Frequency-selective channels are caused by delay spread in the channel. When delay spread is introduced into the channel model, intersymbol interference is observed at the receiver. Intersymbol interference denotes the effect of the channel introducing contamination to the current sample from previous samples. If the communication system does not compensate for this effect, the performance of the link can be degraded significantly. The adverse effects of delay spread can be even more dramatic if a strong interferer that is observed by a multiple-antenna receiver has a channel that is frequency selective. For example, consider a single-antenna interferer. Without delay spread, a capable multiple-antenna receiver can mitigate the effects of the interference. In channels with significant delay spread, the rank of the interference spatial receive covariance matrix can grow from rank-1 to full rank, because each receive symbol can contain contributions from multiple transmit symbols at various relative delays propagation through channels that cause independent spatial responses. Without changing the processing approach, this full-rank interference covariance matrix can overwhelm the communications link.

The frequency-selective channel can be represented in the frequency domain by employing a channel representation with coefficients at various frequencies, or in the time domain by employing a channel representation with coefficients at various delays (delay taps). To complicate the channel problem, if the channel is not static because of the motion of the transmitter, receiver, or scatterers, then compensating for delay spread can be more difficult.

A receiver cannot decode a signal if it is not aware of the existence of the transmitted signal. Furthermore, if a transmitter and receiver are not aligned in time and frequency, then the transmitted signal will not make any sense to the receiver. Consequently, in order to establish a wireless communication link, the receiver must find or acquire the transmitted signal, and some sort of synchronization in time and frequency between the transmitter and receiver must occur. In this chapter, the process of acquisition and synchronization is simply denoted synchronization. In order for two nodes to be synchronized, they must agree on both the carrier frequency and timing. In this discussion, it is assumed that any frequency errors are small enough that frequency synchronization can be achieved after temporal synchronization. Extensions to the discussions provided here would enable joint temporal and spectral synchronization. In situations in which coherence is required for long durations, frequency is of greater importance [244], and the techniques discussed in this chapter need to be modified to address this sensitivity.

The performance of synchronization or acquisition techniques is often characterized in terms of probability of detecting the signal of interest given that it is there versus the probability of falsely “detecting” a signal (a false alarm) given that the signal is absent. The function relating these two probabilities for some test statistic is often denoted the receiver operating characteristic (ROC) curve that is discussed in Section 3.7.3.

The need for medium-access control

In wireless and certain wired networks, multiple users share the same physical medium. Data communication rates in networks can often be improved by using medium-access control (MAC) protocols, whereby multiple users share the medium in a controlled manner such that the adverse effects of their interfering signals is reduced. A general treatment of this topic can be found in Reference [21]. The main reason for improved data rates with medium-access control is that communication in noise typically tends to be at much higher data rates than communication in interference if the data rates are a function of the signal-to-interference-plus-noise ratio (SINR).

Earlier in the book, we introduced multiple-access schemes such as frequency-division-multiple access (FDMA), time-division-multiple access (TDMA), code-division-multiple access (CDMA) and space-division-multiple access (SDMA). Each of these multiple-access schemes attempts to reduce interference by ensuring that multiple links operate in orthogonal or approximately orthogonal spaces, such as by time or frequency division. We did not, however, describe in much detail how the assignments of frequency bands, time slots, or spatial dimensions to users are made.

In cellular telephone networks, the assignments of links to time slots, frequency bands, or codes can be made by the base station, which controls the behavior of the mobile units in its own cell. The network topology (where there is a central control node) and the connection-oriented nature of telephone links where links stay operational for long periods (seconds or minutes) make this an attractive approach.

As is true for single-input single-output (SISO) communication links, there are many approaches for coding multiple-input multiple-output (MIMO) systems in an attempt to approach the channel capacity. Many of the space-time coding approaches have analogs in the SISO regime. The multiple antennas of MIMO systems enable spatial diversity, increased data rates, and interference suppression. In general, there are trades in these characteristics, depending upon the coding and receiver approach. One of the most important trade-offs in this regard is the trade-off between data rate and diversity whereby the data communication rate is reduced to improve probability of error or outage. That is to say, a fraction of the data rate is sacrificed to improve robustness.

There have been numerous contributions in the field of space-time coding. The major contributions include References [8, 99, 307, 305, 306, 361, 362, 314, 57, 58, 292, 86, 207, 166, 119, 87, 222, 270, 42, 43, 269, 226, 196]. Of particular note are Reference [8] which introduced what is now known as the Alamouti code, a simple and elegant space-time code for systems with two transmitter antennas, Reference [307] which introduced systematic methods to evaluate space-time codes, and Reference [361] which analyzed the fundamental trade-offs between diversity and rate in multiantenna systems.

Introduction

In this chapter, we analyze the performance of networks with two multiantenna transmit nodes and two multiantenna receive nodes. The canonical 2 × 2 network is illustrated in Figure 12.1. Transmitter 1, equipped with nt1 antennas, wishes to communicate with receiver 1 which has nr1 antennas, and transmitter 2, equipped with nt2 antennas, wishes to communicate with receiver 2, which has nr2 antennas. The signal from transmitter 1 acts as interference to receiver 2 and vice versa.

Even for this simple network, fundamental capacity results are still unknown. For instance, the capacity region of the 2 × 2 network even in the SISO case under general assumptions is unknown. For certain special cases, it is possible to derive the capacity of such channels, in particular when the interfering signals are strong such as in References [52, 272] and [282]. Most works in the literature have focused on deriving outer bounds to the capacity region such as References [177, 224, 10], and [89] for SISO systems, and References [243] and [281] for MIMO systems. Achievable rates of such networks under different sets of assumptions, such as in References[135, 271, 59] and [283], have also been found. Additionally, in Reference [89], the capacity region of the SISO Gaussian interference channel is derived to within one bit/second/Hz using an achievable rate region based on the Han–Kobayashi scheme introduced in Reference [135], and on novel outer bounds. Recently, the interference channel has been analyzed in the high SNR regime where interference-alignment introduced in Reference [50] has been shown to provide enormous network-wide performance improvements.

In many texts on communication, including this one, it is often assumed that transmitters and receivers are ideal. This assumption is never true. In practice, to get a communication system to operate, more time and effort are often invested in compensating for imperfections than in designing the ideal communication system. Furthermore, many system and algorithm design choices can increase or decrease the sensitivity of the communications system to the imperfections of the constituent components.

In this chapter, a few practical issues are addressed, including noise models, noise figure, power consumption, antennas, local oscillators, and dynamic range. These are only a small fraction of all practical issues faced by radio designers, but these are presented to sensitize the system designer to these potential issues. Many of these issues are addressed with greater depth in texts such as References [261, 252].

Antennas

In order for signals to be radiated or received, the electromagnetic energy is coupled through an antenna [15]. As discussed in Section 5.1, antennas can significantly affect the signal being transmitted or received. Each antenna has some directional and polarimetric response. In addition, the antenna has a frequency response. In the context of this text, the effects of the antennas are typically folded into the channel. Finally, if the impedance of the antenna is not matched well to the transmitter or receiver, then inefficiencies are introduced. There is constant drive to reduce the size of radio systems.

The nomenclature of cognitive radio, suggested in Reference [219], indicates the concept of a radio that is flexible in terms of its strategy or etiquette so that it can respond to the needs of the users and environment. The fundamental notion of the cognitive radio is that it is aware of its users and environment and makes decisions that maximize the link performance while minimizing adverse effects on other links in its or a legacy network [94].

A significant driver to the investigation of cognitive radios is the observation that allocated spectrum is not always utilized well [6]. Some researchers in cognitive radios focus on the technology used to describe the logic and rules used by the radio [153]. Game-theoretic models for developing dynamic spectrum access techniques have been investigated [161], and punishment techniques have been considered for users that behave poorly [354]. Conversely, some research focuses on information-theoretic investigations [80, 329] that often assume some level of cooperation between cognitive radios. At a deeper level, an aggressive definition would be for a cognitive radio to try strategies and then learn from them how well a particular strategy works, although we will not require that high level of cognition.

As an aside, it has been common in engineering literature to require that a cognitive system be of greater sophistication than an adaptive system. Amusingly, the engineering literature is at odds with the typical usage of these terms. As one can immediately recognize, a system must be cognizant of its environment before it can be adaptive to it. Consequently, typical usage of these terms would indicate that adaptive systems must have greater sophistication than cognitive systems. However, we will not attempt to correct this abuse of the language.

By using the diversity made available by multiple-antenna communications, links can be improved [349, 109]. For example, the multiple degrees of freedom can be used to provide robustness through channel redundancy or increased data rates by exploiting multiple paths through the environment simultaneously. These advantages can even potentially be employed to reduce the probability of interception [1]. Knowledge of the channel can even be used to generate cryptographic keys [332].

The basic concept of a multiple-input multiple-output (MIMO) wireless communication link is that a single source distributes data across multiple transmit antennas, as seen in Figure 8.1. Potentially independent signals from the multiple transmit antennas propagate through the environment, typically encountering different channels. The multiple-antenna receiver then disentangles the signal from the multiple transmitters [99, 209, 308, 33, 116, 258, 84]. There is a wide range of approaches for distributing the data across the transmitters and in implementations for the receiver.

While MIMO communication can operate in line-of-sight environments (at the expense of some of the typical assumptions used in MIMO communications), the most common scenario for MIMO communications is to operate in an environment that is characterized by complicated multipath scattering. Consequently, most, if not all, of the energy observed at the receive array impinges upon the array from directions different from the direction to the source. Consequently, the line-of-sight environment assumption employed in Chapters 6 and 7 is not valid for most applications of MIMO communications.

For most wireless communications, channels (what happens between the transmitter and receiver) are complicated things. For the sake of introduction, in this section we consider a single transmit antenna and receive antenna, residing in a universe without scatterers or blockage.

Antennas

The study and design of antennas is a rich field [15]. Here, we focus on a small set of essential features. The first important concept is that antennas do not radiate power uniformly in direction or in polarization. The radiated power as a function of direction is denoted the radiation pattern. If the antenna is small compared with the wavelength (for example, if the antenna fits easily within radius of a 1/8 wavelength), then the shape of the radiation pattern is relatively smooth. However, if the antenna is large compared with the wavelength, then the radiation pattern can be complicated. Antenna patterns are often displayed in terms of decibels relative to a notional isotropic antenna (denoted dBi). The notional isotropic antenna has the same gain over all 4π of solid angle. Gain is an indication of directional preference in the transmission and reception of power. The axisymmetric radiation pattern for an electrically small (small compared with a wavelength) dipole antenna is displayed in Figure 5.1. In the standard spherical coordinates of r, θ, φ, which correspond to the radial distance, the polar angle, and the azimuthal angle, respectively, the far-field electric field is limited to components along the direction of θ, denoted eθ.

Communication stack

For convenience in design, the operations of radios are often broken into a number of functional layers. The standard version of this stack is referred to as the open systems interconnection (OSI) model [291], as seen in Figure 4.1. The model has two groups of layers: host and media. The host layers are the application, presentation, session, and transport layers. The media layers are the network, data-link, and physical layers. In many radio systems, some of these layers are trivial or the division between the layers may be blurred. The OSI stack is commonly interpreted in terms of wired networks such as the internet. Depending upon the details of an implementation, various tasks may occupy different layers. Nonetheless, the OSI layered architecture is useful as a common reference for discussing radios. In this text, the media layers are of principal importance.

The network layer indicates how data are routed from an information source to a sink node, as seen in Figure 4.2. In the case of a network with two nodes, this routing is trivial. In the case of an ad hoc wireless network, the routing may be both complicated and time varying. The network layer may break a data sequence at the source node into smaller blocks and then reassemble the data sequence at the sink node. It also may provide notification of errors to the transport layer.

Although angle estimation of a source is not typically of significant interest in communication systems, because angle-of-arrival estimation is commonly addressed in treatments of multiple-antenna systems, we will consider it here in this chapter. In addition, some of the tools and intuition are helpful when considering adaptive multiple-antenna receivers. Furthermore, there are special cases of communications systems for which line-of-sight propagation is valid and for which angle-of-arrival estimation is of value. Angle estimation to the source is sometimes denoted direction finding. There is a large body of work addressing this topic [294, 223], and numerous useful approaches (for example, those in References [16, 266]), many of which will be skipped for brevity. In general, the direction to the source requires both azimuthal and elevation information, but for most examples here it is assumed that the source is in the plan of the array, so only azimuthal information encoded in the angle φ is required. A few approaches are considered here as an introduction to the area.

Within this chapter, it is assumed that any multipath scattering is minimal and that the signal is not dispersive across the array; that is, the array is small compared with the speed of light divided by the bandwidth of the signal. This assumption is sometimes denoted the narrowband signal assumption. Furthermore, to simplify the introduction, it is assumed that the direction can be characterized by a single angle φ. Here it is assumed that multiple samples of the received signal are available. The number of samples is denoted ns.