The performance limitation of any wireless network will always be at the physical layer, because, fundamentally, the amount of information that can be transferred between two locations is limited by the availability of spectrum, the laws of electromagnetic propagation, and the principles of information theory.

There are three basic ways in which the efficiency of a wireless network may be improved: (i) deploying access points more densely; (ii) using more spectrum; and (iii) increasing the spectral efficiency, that is, the number of bits that can be conveyed per second in each unit of bandwidth. While future wireless systems and standards are likely to use an ever-increasing access point density and use new spectral bands, the need for maximizing the spectral efficiency in a given band is never going to vanish.

The use of multiple antennas, also known as multiple-input, multiple-output (MIMO) technology, is the only viable approach for substantial improvement of spectral efficiency. While mostly developed during the last two decades, it is noteworthy that a basic idea behind MIMO is almost a century old: in [1], directional beamforming using an antenna array was suggested to permit more aggressive frequency reuse of scarce spectrum – in this case, very low frequency – for transoceanic communication.

MIMO technology is logically classified into one of three categories, whose development occurred during roughly disjoint epochs: Point-to-Point MIMO, Multiuser MIMO, and Massive MIMO. This book is about Massive MIMO, which arguably will be the ultimate embodiment of MIMO technology. The following sections explain these incarnations of MIMO and their important differences. This treatment is intended to be a quick overview, and subsequent chapters will expand upon the concepts introduced here.

Point-to-Point MIMO

Point-to-Point MIMO emerged in the late 1990s [2–11] and represents the simplest form of MIMO: a base station equipped with an antenna array serves a terminal equipped with an antenna array; see Figure 1.1. Different terminals are orthogonally multiplexed, for example via a combination of time- and frequency-division multiplexing. In what follows, we summarize some basic facts about Point-to-Point MIMO. More details, along with derivations of all formulas given here, are provided in Section C.3.

This chapter treats the single-cell scenario of Section 2.2.1, where a base station uses an array of M antennas to communicate simultaneously with K active terminals. A great deal of Massive MIMO phenomenology surfaces in this scenario: the effects of noise, channel non-orthogonality, and channel estimation errors; the details of multiplexing and de-multiplexing; near/far effects; and the significance of power control.

Throughout the chapter, we consider only zero-forcing and maximum-ratio processing. While there are somewhat better performing alternatives: MMSE on the uplink, and suitably optimized regularized zero-forcing on the downlink [30], there are no closed-form non-asymptotic expressions available for their performance. Moreover, zero-forcing and maximum-ratio themselves tend to be optimal under high- and low-SINR conditions respectively.

Uplink Pilots and Channel Estimation

Learning the channel at the base station is a critical operation. As we have seen, a wideband channel can be decomposed into coherence intervals of duration Tc seconds and bandwidth Bc Hz. Every such interval offers τc = BcTc independent uses of a frequency-flat channel as modeled in Section 2.2.1. Figure 2.3(b) illustrates the three activities that occupy each coherence interval: uplink data transmission, uplink pilot transmission, and downlink data transmission. In every coherence interval, the terminals use τp of the τc available samples to transmit pilots that are known at both ends of the link, and from which the base station estimates the channels.

Orthogonal Pilots

Each coherence intervalmust host K pilot waveforms, and in order for them not to interfere, they have to be mutually orthogonal. Henceforth, we assume that the terminals are assigned mutually orthogonal pilot sequences of length τp, where τc ≥ τp ≥ K. Any set of orthogonal pilots with the same energies yield the same performance. The significance of τp is to quantify how much energy each terminal spends on pilots in each coherence interval. In principle, any τp samples in the uplink part of the coherence interval can be used for pilots. In practice, transmitters are typically peak-power limited, so constant-modulus signals, such as orthogonal sinewaves, make ideal pilots.

The case studies in this chapter are of two types: first, a single isolated cell for rural broadband fixed access (Section 6.1); second, multi-cell deployments for dense urban and suburban mobile access (Section 6.3). We model all important physical phenomena, including randomness of terminal locations, path loss, and shadow fading, and use the capacity expressions derived in Chapters 3–5. These expressions account for the effects of intra- and inter-cell interference, channel estimation errors, and the cost of pilot transmission. While all capacity bounds in Chapters 3 and 4 are rigorous and all algorithms in Chapter 5 provide exact solutions to precise optimization problems, in the multi-cell design examples some heuristic algorithms are needed for terminal-to-base station assignment, pilot assignment, and power control; we describe these algorithms in Section 6.2.

Tables 6.1–6.3 summarize all parameters used in the three design examples, and the resulting performance. The numbers given in Tables 6.2 and 6.3 represent 95% likely values (over the randomness associated with the large-scale fading), for the coverage probabilities specified in Table 6.1. Specifically, in the rural scenario, all 3000 homes obtain 20 Mb/s in the downlink and 10 Mb/s in the uplink, i.e., the coverage probability is 100%; Tables 6.2 and 6.3 list the numbers of antennas that are needed, with 95% probability, to offer this service. In the mobile access scenario, the coverage probability is 95% – that is, 5% of all terminals are dropped from service. The throughput numbers in Tables 6.2 and 6.3 represent the 95% likely throughput for the terminals that remain in service. For the mobile access scenarios, therefore, the overall reliability is equal to 0.95 × 0.95.

Single-Cell Deployment Example: Fixed Broadband Access in Rural Area

A single Massive MIMO base station serves 3000 homes in a rural area with data rates comparable to cable- or fiber-based access. We assume an isolated cell, for example a rural town, which is therefore free from inter-cell interference.

The capacity bounds derived in Chapter 3 were chosen for their utility and tractability. However, these are not the only possible bounds. In this appendix, we derive alternative bounds that, in some cases, may be tighter than their counterparts in Chapter 3.

Uplink Zero-Forcing

In the uplink, the variances of the effective noises in (3.23) and (3.32) depend on the channel estimate, which itself is known by the base station. As a result, the corresponding capacity bounds entail inconvenient expectations of logarithms of stochastic quantities. In Chapter 3, we circumvented this difficulty via the “use and forgetCSI” trick,which converted the problem into one that was amenable to the bounding technique of Section 2.3.2. We point out that the same issue does not occur in the case of the downlink, where the terminal is ignorant of the channel estimate.

Capacity Lower Bound via Jensen's Inequality

Jensen's inequality – see Section C.1 – circumvents the difficulty of taking the expectation of a logarithm by bringing the expectation inside the logarithm. We apply Jensen in two ways: first to produce a lower bound on the expectation, second to produce an upper bound.

The application of Jensen's inequality to (3.26), in the form (C.3), yields

where we have used the identity (B.1). This argument merely reproduces the earlier “use and forget CSI” bound in (3.28), but it is interesting that radically different approaches yield the same bound.

Tightness of the Lower Bound (3.28)

We now show that the expression (3.28) is typically an excellent approximation to (3.26), by finding an upper bound on the expectation of the logarithm (3.26). To this end, we apply Jensen's inequality in the form (C.2) to (3.26),

where we have also used the identity (B.2).

We now extend the single-cell performance analysis of Chapter 3 to the multi-cell setting as modeled in Section 2.2.2. The activities in the various cells occur synchronously, but otherwise there is no cell-to-cell cooperation, with the exception of pilot assignment and possibly power control. Each base station serves its own terminals. Data transmission activities are identical to their single-cell counterparts. The channel estimation, however, has to account for reuse of pilots in other cells.

The culmination of this chapter is Table 4.1 that summarizes effective SINRs analogous to Table 3.1 in the single-cell case. Compared with the single-cell SINRs, the multi-cell SINRs contain additional effective noise terms that correspond to inter-cell interference of two types: non-coherent interference that is independent of the number of base station antennas, and coherent interference that scales with the number of base station antennas.

Throughout this chapter, we denote the home cell by the index l. Although the terminals within each cell have mutually orthogonal pilots, some reuse of pilots from cell to cell is permitted. The assumption is that for any two distinct cells, the pilot sequences are either perfectly orthogonal from cell to cell, or are perfectly replicated. Cells that use the same pilots as the home cell cause pilot contamination. We call these cells contaminating cells and denote the set of their indices by Pl, where by definition Pl also includes the home cell l. For all l ʹ ∈ Pl, the kth terminal in the lʹth cell is assigned the same kth pilot sequence.

We use the capacity bounding techniques of Chapter 3, but in our treatments of the uplink we immediately adopt the “use and forget” technique, thereby skipping the forms of capacity bounds that require taking expectations of logarithms.

Uplink Pilots and Channel Estimation

During the training phase, the terminals in each cell transmit pilot sequences, as in (3.2). Upon reception of the pilots, the base station performs a de-spreading operation. Because of pilot reuse, the resulting signal is a linear combination of channel matrices from all cells that share the same pilot sequences.