8×8 antenna dual-polarized antenna arrays and 128 antenna ports, and ‘mMIMO w/ nulls’ complements the latter with additional space-domain inter-cell interference suppression techniques.

Overall, the efforts towards realizing aerial wireless networks are just commencing, and massive MIMO will likely play a key role. In the exciting era of fly-and-connect, we must revisit our understanding of cellular networks and develop novel architectures and techniques, catering not only for roads and buildings, but also for the sky.