Armed with the theoretical limits of MIMO wireless performance from Chapter 2, we now embark on the design of specific system blocks. At the transmitter, two major MIMO processing components at the symbol level are precoding and spacetime coding. Precoding, the last digital processing block at the transmitter (see Figure 1.2), is a technique that exploits the channel information available at the transmitter. Such information is generally referred to as transmit channel side information, or CSIT (this definition is more general than that in Chapter 2). In MIMO wireless, spatial CSIT is particularly useful in enhancing system performance. Space-time coding, on the other hand, assumes no CSIT and focuses on enhancing reliability through diversity. In addition to these two components, regular channel coding is required for bit-level protection. This chapter focuses on precoding design, and space-time coding is discussed in Chapter 4.

CSIT helps to increase the transmission rate, to enhance coverage, and to reduce receiver complexity in MIMO wireless systems. Many forms of CSIT exist. Exact channel knowledge at each time instance, or perfect CSIT, is ideal; but it is often difficult to acquire in a time-selective fading channel. CSIT is more likely to be available as a channel estimate with an associated error covariance, which reduce in the limit to the channel statistics, such as the channel mean and covariance. Such CSIT encompasses several models discussed in Chapter 2, including perfect CSIT and CDIT. Other partial CSIT forms can involve only parametric channel information, such as the channel condition number or the Ricean K factor.

Introduction

Chapter 1 introduced the basic concepts behind multiple-input multiple-output (MIMO) communications along with their performance advantages. In particular, we saw that MIMO systems provide tremendous capacity gains, which has spurred significant activity to develop transmitter and receiver techniques that realize these capacity benefits and exploit diversitymultiplexing trade-offs. In this chapter we will explore in more detail the Shannon capacity limits of single- and multi-user MIMO systems. These fundamental limits dictate the maximum data rates that can be transmitted over the MIMO channel to one or more users (not in outage) with asymptotically small error probability, assuming no constraints on the delay or the complexity of the encoder and decoder. Much of the initial excitement about MIMO systems was due to pioneering work by Foschini and Telatar predicting remarkable capacity growth for wireless systems with multiple antennas when the channel exhibits rich scattering and its variations can be accurately tracked. This promise of exceptional spectral efficiency almost “for free,” also studied in earlier work by Winters, resulted in an explosion of research and commercial activity to characterize the theoretical and practical issues associated with MIMO systems. However, these predictions are based on somewhat unrealistic assumptions about the underlying time-varying channel model and how well it can be tracked at the receiver as well as at the transmitter. More realistic assumptions can dramatically impact the potential capacity gains of MIMO techniques. This chapter provides a comprehensive summary of MIMO Shannon capacity for both single- and multi-user systems with and without fading under different assumptions about what is known at the transmitter(s) and receiver(s).

Introduction

The preceding chapter considered the design of receivers for MIMO systems operating as single-user systems. Increasingly however, as noted in Chapters 2 and 4, wireless communication networks operate as shared-access systems in which multiple transmitters share the same radio resources. This is due largely to the ability of shared-access systems to support flexible admission protocols, to take advantage of statistical multiplexing, and to support transmission in unlicensed spectrum. In this chapter we will extend the treatment of Chapter 5 to consider receiver structures for multi-user, and specifically, multiple-access MIMO systems. We will also generalize the channel model considered to include more general situations than the flat-fading channels considered in Chapter 5. To treat these problems, we will first describe a general model for multi-user MIMO signaling, and then discuss the structure of optimal receivers for this signal model. This model will generally include several sources of interference arising in MIMO wireless systems, including multiple-access interference caused by the sharing of radio resources noted above, inter-symbol interference caused by dispersive channels, and inter-antenna interference caused by the use of multiple transmit antennas. Algorithms for the mitigation of all of these types of interference can be derived in this common framework, leading to a general receiver structure for multi-user MIMO communications over frequency-selective channels. As we shall see, these basic algorithms will echo similar algorithms that have been described in Chapters 3 and 5. Since optimal receivers in this situation are often prohibitively complex, the bulk of the chapter will focus on useful lower complexity sub-optimal iterative and adaptive receiver structures that can achieve excellent performance in mitigating interference in such systems. This discussion is organized as follows.

Introduction

This chapter is devoted to MIMO receivers, with special focus on single-user systems and frequency-flat channels (multi-user systems and more general channels will be the subject of the next chapter). We start with a brief discussion of uncoded MIMO systems, describing their optimum (maximum-likelihood, ML) receivers. Since these may exhibit a complexity that makes them unpractical, it is important to seek receivers that achieve a close-to-optimum performance while keeping a moderate complexity: these would remove the practical restriction to small signal constellations or few antennas. Linear receivers and receivers based on the sphere-detection algorithm are examined as possible solutions to the complexity problem. Next, we study iterative processing of received signals. We introduce here the idea of factor graphs. Their use offers a versatile tool, allowing one to categorize in a simple way the approximations on which MIMO receivers and their algorithms are based. In addition, they yield a “natural” way for the description of iterative (turbo) algorithms, and of their convergence properties through the use of EXIT-charts. Using factor graphs, we describe iterative algorithms for the reception of MIMO signals, along with some noniterative schemes that can be easily developed by using the factor-graph machinery.

A basic assumption in this chapter is that channel state information (i.e., the values taken on by all path gains) is available at the receiver, while the transmitter knows the channel distribution (i.e., the joint probability density function of the channel gains). In addition, the channel is quasi-static (i.e., it remains constant throughout the transmission of a whole data frame or codeword), and the transmitted signals are two-dimensional.

Wireless is one of the most rapidly developing technologies in our time, with dazzling new products and services emerging on an almost daily basis. These developments present enormous challenges for communications engineers, as the demand for increased wireless capacity grows explosively. Indeed, the discipline of wireless communications presents many challenges to designers that arise as a result of the demanding nature of the physical medium and the complexities in the dynamics of the underlying network. The dominant technical issue in wireless communications is that of multipath-induced fading, namely the random fluctuations in the channel gain that arise due to scattering of transmitted signals from intervening objects between the transmitter and the receiver. Multipath scattering is therefore commonly seen as an impairment to wireless communication. However, it can now also be seen as providing an opportunity to significantly improve the capacity and reliability of such systems. By using multiple antennas at the transmitter and receiver in a wireless system, the rich scattering channel can be exploited to create a multiplicity of parallel links over the same radio band, and thereby to either increase the rate of data transmission through multiplexing or to improve system reliability through the increased antenna diversity. Moreover, we need not choose between multiplexing and diversity, but rather we can have both subject to a fundamental tradeoff between the two.

This book addresses multiple-input/multiple-output (MIMO) wireless systems in which transmitters and receivers may have multiple antennas. Since the emergence of several key ideas in this field in the mid-1990s, MIMO systems have been one of the most active areas of research and development in the broad field of wireless communications.

Introduction

The essential feature of wireless transmission is the randomness of the communication channel which leads to random fluctuations in the received signal commonly known as fading. This randomness can be exploited to enhance performance through diversity. We broadly define diversity as the method of conveying information through multiple independent instantiations of these random fades. There are several forms of diversity; our focus in this chapter will be on spatial diversity through multiple independent transmit/receive antennas. Information theory has been used to show that multiple antennas have the potential to dramatically increase achievable bit rates, thus converting wireless channels from narrow to wide data pipes.

The earliest form of spatial transmit diversity is the delay diversity scheme proposed in where a signal is transmitted from one antenna, then delayed one time slot, and transmitted from the other antenna. Signal processing is used at the receiver to decode the superposition of the original and time-delayed signals. By viewing multiple antenna diversity as independent information streams, more sophisticates transmission (coding) scheme can be designed to get closer to theoretical performance limits. Using this approach, we focus on space-time coding (STC) schemes defined by Tarokh et al. and Alamouti, which introduce temporal and spatial correlation into the signals transmitted from different antennas without increasing the total transmitted power or the transmission bandwidth. Therefore is, in fact, a diversity gain that results from multiple paths between the base-station and the user terminal, and a coding gain that results from how symbols are correlated across transmit antennas.

Although we have seen that most of the MC signals have peaks of value about √n ln n, there are plenty of signals with maxima of order √n. This chapter is devoted to methods of constructing such signals. I begin with relating the maxima in signals to the distribution of their a periodic correlations (Theorem 7.2). Then I describe in Section 7.2 the Rudin–Shapiro sequences over {−1, 1}, guaranteeing a PMEPR of at most 2 for n being powers of 2. They appear in pairs, where each one of the sequences possesses the claimed property. The Rudin–Shapiro sequences are representatives of a much broader class of complementary sequences discussed in Section 7.3. The signals defined by these sequences also have a PMEPR not exceeding 2, while existing for a wider spectrum of lengths. In Section 7.4, I introduce complementary sets of sequences. The number of sequences in the sets can be more than two, and the corresponding sequences have a PMEPR not exceeding the number of sequences in the set. In Section 7.5, I generalize the earlier derived results to the polyphase case, and describe a general construction of complementary pairs and sets stemming from cosets of the first-order Reed–Muller codes within the second-order Reed–Muller codes. Another idea in constructing sequences with low PMEPR is to use vectors defined by evaluating the trace of a function over finite fields or rings. This topic is explored in Section 7.6 using estimates for exponential sums. Finally, in Sections 7.7 and 7.8, I study two classes of sequences, M-sequences and Legendre sequences, guaranteeing PMEPR of order at most (ln n)2.

In this chapter, I introduce the main issues we will deal with in the book. In Section 2.1, I describe a multicarrier (MC) communication system. I introduce the main stages that the signals undergo in MC systems and summarize advantages and drawbacks of this technology. Section 2.2 deals with formal definitions of the main notions related to peak power: peak-to-average power ratio, peak-to-mean envelope power ratio, and crest factor. In Section 2.3, I quantify the efficiency of power amplifiers and its dependence on the power of processed MC signals. Section 2.4 introduces nonlinear characteristics of power amplifiers and describes their influence on the performance of communication systems.

Model of multicarrier communication system

The basic concept behind multicarrier (MC) transmission is in dividing the available spectrum into subchannels, assigning a carrier to each of them, and distributing the information stream between subcarriers. Each carrier is modulated separately, and the superposition of the modulated signals is transmitted. Such a scheme has several benefits: if the subcarrier spacing is small enough, each subchannel exhibits a flat frequency response, thus making frequency-domain equalization easier. Each substream has a low bit rate, which means that the symbol has a considerable duration; this makes it less sensitive to impulse noise. When the number of subcarriers increases for properly chosen modulating functions, the spectrum approaches a rectangular shape. The multicarrier scheme shows a good modularity. For instance, the subcarriers exhibiting a disadvantageous signal-to-noise ratio (SNR) can be discarded. Moreover, it is possible to choose the constellation size (bit loading) and energy for each subcarrier, thus approaching the theoretical capacity of the channel.

Multicarrier (MC) modulations such as orthogonal frequency division multiplexing (OFDM) and discrete multitone (DMT) are efficient technologies for the implementation of wireless and wireline communication systems. Advantages of MC systems over single-carrier ones explain their broad acceptance for various telecommunication standards (e.g., ADSL, VDSL, DAB, DVB, WLAN, WMAN). Yet many more appearances are envisioned for MC technology in the standards to come. A relatively simple implementation is possible for MC systems. Low complexity is due to the use of fast discrete Fourier transform (DFT), avoiding complicated equalization algorithms. Efficient performance of MC modulation is especially vivid in channels with frequency selective fading and multipath. Nonetheless, still a major barrier for implementing MC schemes in low-cost applications is its nonconstant signal envelope, making the transmission sensitive to nonlinear devices in the communication path. Amplifiers and digital-to-analog converters distort the transmit signals leading to increased symbol error rates, spectral regrowth, and reduced power efficiency compared with single carrier systems. Naturally, the transmit signals should be restricted to those that do not cause the undesired distortions. Areasonable measure of the relevance of the signals is the ratio between the peak power values to their average power (PAPR). Thus the goal of peak power control is to diminish the influence of transmit signals with high PAPR on the performance of the transmission system. Alternatives are either the complete exclusion of such signals or an essential decrease in the probability of their appearance.

In this chapter, I consider methods of decreasing peak power in MC signals. The simplest method is to clip the MC signal deliberately before amplification. This method is very simple to implement and provides essential PMEPR reduction. However, it suffers some performance degradation, as estimated in Section 8.1. In selective mapping (SLM), discussed in Section 8.2, one favorable signal is selected from a set of different signals that all represent the same information. One possibility for SLM is to choose the best signal from those obtained by inverting any of the coordinates of the coefficient vector. The method of deciding which of the coordinates should be inverted is described in Section 8.3. Further, in Section 8.4 a modification of SLM is analyzed. There the favorable vector is chosen from a coset of a code of given strength. Trellis shaping, where the relevant modification is chosen based on a search on a trellis, is described in Section 8.5. In Section 8.6, the method of tone injection is discussed. Here, instead of using a constellation point its appropriately shifted version can be used. In active constellation extension (ACE), described in Section 8.7, some of the outer constellation points can be extended, yielding PMEPR reduction. In Section 8.8, a method of finding a constellation in the frequency domain is described, such that the resulting region in the time domain has a low PMEPR. In partial transmit sequences (PTS), the transmitted signal is made to have a low PMEPR by partitioning the information-bearing vector to sub-blocks followed by multiplying by a rotating factor the coefficients belonging to the same sub-block.

In this chapter I collect the basic mathematical tools which are used throughout the book. Most of them are given with rigorous proofs. I mainly concentrate here on results and methods that do not appear in the standard engineering textbooks and omit those that happen to be common technical knowledge. On the other hand, I have included some material that is not directly used in further arguments, but I feel that it might prove useful in further research on peak power control problems. It should be advised that the chapter is mainly for reference purposes and may be omitted in the first reading.

The chapter is organized as follows. Section 3.1 deals with harmonic analysis. In Section 3.1.1 I describe the Parseval equality and its generalizations. Section 3.1.2 introduces some useful trigonometric relations. Chebyshev polynomials and interpolation are described in Section 3.1.3. Finally, in Section 3.1.4, I prove Bernstein's inequality relating the maximum of the absolute value of a trigonometric polynomial and its derivative. In Section 3.2 I deal with some notions related to probability. I prove the Chernoff bound on the probability of deviations of values of random variables. In Section 3.3, I introduce tools from algebra. In Section 3.3.1 groups, rings, and fields are defined. Section 3.3.2 describes exponential sums in finite fields and rings. A short account of results from coding theory is presented in Section 3.4. Section 3.4.1 deals with properties of the Hamming space. In Section 3.4.2, definitions related to error-correction codes are introduced. Section 3.4.3 deals with the distance distributions of codes. In Section 3.4.4, I analyze properties of Krawtchouk polynomials playing an important role in the Mac Williams transform of the distance distributions.

In many situations it is beneficial to deal with a discrete-time “sampled” version of multicarrier signals. This reduction allows passing from the continuous setting to an easier-to-handle discrete one. However, we have to estimate the inaccuracies stemming from the approach. In this chapter, I analyze the ratio between the maximum of the absolute value of a continuous MC signal and the maximum over a set of the signal's samples. We start with considering the ratio when the signal is sampled at the Nyquist frequency, i.e. the number of sampling points equals the number of tones. In this case I show that the maximum of the ratio over all MC signals grows with the number of subcarriers (Theorem 4.2). However, if one computes a weighted sum of the maximum of the signal's samples and the maximum of the signal derivative's samples the ratio already is, at most, a constant (Theorem 4.5). I further show that actually the ratio depends on the maximum of the signal; the larger the maximum is the smaller is the ratio (Theorem 4.6). An even better strategy is to use over sampling. Then the ratio becomes constant tending to 1 when the over sampling rate grows (Theorems 4.8, 4.9, 4.10, and 4.11). Furthermore, I tackle the case when we have to use the maximum estimation, projections on specially chosen measuring axes instead of the absolute values of the signal (Theorem 4.14). Finally, I address the problem of relation between the PAPR and the PMEPR and show that the PMEPR estimates the PAPR quite accurately for large values of the carrier frequency (Theorem 4.19).