Category Archives: Commentary

How to Normalize a Precoding Matrix?

The transmitted signal \mathbf{x} from an M-antenna base station can consist of multiple information signals that are transmitted using different precoding (e.g., different spatial directivity). When there are K unit-power data signals s_1,\ldots,s_K intended for K different users, the transmitted signal can be expressed as

(1)   \begin{equation*}\mathbf{x} = \sum_{i=1}^{K} \mathbf{w}_i s_i,\end{equation*}

where \mathbf{w}_1,\ldots,\mathbf{w}_K are the M-dimensional precoding vectors assigned to the different users. The direction of the vector \mathbf{w}_i determines the spatial directivity of the signal s_i, while the squared norm \|\mathbf{w}_i\|^2 determines the associated transmit power. Massive MIMO usually means that M\gg K.

When selecting the precoding vectors, we need to make sure that we are not using too much transmit power. If the maximum power is P and we define the M \times K precoding matrix

(2)   \begin{equation*} \mathbf{W} = [\mathbf{w}_1 \, \, \ldots \,\, \mathbf{w}_K],\end{equation*}

then we need to make sure that the squared Frobenius norm of \mathbf{W} equals the maximum transmit power:

(3)   \begin{equation*} \| \mathbf{W} \|_F^2 = P.\end{equation*}

In the Massive MIMO literature, there are two popular methods to achieve that: matrix normalization and vector normalization. The papers [Ref1], [Ref2] consider both methods, while other papers only consider one of the methods. The main idea is to start from an arbitrarily selected precoding matrix  \mathbf{F} = [\mathbf{f}_1 \, \, \ldots \,\, \mathbf{f}_K] and then adapt it to satisfy the power constraint in (3).

Matrix normalization: In this case, we take the matrix \mathbf{F} and scales all the entries with the same number, which is selected to satisfy (3). More precisely, we select

(4)   \begin{equation*}\mathbf{W} = \frac{\sqrt{P}}{\|\mathbf{F} \|_F} \mathbf{F}.\end{equation*}

Vector normalization: In this case, we first normalize each column in \mathbf{F} to have unit norm and then scale them all with \sqrt{P/K} to satisfy (3). More precisely, we select

(5)   \begin{equation*}\mathbf{W} = \sqrt{\frac{P}{K}} \left[ \frac{\mathbf{f}_1}{\| \mathbf{f}_1\|} \, \, \ldots \,\, \frac{\mathbf{f}_K}{\| \mathbf{f}_K\|} \right].\end{equation*}

Which of the two normalizations should be used?

This is a question that I receive now and then, so I wrote this blog post to answer it once and for all. My answer: none of them!

The problem with matrix normalization is that the method that was used to select \mathbf{F} will determine how the transmit power is allocated between the different signals/users. Hence, we are not in control of the power allocation and we cannot fairly compare different precoding schemes. For example, maximum-ratio (MR) allocates more power to users with strong channels than users with weak channels, while zero-forcing (ZF) does the opposite. Hence, if one tries to compare MR and ZF under matrix normalization, the different power allocations will strongly influence the results.

This issue is resolved by vector normalization. However, the problem with vector normalization is that all users are assigned the same amount of power, which is undesirable if some users have strong channels and others have weak channels. One should always make a conscious decision when it comes to power allocation between users.

What we should do instead is to select the precoding matrix as

(6)   \begin{equation*}\mathbf{W} =  \left[ \sqrt{p_1} \frac{\mathbf{f}_1}{\| \mathbf{f}_1\|} \, \, \ldots \,\, \sqrt{p_K} \frac{\mathbf{f}_K}{\| \mathbf{f}_K\|} \right],\end{equation*}

where p_1,\ldots,p_K are variables representing the power assigned to each of the users. These should be carefully selected to maximize some performance goals of the network, such as the sum rate, proportional fairness, or max-min fairness. In any case, the power allocation must be selected to satisfy the constraint

(7)   \begin{equation*} \| \mathbf{W} \|_F^2 =  \sum_{i=1}^{K} p_i = P.\end{equation*}

There are plenty of optimization algorithms that can be used for this purpose. You can find further details, examples, and references in Section 7.1 of my book Massive MIMO networks.

Channel Sparsity in Massive MIMO

Channel estimation is critical in Massive MIMO. One can use the basic least-squares (LS) channel estimator to learn the multi-antenna channel from pilot signals, but if one has prior information about the channel’s properties, that can be used to improve the estimation quality. For example, if one knows the average channel gain, the linear minimum mean-squared error (LMMSE) estimator can be used, as in most of the literature on Massive MIMO.

There are many attempts to exploit further channel properties, in particularly channel sparsity is commonly assumed in the academic literature. I have recently received several questions about this topic, so I will take the opportunity to give a detailed answer. In particular, this blog post discusses temporal and spatial sparsity.

Temporal sparsity

This means that the channel’s impulse response contains one or several pulses with zeros in between. These pulses could represent different paths, in a multipath environment, which are characterized by non-overlapping time delays. This does not happen in a rich scattering environment with many diffuse scatterers having overlapping delays, but it could happen in mmWave bands where there are only a few reflected paths.

If one knows that the channel has temporal sparsity, one can utilize such knowledge in the estimator to determine when the pulses arrive and what properties (e.g., phase and amplitude) each one has. However, several hardware-related conditions need to be satisfied. Firstly, the sampling rate must be sufficiently high so that the pulses can be temporally resolved without being smeared together by aliasing. Secondly, the receiver filter has an impulse response that spreads signals out over time, and this must not remove the sparsity.

Spatial sparsity

This means that the multipath channel between the transmitter and receiver only involves paths in a limited subset of all angular directions. If these directions are known a priori, it can be utilized in the channel estimation to only estimate the properties (e.g., phase and amplitude) in those directions. One way to determine the existence of spatial sparsity is by computing a spatial correlation matrix of the channel and analyze its eigenvalues. Each eigenvalue represents the average squared amplitude in one set of angular directions, thus spatial sparsity would lead to some of the eigenvalues being zero.

Just as for temporal sparsity, it is not necessary that spatial sparsity can be utilized even if it physically exists. The antenna array must be sufficiently large (in terms of aperture and number of antennas) to differentiate between directions with signals and directions without signals. If the angular distance between the channel paths is smaller than the beamwidth of the array, it will smear out the paths over many angles. The following example shows that Massive MIMO is not a guarantee for utilizing spatial sparsity.

The figure below considers a 64-antenna scenario where the received signal contains only three paths, having azimuth angles -20°, +30° and +40° and a common elevation angle of 0°. If the 64 antennas are vertically stacked (denoted 1 x 64), the signal gain seems to be the same from all azimuth directions, so the sparsity cannot be observed at all. If the 64 antennas are horizontally stacked (denoted 64 x 1), the signal gain has distinct peaks at the angles of the three paths, but there are also ripples that could have hidden other paths. A more common 64-antenna configuration is a 8 x 8 planar array, for which only two peaks are visible. The paths 30° and 40° are lumped together due to the limited resolution of the array.

Figure: The received signal gain that is observed from different azimuth angles, using different array geometries. The true signal only contains three paths, which are coming from the azimuth angles -20°, +30° and +40°.

In addition to have a sufficiently high spatial resolution, a phase-calibrated array might be needed to make use of sparsity, since random phase differences between the antennas could destroy the structure.

Do we need sparsity?

There is no doubt that temporal and spatial sparsity exist, but not every channel will have it. Moreover, the transceiver hardware will destroy the sparsity unless a series of conditions are satisfied. That is why one should not build a wireless technology that requires channel sparsity because then it might not function properly for many of the users. Sparsity is rather something to utilize to improve the channel estimation in certain special cases.

TDD-reciprocity based Massive MIMO, as proposed by Marzetta and further considered in my book Massive MIMO networks, does not require channel sparsity. However, sparsity can be utilized as an add-on when available. In contrast, there are many FDD-based frameworks that require channel sparsity to function properly.

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

Massive MIMO Enables Fixed Wireless Access

The largest performance gains from Massive MIMO are achieved when the technology is used for spatial multiplexing of many users. These gains can only be harnessed when there actually are many users that ask for data services simultaneously. I sometimes hear the following negative comments about Massive MIMO:

  1. The data traffic is so bursty that there seldom are more than one or two users that ask for data simultaneously.
  2. When there are multiple users, the uplink SNR is often too poor to get the high quality channel state information that is needed to truly benefit from spatial multiplexing.

These points might indeed be true in current cellular networks, but I believe the situation will change in the future. In particular, the new fixed wireless access services require that the network can simultaneously deliver high-rate services to many customers. The business case for these service rely strongly on Massive MIMO and spatial multiplexing, so that one base station site can guarantee a certain data rate to as many customers as possible (just as fiber and cable connections can). The fixed installation of the customer equipment means that channel state information is much easier to acquire (due to better channel conditions, higher transmit power, and absence of mobility). The following video from Ericsson touches upon some of these aspects:

Reconfigurable Reflectarrays and Metasurfaces

In the research on Beyond Massive MIMO systems, a number of new terminologies have been introduced with names such as:

  1. Reconfigurable reflectarrays;
  2. Software-controlled metasurfaces;
  3. Intelligent reflective surfaces.

These are basically the same things (and there are many variations on the three names), which is important to recognize so that the research community can analyze them in a joint manner.

Background

The main concept has its origin in reflectarray antennas, which is a class of directive antennas that behave a bit like parabolic reflectors but can be deployed on a flat surface, such as a wall. More precisely, a reflectarray antenna takes an incoming signal wave and reflects it into a predetermined spatial direction, as illustrated in the following figure:

Figure 1: A reflectarray antenna (also known as metasurface and intelligent reflective surface) takes an incoming wave and reflects it as a beam in a particular direction (or towards a spatial point).

Instead of relying on the physical shape of the antenna to determine the reflective properties (as is the case for parabolic reflectors), a reflectarray consists of many reflective elements that impose element-specific time delays to their reflected signals. These elements are illustrated by the dots on the surface in Figure 1. In this way, the reflected wave is beamformed and the reflectarray can be viewed as a passive MIMO array. The word passive refers to the fact that the radio signal is not generated in the array but elsewhere. Since a given time delay corresponds to a different phase shift depending on the signal’s frequency, reflectarrays are primarily useful for reflecting narrowband signals in a single direction.

Reconfigurability

Reconfigurable reflectarrays can change the time delays of each element to steer the reflected beam in different directions at different points in time. The research on this topic has been going on for decades; the book “Reflectarray Antennas: Analysis, Design, Fabrication, and Measurement” from 2014 by Shaker et al. describes many different implementation concepts and experiments.

Recently, there is a growing interest in reconfigurable reflectarrays from the communication theoretic and signal processing community. This is demonstrated by a series of new overview papers that focus on applications rather than hardware implementations:

The elements in the reflecting surface in Figure 1 are called meta-atoms or reflective elements in these overview papers. The size of a meta-atom/element is smaller than the wavelength, just as for conventional low-gain antennas. In simple words, we can view an element as an antenna that captures a radio signal, keeps it inside itself for a short time period to create a time-delay, and then radiates the signal again. One can thus view it as a relay with a delayand-forward protocol. Even if the signals are not amplified by a reconfigurable reflectarray, there is a non-negligible energy consumption related to the control protocols and the reconfigurability of the elements.

It is important to distinguish between reflecting surfaces and the concept of large intelligent surfaces with active radio transmitters/receivers, which was proposed for data transmission and positioning by Hu, Rusek, and Edfors. This is basically an active MIMO array with densely deployed antennas.

What are the prospects of the technology?

The recent overview papers describe a number of potential use cases for reconfigurable reflectarrays (metasurfaces) in wireless networks, such as range extension, improved physical layer security, wireless power transfer, and spatial modulation. From a conceptual perspective, it is indeed an exciting prospect to build future networks where not only the transmitter and receiver algorithms can be optimized, but the propagation environment can be partially controlled.

However, the research on this topic is still in its infancy. It is of paramount importance to demonstrate practically important use cases where reconfigurable reflectarrays are fundamentally better than existing methods. If it should be economically feasible to turn the technology into a commercial reality, we should not look for use cases where a 10% gain can be achieved but rather a 10x or 100x gain. This is what Marzetta demonstrated with Massive MIMO and this is also what it can deliver in 5G.

I haven’t seen any convincing demonstrations of such use cases of reflectarray antennas (metasurfaces) thus far. On the contrary, my new paper “Intelligent Reflecting Surface vs. Decode-and-Forward: How Large Surfaces Are Needed to Beat Relaying?” shows that the new technology can indeed provide range extension, but a basic single-antenna decode-and-forward relay can outperform it unless the surface is very large. There is much left to do on this topic!

A case against Massive MIMO?

I had an interesting conversation with a respected colleague who expressed some significant reservations against massive MIMO. Let’s dissect the arguments. 


The first argument against massive MIMO was that most traffic is indoors, and that deployment of large arrays indoors is impractical and that outdoor-to-indoor coverage through massive MIMO is undesirable (or technically infeasible). I think the obvious counterargument here is that before anything else, the main selling argument for massive MIMO is not indoor service provision but outdoor macrocell coverage: the ability of TDD/reciprocity based beamforming to handle high mobility, and efficiently suppress interference thus provide cell-edge coverage. (The notion of a “cell-edge” user should be broadly interpreted: anyone having poor nominal signal-to-interference-and-noise ratio, before the MIMO processing kicks in.) But nothing prevents massive MIMO from being installed indoors, if capacity requirements are so high that conventional small cell or WiFi technology cannot handle the load. Antennas could be integrated into walls, ceilings, window panes, furniture or even pieces of art. For construction of new buildings, prefabricated building blocks are often used and antennas could be integrated into these already at their production. Nothing prevents the integration of thousands of antennas into natural objects in a large room.

Outdoor-to-indoor coverage doesn’t work? Importantly, current systems provide outdoor-to-indoor coverage already, and there is no reason Massive MIMO would not do the same (to the contrary, adding base station antennas is always beneficial for performance!). But yet the ideal deployment scenario of massive MIMO is probably not outdoor-to-indoor so this seems like a valid point, partly. The arguments against the outdoor-to-indoor are that modern energy-saving windows have a coating that takes out 20 dB, at least, of the link budget. In addition, small angular spreads when all signals have to pass through windows (maybe except for in wooden buildings) may reduce the rank of the channel so much that not much multiplexing to indoor users is possible. This is mostly speculation and not sure whether experiments are available to confirm, or refute it.

Let’s move on to the second argument. Here the theme is that as systems use larger and larger bandwidths, but can’t increase radiated power, the maximum transmission distance shrinks (because the SNR is inversely proportional to the bandwidth). Hence, the cells have to get smaller and smaller, and eventually, there will be so few users per cell that the aggressive spatial multiplexing on which massive MIMO relies becomes useless – as there is only a single user to multiplex. This argument may be partly valid at least given the traffic situation in current networks. But we do not know what future requirements will be. New applications may change the demand for traffic entirely: augmented or virtual reality, large-scale communication with drones and robots, or other use cases that we cannot even envision today.

It is also not too obvious that with massive MIMO, the larger bandwidths are really required. Spatial multiplexing to 20 terminals improves the bandwidth efficiency 20 times compared to conventional technology. So instead of 20 times more bandwidth, one may use 20 times more antennas. Significant multiplexing gains are not only proven theoretically but have been demonstrated in commercial field trials. It is argued sometimes that traffic is bursty so that these multiplexing gains cannot materialize in practice, but this is partly a misconception and partly a consequence of defect higher-layer designs (most importantly TCP/IP) and vested interests in these flawed designs. For example, for the services that constitute most of the raw bits, especially video streaming, there is no good reason to use TCP/IP at all. Hopefully, once the enormous potential of massive MIMO physical layer technology becomes more widely known and understood, the market forces will push a re-design of higher-layer and application protocols so that they can maximally benefit from the massive MIMO physical layer.  Does this entail a complete re-design of the Internet? No, probably not, but buffers have to be installed and parts of the link layer should be revamped to maximally use the “wires in the air”, ideally suited for aggressive multiplexing of circuit-switched data, that massive MIMO offers.

Reproducible Research: Best Practices and Potential Misuse

In the May issue of the IEEE Signal Processing Magazine, you can read the most personal article that I have written so far. It is entitled “Reproducible Research: Best Practices and Potential Misuse” and is available on IEEEXplore and ArXiv.org. In this article, I share my experiences of making simulation code openly available.

I started with doing that in 2012, the year after I received my Ph.D. degree. It was very uncommon to make code publicly available in the MIMO field at that time, but I think we are definitely moving in the right direction. For example, the hype around machine learning has encouraged people to create open datasets and to share Python code. The Machine Learning for Communications Emerging Technologies Initiative by IEEE ComSoc has recently created a website with simulation code, which contains tens of contributions from many different authors. A few of them are related to Massive MIMO!

One important side-effect of making my code available is that I force myself to write the code as cleanly as possible. This is incredibly useful if you are going to reuse parts of the code in future publications. For example, when I wrote the paper “Making Cell-Free Massive MIMO Competitive With MMSE Processing and Centralized Implementation” earlier this year, I could reuse a lot of the code from my book Massive MIMO Networks. I was amazed by how little time it actually took to generate the simulations for that paper. The simulation setup is entirely different, but I could anyway reuse many of the signal processing and optimization algorithms that I had implemented earlier.

When Will Hybrid Beamforming Disappear?

There has been a lot of fuss about hybrid analog-digital beamforming in the development of 5G. Strangely, it is not because of this technology’s merits but rather due to general disbelief in the telecom industry’s ability to build fully digital transceivers in frequency bands above 6 GHz. I find this rather odd; we are living in a society that becomes increasingly digitalized, with everything changing from being analog to digital. Why would the wireless technology suddenly move in the opposite direction?

When Marzetta published his seminal Massive MIMO paper in 2010, the idea of having an array with a hundred or more fully digital antennas was considered science fiction, or at least prohibitively costly and power consuming. Today, we know that Massive MIMO is actually a pre-5G technology, with 64-antenna systems already deployed in LTE systems operating below 6 GHz. These antenna panels are very commercially competitive; 95% of the base stations that Huawei are currently selling have at least 32 antennas. The fast technological development demonstrates that the initial skepticism against Massive MIMO was based on misconceptions rather than fundamental facts.

In the same way, there is nothing fundamental that prevents the development of fully digital transceivers in mmWave bands, but it is only a matter of time before such transceivers are developed and will dominate the market. With digital beamforming, we can get rid of the complicated beam-searching and beam-tracking algorithms that have been developed over the past five years and achieve a simpler and more reliable system operation, particularly, using TDD operation and reciprocity-based beamforming.

Figure 1: Photo of the experimental equipment with 24 digital transceivers that was used by NEC. It uses 300 MHz of bandwidth in the 28 GHz band.

I didn’t jump onto the hybrid beamforming research train since it already had many passengers and I thought that this research topic would become irrelevant after 5-10 years. But I was wrong – it now seems that the digital solutions will be released much earlier than I thought. At the 2018 European Microwave Conference, NEC Cooperation presented an experimental verification of an active antenna system (AAS) for the 28 GHz band with 24 fully digital transceiver chains. The design is modular and consists of 24 horizontally stacked antennas, which means that the same design could be used for even larger arrays.

Tomoya Kaneko, Chief Advanced Technologist for RF Technologies Development at NEC, told me that they target to release a fully digital AAS in just a few years. So maybe hybrid analog-digital beamforming will be replaced by digital beamforming already in the beginning of the 5G mmWave deployments?

Figure 2: Illustration of what is found inside the AAS box in Figure 1. There are 12 horizontal cards, with two antennas and transceivers each. The dimensions are 308 mm x 199 mm.

That said, I think that the hybrid beamforming algorithms will have new roles to play in the future. The first generations of new communication systems might reach faster to the market by using a hybrid analog-digital architecture, which require hybrid beamforming, than waiting for the fully digital implementation to be finalized. This could, for example, be the case for holographic beamforming or MIMO systems operating in the sub-THz bands. There will also remain to exist non-mobile point-to-point communication systems with line-of-sight channels (e.g., satellite communications) where analog solutions are quite enough to achieve all the necessary performance gains that MIMO can provide.